Skeletal graphs of the two interpenetrating labyrinths of a TPMS
and Voronoi polyhedra that enclose the vertices of the graph are useful
for representing the symmetry and topology of some examples of TPMS,
especially when the graph edges for each labyrinth are symmetrically equivalent.
These geometrical constructions do not yield analytic solutions for the surfaces, however.
(All stereoscopic image pairs are arranged for 'cross-eyed' viewing.)
Fig. E1.0a
A dual pair of diamond skeletal graphs
Fig. E1.0b
A dual pair of primitive cubic skeletal graphs
Curiously, the set of vertices of a dual pair of diamond (D) graphs
and the set of vertices of a dual pair of primitive cubic (P) graphs
are identical.
The lattice for this set is b.c.c (body-centered cubic).
The Voronoi polyhedron
for a vertex in this set is the truncated octahedron.
I know of no other example of a pair of
directly congruent dual graphs with cubic lattice symmetry.
1. Dual diamond graphs (skeletal graphs of the D surface)
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A triply-periodic minimal surface (TPMS) that is embedded, i.e., free of self-intersections,
partitions space into a pair of disjoint labyrinths.
If the labyrinths are congruent — either directly or oppositely — the surface is called balanced.
The D (diamond) surface of H.A. Schwarz, which is a balanced TPMS, is shown in Fig. E1.1a.
Fig. E1.1a
Four translation fundamental domains of Schwarz's D surface
(genus 3)
For some purposes it is convenient to represent a TPMS of low genus by a surrogate
with plane faces — a triply-periodic
polyhedron with the same symmetry and topology as the surface.
The two labyrinths of the TPMS and of its surrogate may be represented by a pair of triply-periodic
skeletal graphs that have the same symmetry as the TPMS and its surrogate. Every edge in these graphs
joins a pair of vertices that lie at centers of symmetry of the TPMS.
A simple example of a surrogate of the D surface is the triply-periodic
quasi-regular skew polyhedron (6.4)2,
which is derived from the Coxeter-Petrie regular skew polyhedron {6,4|4}
(cf. Fig. E1.35c). It is shown in Fig. 1.1b.
Fig. E1.1b
(6.4)2, a surrogate of Schwarz's D surface (cf. Fig. E1.1a).
The red and green skeletal graphs are both replicas of the diamond graph.
Its edges join the sites of adjacent carbon atoms in diamond.
The
Voronoi polyhedron
of a vertex of the union of the two dual skeletal graphs
is the truncated octahedron.
The vertices of the dual skeletal graphs in Fig. E1.1b
lie at the centers of the chambers of the respective labyrinths.
Each chamber in (6.4)2 is a truncated octahedron from
which a tetrahedrally arranged subset of four hexagons has been removed.
Hence the boundary of (6.4)2 is composed of four regular hexagons and six squares
(cf. Fig. E1.1c).
Fig. E1.1c
The two differently oriented chambers in (6.4)2 (cf. Fig. E1.1b).
The four faces incident at each vertex of (6.4)2 are arranged
in cyclic order 6.4.6.4 — hence the name (6.4)2.
In each of the two labyrinths, there are two differently oriented varieties of chambers. They are related
by a quarter-turn about any one of the three Cartesian axes.
In each labyrinth, adjacent chambers related by
a translation of type [111] are of opposite variety.
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Fig. E1.1d
Fig. E1.1e
Fig. E1.1f
Fig. E1.1g
Like (6.4)2, the Coxeter-Petrie triply-periodic regular skew polyhedron {6,4|4}
(cf. Figs. E1.2a,c) has the same topology and symmetry as
Schwarz's diamond surface D (cf. Fig. E1.2b).
For this example, in contrast to the case of (6.4)2,
only one orientation of the unit call is required (cf. Fig. E1.2c).,
since it is a translation fundamental region.
Fig. E1.2a
A lattice fundamental region of {6,4|4}
The lattice is f.c.c.
Fig. E1.2b
A lattice fundamental region of Schwarz's diamond surface D
Fig. E1.2c
Thirteen lattice fundamental regions of {6,4|4}
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Dual pairs of diamond graphs are shown below in both 'medium thick' and 'thick' versions. In the thick
version (cf. Figs. E1.3e-h), the diameter d of the cylindrical tubes is the largest possible, consistent
with the requirement that the dual graphs not overlap.
Overlap occurs when the ratio d/e ≥ 21/2/2 (~.707),
where e is the edge length of an ideally thin skeletal graph.
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Medium thick diamond graphs
oblique view
Fig. E1.3a
Fig. E1.3b
Fig. E1.3c
graph 1
graph 2
graphs 1 and 2
Fig. E1.3d
Orthogonal projection of graphs 1 and 2 on [111] plane
Thick diamond graphs
oblique view
Fig. E1.3e
Fig. E1.3f
Fig. E1.3g
graph 1
graph 2
graphs 1 and 2
Fig. E1.3h
Orthogonal projection of graphs 1 and 2 on [111] plane
Fig. E1.3i
(6.4)2 — a triply-periodic quasi-regular polyhedron
that has the same topology and symmetry as
Schwarz's diamond surface D
Fig. E1.3j
(6.4)2 with embedded [skinny] dual graphs
Fig. E1.3k
(6.4)2 with embedded [fat] dual graphs
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2. Dual simple cubic graphs (skeletal graphs of the P surface)
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Dual pairs of simple cubic graphs are shown below in both 'medium thick' and 'thick' versions. In the thick
version (cf. Figs. E1.4e-h), the diameter d of the cylindrical tubes is the largest possible, consistent
with the requirement that the dual graphs not overlap.
Overlap occurs when the ratio d/e ≥ 1/2,
where e is the edge length of an ideally thin skeletal graph.
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Medium thick simple cubic graphs
oblique view
Fig. E1.4a
Fig. E1.4b
Fig. E1.4c
graph 1
graph 2
graphs 1 and 2
Fig. E1.4d
Orthogonal projection of graphs 1 and 2 on [100] plane
Thick simple cubic graphs
oblique view
Fig. E1.4e
Fig. E1.4f
Fig. E1.4g
graph 1
graph 2
graphs 1 and 2
Fig. E1.4h
Orthogonal projection of graphs 1 and 2 on [100] plane
Fig. E1.4i
The Coxeter-Petrie triply-periodic regular skew polyhedron {6,4|4},
which has the same topology and symmetry as
Schwarz's primitive surface P
Fig. E1.4j
{6,4|4} with embedded [skinny] dual graphs
Fig. E1.4k
{6,4|4} with embedded [fat] dual graphs
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{6,4|4} is a is an infinite regular polyhedron.
There is only one variety of chamber — a truncated octahedron whose
six square faces have been removed.
Hence it is bounded by eight regular hexagons.
At the center of every chamber there is a vertex of one of the dual skeletal graphs
illustrated in Figs. E1.4a-h.
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3. Dual Laves graphs (skeletal graphs of the G surface)
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The two intertwined skeletal graphs of the gyroid,
in a Shapeways 3D printed version designed by virtox.
Click here for a video of 'Bones',
an animated view of these graphs,
also made at Shapeways by virtox.
Dual pairs of Laves graphs are shown below in 'thin', 'medium thick', and 'thick' versions. In the thick
version (cf. Figs. E1.5i-l), the diameter d of the cylindrical tubes is the largest possible,
consistent with the requirement that the dual pair of tubular graphs not overlap.
Overlap occurs when the ratio d/e ≥ 31/2/2 (~.866),
where e is the edge length of an ideally thin skeletal graph.
The fact that the d/e ratio
is significantly larger for the pair of thick Laves graphs than it is
for the thick simple cubic and thick diamond graph pairs
suggests (but does not prove) that the thick Laves graphs occupy a larger fraction of space.
In order to make the comparison precise, it would be necessary to take into account
the detailed geometry in the neighborhood of the intersections of the
cyclindrical tubes. I have not done this.
In order to display the pairs of intertwined graphs as clearly as possible,
views are shown for each of the three principal 'crystallographic'
directions: [100], [111], and [110].
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Thin Laves graphs
[100] view
Fig. E1.5a
Fig. E1.5b
Fig. E1.5c
graph 1
graph 2
graphs 1 and 2
Fig. E1.5d
Orthogonal projection of graphs 1 and 2
[100] view
Medium thick Laves graphs
[100] view
Fig. E1.5e
Fig. E1.5f
Fig. E1.5g
graph 1
graph 2
graphs 1 and 2
Fig. E1.5h
Orthogonal projection of graphs 1 and 2
[100] view
Thick Laves graphs
[100] view
Fig. E1.5i
Fig. E1.5j
Fig. E1.5k
graph 1
graph 2
graphs 1 and 2
Fig. E1.5l
Orthogonal projection of graphs 1 and 2
[100] view
Thin Laves graphs
[111] view
Fig. E1.5m
Fig. E1.5n
Fig. E1.5o
graph 1
graph 2
graphs 1 and 2
.jpg)
Fig. E1.5p
Orthogonal projection of graphs 1 and 2
view: [111]
Medium thick Laves graphs
[111] view
Fig. E1.5q
Fig. E1.5r
Fig. E1.5s
graph 1
graph 2
graphs 1 and 2
Fig. E1.5t
Orthogonal projection of graphs 1 and 2
[111] view
Thick Laves skeletal graphs
[111] view
Fig. E1.5u
Fig. E1.5v
Fig. E1.5w
graph 1
graph 2
graphs 1 and 2
Fig. E1.5x
Orthogonal projection of graphs 1 and 2
view: [111]
Thin Laves skeletal graphs
[110] view
Fig. E1.6a
Fig. E1.6b
Fig. E1.6c
graph 1
graph 2
graphs 1 and 2
Fig. E1.6d
Orthogonal projection of graphs 1 and 2
view: [110]
Medium thick Laves skeletal graphs
[110] view
Fig. E1.6e
Fig. E1.6f
Fig. E1.6g
graph 1
graph 2
graphs 1 and 2
Fig. E1.6h
Orthogonal projection of graphs 1 and 2
view: [110]
Thick Laves skeletal graphs
[110] view
Fig. E1.6i
Fig. E1.6j
Fig. E1.6k
graph 1
graph 2
graphs 1 and 2
Fig. E1.6l
Orthogonal projection of graphs 1 and 2
view: [110]
Fig. E1.7
Straw model of the pair of dual Laves skeletal graphs (1960)
view: [100]
Additional stereo images of the Laves graph
Fig. E1.8
The 'clockwise' Laves graph,
skeletal graph of one labyrinth of the G surface
Fig. E1.9
The 'counter-clockwise' Laves graph,
skeletal graph of the other labyrinth of the G surface
Fig. E1.10
The enantiomorphic skeletal graphs of the two disjoint labyrinths of the G surface
Fig. E1.11
Orthogonal projection on [100] plane of the enantiomorphic Laves graphs
Fig. E1.12
Another view of the 'counter-clockwise' Laves graph
4. Other pairs of dual skeletal graphs
Fig. E1.13a
Fig. E1.13b
The dual skeletal graphs of a hypothetical but
nonexistent embedded TPMS called TO-TD
(TO stands for truncated octahedron,
the interstitial cage of the blue graph.
TD stands for tetragonal disphenoid,
the interstitial cage of the orange graph.)
blue graph: degree 4
orange graph: degree 14
The blue graph is
symmetric.
The orange graph is
regular
but not symmetric.
The edges of the blue graph are all symmetrically equivalent.
The edges of the orange graph are clearly not all symmetrically eqivalent.
They are not even all of the same length.
If TO-TD existed,
it would be tiled by surface patches S with curved edges
(mirror-symmetric plane lines of curvature).
In 1974, I used an experimental method to test for the existence of TO-TD.
This extremely tedious method is based on the use of a laser to measure the orientation
of the tangent plane near the boundary of a straight-edged soap film S' that is the
adjoint of S. The results demonstrated that it is impossible to 'kill the periods'.
Hence TO-TD does not exist.
In 2001, Ken Brakke used his Surface Evolver program to confirm this conclusion with
enormously greater accuracy (and speed!) than is possible with the soap film-laser technique.
Fig. E1.13c
The dual pair of skeletal graphs for another
hypothetical but nonexistent embedded TPMS
The green vertices define the sites of the Cu atoms,
and the blue vertices define the sites of the Mg atoms
in the binary alloy Cu2Mg, which has the structure called
Cubic Laves phase C15.
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I call the Cu graph FCC6(II).
The Mg graph is the diamond graph (cf. Fig. E1.3d).
Both graphs are
symmetric.
The interstitial cavities in the Cu graph are of two kinds:
small tetrahedral cages and
large truncated tetrahedral cages.
All the interstitial cavities in the Mg graph are identical:
the expanded regular tetrahedron (ERT)
(cf. Fig. E2.19c, E2.20).
In 2001, Ken Brakke used his Surface Evolver program to demonstrate
that it is impossible to kill periods for this hypothetical surface.
Hence it is almost certainly safe to conclude that the surface does not exist.
Note that in this example, in contrast to other pairs of dual graphs treated here,
it is not true that for both graphs, every interstitial cavity of the graph contains a vertex of the dual graph.
(The small tetrahedral cages of the Cu graph do not contain any vertex of the Mg graph.)
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E2. Mathematical Preliminaries
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The Gauss map
.jpg)
Fig. E2.1
(stereo image)
Curved triangular Flächenstück ABC
of Schwarz's Diamond surface
Fig. E2.2
(stereo image)
Riemann sphere
(unit sphere)
The elementary minimal surface Flächenstück ABC shown in Fig. E0.1 is mapped
onto the spherical triangle ABC on the Riemann sphere shown in Fig. E0.2 by the
Gauss map.
Each point on the minimal surface is mapped onto a point on the
Riemann sphere that has the same normal vector.
The red arrows at points A, B, and C indicate
the directions of the surface normal vectors.
There are twelve replicas of the Flächenstück ABC in the skew
hexagonal face EDAE'D'A' of Schwarz's D surface (cf. Fig. E0.1), but
there are only six corresponding spherical triangles in the large spherical triangle
AED on the Riemann sphere (cf. Fig. E0.2). The two Flächenstücke ABC and A'B'C,
for example, are both mapped onto the same spherical triangle. An entire lattice fundamental
region covers the Riemann sphere twice. As a consequence the mapping defines a two-sheeted
Riemann surface, with branch points at the eight 'cube corner' points like C.
Fig. E2.3
Stereographic projection onto the complex plane
of the elementary triangular Flächenstück ABC
of Fig. E2.2
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Triply periodic minimal surfaces are infinitely-multiply-connected,
but it is nevertheless easy to characterize the topological complexity of every example of such a surface by computing the
genus p of a single lattice fundamental domain.
Except where it is specifically stated to the contrary, it will be assumed in all that follows that TPMS refers to an
embedded surface,
i.e., one that is free of transverse self-intersections.
Since the smallest posssible value for the genus is three,
Schwarz's P and D surfaces are members of a very small select group of topologically simplest examples of TPMS.
Below is a recipe for computing the genus of a TPMS.
It is based on one of Gauss 's most astounding discoveries, the Gauss-Bonnet theorem, which links the topology and the geometry of a surface.
One can very crudely express the essence of the Gauss-Bonnet theorem in this context by saying that
the larger the value of the integrated Gaussian curvature for one lattice fundamental region of the surface,
the steeper the saddle-like surface contours, and — therefore — the larger the number of tubular 'handles' in the surface
as it 'bends around' this way and that.
On p. 233 of the 13th edition of 'Mathematical Recreations and Essays' by W.W. Rouse Ball and H.S.M. Coxeter,
the authors use Euler's
formula
F − E + V = 2,
which relates the number F of faces, the number E of edges, and the number V of vertices of a convex polyhedron
to prove that adding a handle to an orientable surface reduces the Euler-Poincaré characteristic
Χ = 2 − 2p by 2 and therefore increases the genus
p by 1. The proof simply updates the values of F, E, and V
after two different n-gons of a map on the surface are joined by a 'bent prism' (which is a convenient device
for representing a handle). F is increased by n − 2, E is increased by n, and V remains unchanged.
Since
X = F − E + V,
Χ is reduced by 2.
.jpg)
.jpg)
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Fig. E2.4
Recipe for calculating the genus
of one lattice fundamental domain of a TPMS,
applied to Schwarz's P surface
Another way to calculate the genus is to
substitute for dG from Eq. 5 in the equation p =1− dG.
|dG| is equal to the number of times the Gauss map
of the minimal surface ('Gauss image') covers the Riemann sphere.
For Schwarz's P surface, dG= − 2.
The sign of dG for surfaces of negative Gaussian curvature,
like minimal surfaces, is negative because the sense
of a geodesic edge-circuit on the surface is opposite
to that of its Gauss image on the Riemann sphere.
If you're not familiar with the Gauss map,
look here.
For discussion of the Euler-Poincaré characteristic Χ=2 − 2p,
look here.
For information about the Gauss-Bonnet theorem,
look here.
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E3. The D-G-P family of surfaces
Fig. E3.1a
A page from Schwarz's Collected Works
(E3a) H. A. Schwarz's diamond surface D
In 1966 I named this surface 'diamond' because its two interwined labyrinths, which are congruent,
each have the shape of an inflated tubular version of the familiar diamond graph (cf. Figs. E1.3d to E1.3k).
Fig. E3.1b
Fig. E3.1c
Fig. E3.1d
Three of H. A. Schwarz's illustrations of the D surface in
in his Gesammelte Mathematische Abhandlungen,
Springer Verlag, 1890
.jpg)
Fig. E3.1e
Stereo view of the linear asymptotics
embedded in the 'crossed triangles D-catenoid'
wire-frame of Schwarz's diamond surface D
(cf. Fig. E3.1d)
The ratio 2h/λ of the triangle separation 2h
to the triangle edge length λ is equal to √ 6 / 6 (~.408).
.jpg)
Fig. E3.1f
Orthogonal projection of the linear asymptotics
embedded in the 'crossed triangles D-catenoid'.
Fig. E3.1g
Eight translation fundamental domains of Schwarz's D surface
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The lattice for D is face-centered cubic (f.c.c.),
and the translation fundamental domain has genus 3.
The edges of the skeletal graph of degree 4 for each of the two congruent labyrinths
correspond to nearest-neighbor links in the diamond crystal structure.
As a sort of metaphor for the 'pipejoint' module of D in Fig. E3.1h, i, j, imagine that you are
inside a spherical soap bubble at the center of a rhombic dodecahedron.
Now deform the bubble by blowing against its interior surface
in the six tetrahedral directions [1,1,1], [-1,-1,1], [1,-1,-1], [-1,1,-1] simultaneously,
forming four cylindrical tubules attached symmetrically to the inside of the rhombic dodecahedron around four of its eight trigonal corners.
Fig. E3.1h
A rhombic dodecahedral translation fundamental domain
of Schwarz's diamond
triply periodic minimal surface D
Fig. E3.1i
A translation fundamental domain of D
larger image
Fig. E3.1j
A translation fundamental domain of D on which
closed geodesics (the red curves) are inscribed.
These geodesics are not plane curves.
larger image
Fig. E3.1k
A regular skew curvilinear hexagon of D,
which is a face of the regular map with holes {6,6|3}
larger image
The inscribed regular skew hexagon with straight edges
is a face of {6,4|4},
another of the three regular maps with holes described in
Generators and Relations for Discrete Groups,
H.S.M. Coxeter and W.O.J. Moser, Springer-Verlag, New York, 1965
and in
Infinite Periodic Minimal Surfaces Without Self-Intersections,
NASA TN D-5541, p. 49.
(E3b) H. A. Schwarz's primitive surface P
In 1966 I named this surface primitive because its two interwined labyrinths, which are congruent,
each have the shape of an inflated tubular version of the familiar primitive (or simple cubic) graph
(cf. Figs. E3.3d to E3.3k).
D and P are adjoint surfaces:
each surface can be mapped into the other by an isometry
(the Bonnet transformation).
Straight lines in one surface are mirror-symmetric
plane lines of curvature (plane geodesics) in the other.
Fig. E3.2a
Stereoscopic view of the linear asymptotics (blue) and plane geodesic curves (green)
in the 'square catenoid' of P (cf. Fig. 1.2d, e, f)
.jpg)
Fig. E3.2b
Stereo view of the linear asymptotics
embedded in the 'crossed triangles P-catenoid'
wire-frame of Schwarz's primitive surface P
(cf. Figs. 1.2c, d, e)
Here the triangles are only half as far apart as the
crossed triangles in Schwarz's D surface (cf. Figs. 1.1d, e, f).
The ratio h/λ of the triangle separation h to the triangle edge length λ
is equal to √ 6 / 12 (~.204).
The P and D surfaces are related by a dilatation along any of the
four [111] directions.
An annular 'crossed-triangles catenoid' (CTC) minimal surface exists
for every value of h/λ less than some allowed maximum value (h/λ)max,
but there are embedded straight lines only in the CTCs of D and P.
The proof depends in part on Schoenflies's theorem (cf. Fig. E2.1),
which proves that there are only six skew quadrilaterals, spanned
by minimal surfaces, that generate TPMS by half-turn rotations
about their edges, i.e., by repeated applications of
Schwarz's reflection principle.
.jpg)
Fig. E3.2c
Orthogonal projection of the linear asymptotics
in the 'crossed triangles P-catenoid' of Fig. E3.2b
Fig. E3.2d
A cubically symmetrical translation fundamental domain
of the
primitive triply periodic minimal surface P
It was discovered and analyzed by H. A. Schwarz in 1866
together with its adjoint surface D (cf. Fig. E3.1h).
Fig. E3.2e
A translation fundamental domain of P
larger image
Fig. E3.2f
Six translation fundamental domains of Schwarz's P surface
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The lattice for P is simple cubic (s.c.), and
the translation fundamental domain has genus 3.
If the two holes in each pair of opposite holes were joined by a hollow tube,
the translation fundamental domain would be transformed
into an object that is homeomorphic to a sphere with three 'handles'.
P is the unstable stationary state of an inflated jungle-gym-like soap film.
Any finite portion of such a soap film can be made stable if threads are stretched along a sufficient number of the embedded straight lines ('linear asymptotics').
As a sort of metaphor for the 'pipejoint' module of P in Figs. E3.2d, e, imagine that you are
inside a spherical soap bubble at the center of a cube.
Now deform the bubble by blowing against its interior surface in the six directions
x, −x, y, −y, z, −z simultaneously,
forming six cylindrical tubules attached symmetrically to the inside of the cube faces around their centers.
P partitions R3 into two congruent interpenetrating labyrinths.
The skeletal graph (NASA TN D-5541, pp. 38-39) of each labyrinth
is the graph of degree 6 whose edges are those of a packing of congruent cubes.
In Figs. E1.4a-h below are images of tubular simple cubic graphs shown in both 'medium thick' and 'thick' versions. In the
thick version, the diameter d of the cylindrical tubes is the largest possible, consistent
with the requirement that the dual pair of tubular graphs not intersect.
Intersection occurs when the ratio d/e ≥ 1/2, where e is the edge length of the [thin] skeletal graph.
The three Coxeter-Petrie infinite regular skew polyhedra
It dawned on me one day in 1966 that
each of these polyhedra is homemorphic to
— and has the same symmetry as —
either the Schwarz surface D
or the Schwarz surface P.
Regular skew polyhedron {6,6|3}
(homeomorphic to D)
Fig. E3.2g
Translation fundamental domain
The lattice of the oriented surface is f.c.c.
Fig. E3.2h
Assembly of thirteen
translation fundamental domains
Regular skew polyhedron {6,4|4}
(homeomorphic to P)
Fig. E3.2i
Translation fundamental domain
The lattice of the oriented surface is s.c.
Fig. E3.2j
Assembly of twenty-seven
translation fundamental domains
Regular skew polyhedron {4,6|4}
(homeomorphic to P)
Fig. E3.2k
translation fundamental domain
The lattice of the oriented surface is s.c.
Fig. E3.2l
Assembly of
twenty-seven translation fundamental domains
For a comprehensive discussion
of the Coxeter-Petrie regular skew polyhedra,
which includes animated graphics, see this
Wikipedia article.
(E3c) The gyroid surface G
Fig. E3.3a
A translation fundamental domain of the gyroid G
The lattice is b.c.c.
G is a minimal surface that is associate (cf. Fig. E1.2m) to the Schwarz P and D surfaces.
It contains neither straight lines nor plane geodesics. Each of the eight curvilinear
hexagons in this translation fundamental domain is related to each of its four
neighboring hexagons by a half-turn about an axis perpendicular
to the surface at the midpoint of their common edge.
Fig. E3.3b
A translation fundamental domain of G
larger pdf image
[100] tunnels in G
Fig. E3.3c
A ninth hexagonal face has been joined to the top of the piece of G shown in Fig. E1.2g.
This orthogonal projection on the [100] plane shows how nearly circular
the projected outline of each [100] tunnel is.
For a higher-resolution image of this orthogonal projection of G, look
here.
For a stereoscopic perspective view, look
here.
[111] tunnels in G
Fig. E3.3d
This orthogonal projection onto the [111] plane of the piece of G in Fig. E1.2g
shows that the [111] tunnels are fatter and less round than the [100] tunnels.
For a high-resolution view, look
here.
For a stereoscopic perspective view, look
here.
Fig. E3.3e
Many translation fundamental domains of G
view: [100] direction
.jpg)
Fig. E3.3f
Another view of the model of G shown in Fig. E1.2k
view: [111] direction
How I got started (1966-1970)
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In the spring of 1966, I 'discovered' the Schwarz surfaces P and D and their connection to the
Coxeter-Petrie regular skew polyhedra. In order to represent the symmetry and
combinatorial structure of both the surfaces and their flat-faced relatives, the Coxeter-Petrie polyhedra,
I employed the metaphorical device of dual skeletal graphs, which I'll call
g1 and g2. These are triply-periodic graphs regarded
as lying centered in the interiors of the two intertwined labyrinths
of these structures.
Now imagine that every edge of g1
is replaced by an infinitely thin hollow tube with walls composed of some soap-film-like material,
and that the space inside the entire connected network of these tubes defines a single
hollow — but shrunken — labyrinth t1. (Assume that t1 has no self-intersections.
Remember: this is a metaphorical concept, not
a rigorous mathematical construction.)
Here is how I described the relation between g1 and g2,
the two skeletal graphs of a TPMS in 1970, on p. 79 of
Infinite Periodic Minimal Surfaces Without Self-Intersections:
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Assume that the skeletal graph is given for one labyrinth of a given intersection-free TPMS.
Let each edge of the skeletal graph be replaced by a thin open tube, and let these tubes be smoothly
joined (without intersections) around each vertex
so that the whole tubular graph forms a single infinitely multiply-connected surface, which contains the
skeletal graph in its interior. Such a tubular graph is globally homeomorphic to the corresponding minimal surface.
If the tubular graph is sufficiently "inflated", it becomes deformed into a dual tubular graph which contains in its interior
the skeletal graph of the other labyrinth of the surface. The "outside" of the first tubular graph is the "inside" of the second tubular graph.
The two tubular graphs of a given TPMS are required to have the same space group as the TPMS, and to correspond, respectively, to two
tubular graphs which are globally homeomorphic to the TPMS.
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Now imagine inflating t1 so that at its summit,
(i) it contains exactly half of space in its interior,
(ii) its surface has zero mean curvature everywhere,
(iii) it has the same symmetry as the configuration
of the two dual skeletal graphs g1 and g2, and
(iv) it has no self-intersections.
At the inflation summit, the surface t1 is an embedded TPMS.
Until it reaches the summit, t1 exhibits the symmetry of g1.
At the summit, it has the symmetry of the union of the [intertwined] g1 and g2.
As the inflation proceeds beyond the summit, the surface exhibits the symmetry of g2.
It eventually shrinks down to the thin tubular graph t2, which has the symmetry of g2.
Curiously, the outside of t1 is transformed into the inside of t2.
In early 1969, the distinguished topologist
Dennis Sullivan
Fig. E1.2m
Dennis Sullivan
was sharing an office at MIT (just across the street from NASA/ERC) with the mathematics professor
Dirk Struik,
Fig. E1.2n
Dirk J. Struik
1894-2000
In 1967, I had the rare privilege of becoming acquainted with Dirk Struik.
We both enjoyed hiking along the nature trails in Concord near my home.
When Prof. Struik was more than 100 years old, I attended his lecture
on the history of mathematics at an AMS meeting in Cincinnati.
He was in tip-top form.
who had shown him a draft copy of my
Infinite Periodic Minimal Surfaces Without Self-Intersections.
Dennis invited me to his office to explain
that the transformation of the tubular graphs g1 and g2 is an example of the classical
Alexander-Pontryagin duality
(which I had never heard of before).
For the Schwarz surface P, the skeletal graphs
g1 and g2 are identical to the 6-valent simple cubic graph
defined by the edges of an ordinary packing of cubes.
For the Schwarz surface D, both g1 and g2 are copies of the 4-valent diamond graph,
whose edges correspond to the nearest neighbor links in the diamond crystal structure.
Both of these graphs are symmetric, i.e.,
there is a group of symmetries that is transitive on all the edges and all the vertices.
I knew of only one other example of a symmetric triply-periodic graph on a cubic lattice — the Laves graph.
In the spring of 1966, I was seized by the notion that there must exist a TPMS with [enantiomorphic] Laves graphs for its skeletal graphs.
I called it the Laves surface L.
Unlike the skeletal graphs of P and D, however, the configuration of two dual Laves graphs has no reflection symmetries, and
its axes of rotational symmetry lie in directions that I determined could not possibly correspond to lines embedded in the surface
I was seeking.
As a consequence, I had no idea how to generate a surface patch bounded by either straight line segments or plane geodesics
a ("Schwarz chain").
By the time I moved to NASA in July 1967, I had made a reasonably thorough study of Schwarz's writings on periodic minimal surfaces,
and I understood the Bonnet associate surface transformation that defines the relation between P and D
(and also the relation between the catenoid and the helicoid).
The brightly colored plastic models of P and D I had constructed were almost literally screaming out to me
that I should explore the territory between these two surfaces (where one surface is bent continuously into the shape of the other),
but I did not hear their screams!
In February 1968, I stumbled accidentally on a very close approximation of the gyroid. I'll call it the pseudo-gyroid.
The models illustrated in Figs. E1.2k, E1.2l, and E1.17 show the final steps in the procedure that led to this pseudo-gyroid.
The resemblance between this virtual doppelgänger and the true gyroid is so close
that with the naked eye it is impossible to tell them apart.
I still had no idea yet that the gyroid is just a surface associate to P and D that happens to be free of self-intersections.
In those days there were not yet any known examples of embedded TPMS derived by examining intermediate stages of the 'morphing'
transformation that bends one minimal surface into its adjoint surface
via the associate surface transformation,
and I didn't have the imagination to think of that possibility.
In 1990, it occurred to
Sven Lidin and Stefan Larsson
to look for an embedded surface among the surfaces associate to
Schwarz's [embedded] surface H
and its self-intersecting adjoint surface,
and they found exactly one — the
lidinoid:
Fig. E1.2o
The lidinoid
Fig. E1.2p
Schwarz's H surface
To return to the gyroid story, in May 1966 — as mentioned above —
I had already begun to suspect that a minimal surface with the symmetry and topology of the gyroid might exist.
My suspicions were based on the fact that the Laves graph
— like the skeletal graph of each labyrinth in Schwarz's P and D surfaces — is
not merely regular (all vertices are of the same degree), but also symmetric
(it is both vertex-transitive and edge-transitive).
In H and CLP, the two other examples of Schwarz's TPMS, which — like P and D — are of genus 3,
the skeletal graphs are merely regular and not symmetric.
My intuition suggested that a symmetric graph is so homogeneous that it is very likely to be the skeletal graph of
a labyrinth of some embedded TPMS. (I eventually discovered that although some of the few known examples of symmetric
triply-periodic graphs are skeletal graphs of labyrinths of such surfaces, by no means all of them are.)
During the spring and summer of 1968, I concentrated on the writing of
a so-called preliminary report (an internal NASA document, not intended for general circulation),
entitled "Expansion-Collapse Transformations on Infinite Periodic Graphs",
NASA/Electronics Research Center Technical Note PM-98 (September 1969),
draft versions of two patent applications, and
computer graphics animations of collapsing graphs.
The considerably less time-consuming one of the two patent drafts was eventually entitled,
"Honeycomb Panels Formed of Minimal Surface Periodic Tubule Layers".
I had discovered no useful ideas about how to prove that the pseudo-gyroid (cf. Fig. E1.17)
was the basis for a bona fide minimal surface.
Blaine Lawson told me in early August that he too had made no progress toward figuring out how to
prove that a skew hexagonal face of the pseudo-gyroid, with its strictly helical edges,
could somehow be analytically continued to generate an embedded periodic minimal surface.
But ever since my first phone conversation with Blaine in late spring, I had found it extremely helpful to
discuss with him a variety of questions concerning minimal surfaces other than the gyroid. I used him
as a sounding board on some of my still tentative ideas about how to derive new examples of embedded TPMS
by
(a) enumerating all the ways of constructing a Schwarz chain as a connected sequence of arcs
on the n faces (one arc on each face) of what is now called a
'Coxeter cell' — a convex polyhedral space-filler related to each of its neighbors by reflection in a face, and,
somewhat later,
(b) 'hybridizing' two TPMS (cf. Fig. E2.71).
In early September 1968, I returned to Cambridge from an AMS summer meeting at Madison, Wisconsin, where
I had used the pseudo-gyroid model shown in Fig. E1.17 to illustrate my 15-minute talk (cf. Fig. E2.10).
I was still calling the surface the 'Laves surface' in those days.
Fig. E2.69a
A souvenir postcard
Lake Mendota,
from the Wisconsin Union Boat House
Madison, Wisconsin
(1968)
It was at this Madison summer meeting that I met several mathematicians
of my father's generation who knew something about minimal surfaces.
I particularly enjoyed meeting
Wolfgang Wasow, who is
shown below in a 1952 photo with
Magnus Hestenes.
(I met Hestenes in 1969 in Tbilisi, Georgia.)
Fig. E2.69b
Wolfgang Wasow (left) and Magnus Hestenes (right)
Fig. E2.69c
H. Blaine Lawson, Jr.
About a week after my return to Cambridge, I used the government WATS line to phone Blaine to ask him
whether he had made any progress toward a proof that the gyroid is a minimal surface. He replied that he hadn't,
because finishing his dissertation had left him little time to think about other matters.
He said he was going to have to abandon work on the problem.
I begged him not to give up, because I felt certain the solution was close at hand
(even though I had no rational grounds for believing that to be the case!).
To change the subject, I told Blaine about the graph collapse transformation I had discovered, and how
it could be 'run backwards' to provide the basis for the design of expandable space-frames.
I had investigated the transformation for graphs associated with
the P, G, and D surfaces (cf. Figs. E2.68b.0, E2.68b.1, and E2.68b.2, for example).
I described what I called 'just a coincidence' (or words to that effect): that
the trajectory of every graph vertex is an ellipse not only in the associate surface transformation of Bonnet
but also in the totally unrelated graph collapse transformation.
I emphasized that there is no fundamental connection between these two transformations.
I described how I had found that of the triply-periodic graphs that are associated with
P and D, either as embedded graphs or skeletal graphs, those that have reflection symmetries
are not candidates for expandable space-frames
because of pairwise collisions of edges (called webs or struts in the space-frame) that occur early in the collapse.
In contrast to this behavior, for all of the twisted graphs
derived from the pseudo-gyroid, including the Laves graph, no such collisons of edges occur. The only collisions are
the ones that would occur in actual physical spaceframes, in which struts collide somewhat before the 'complete collapse' stage
because of their finite thickness.
I had not previously even mentioned graph collapse to Blaine, and it's hardly surprising that he
didn't seem to understand the details of what I said to him.
It was obvious that I hadn't explained the elliptical trajectories coincidence very well, because Blaine's response was something like:
"Are you saying that the gyroid is associate to Schwarz's P and D surfaces?"
I hadn't said that at all, but it hardly mattered, because
at that instant, everything suddenly fell into place. The fog had finally lifted!
Thanks to Blaine's question, I finally understood that
the gyroid is just a surface associate to P and D that
happens to be embedded (free of self-intersections). It is the only such surface,
as I was soon able to confirm by means of simple 'morphing' sketches similar to the computer drawings in Fig. E1.21.
I knew from my few months of elementary (but slightly complicated) engineering analysis of graph collapse transformations
that the surface orientation is the same at corresponding vertices not only on
the hexagonal faces of the Coxeter-Petrie map {6,4|4} on P and D but also on the pseudo-gyroid,
and now I realized what a strong hint pointing to the
Bonnet transformation that had been.
I realized with considerable embarrassment how obvious it should have been to me from the very beginning
that the gyroid is associate to P and D!
After all, I was familiar with the properties of the Bonnet transformation. I had long since traced out
the geometrical relation between the equatorial circle in the catenoid and the central axis of the helicoid,
which I had found illustrated in
Dirk Struik's marvellous
Lectures on Classical Differential Geometry.
I had also sketched the corresponding curves in P and D countless times.
Those relations should have been the clue.
I had also spent days studying not only H. A. Schwarz's Collected Works, but also
Erwin Kreyszig's
Differential Geometry,
Luther Pfahler Eisenhart's
A Treatise on the Differential Geometry of Curves and Surfaces,
and
Barrett O'Neill's
Elementary Differential Geometry.
Although I knew from experience that ideas that should seem obvious are sometimes anything but obvious,
I nevertheless felt stupid when I
realized that I had posed the wrong question from the start.
I felt only slightly less stupid when I discovered that Blaine's response to my harangue about
the elliptical trajectories of the vertices of collapsing graphs
was not actually the result of his understanding that the gyroid was associate to P and D.
He had been justifiably confused by my rambling description of those irrelevant elliptical trajectories.
When I explained to him the evidence for the associate surface relationship,
he agreed that it was a reasonable idea.
I immediately proposed that we publish together an announcement about the gyroid.
He courteously refused, explaining that his crucial question to me was prompted by a misunderstanding of what I was saying.
But I insisted that if he had not asked me that question in precisely those words, it would have been impossible to say how long
it might have taken me
to understand what was going on. He then reluctantly agreed to collaborate on a paper about the gyroid.
Two days later, Dr. Van Atta returned to ERC from an out-of-town trip.
He had been following my struggles with the pseudo-gyroid for months.
As soon as I told him my exciting news,
including my plan to co-publish with Blaine Lawson,
he scolded me in no uncertain terms! He insisted that I phone Blaine and explain that I had made a serious error, and that I must publish alone.
(It was the only time Dr. Van Atta ever displayed anger or impatience toward me.)
Blaine was courteous when I relayed my new message to him, but I realized that my vacillation must have offended him.
Gradually I succeeded in feeling very slightly less stupid than I had at first, after reflecting on the fact that
neither Schwarz, Riemanm, Weierstrass,
nor any of their successors seem to have suspected the existence of an embedded surface associate to P and D,
in spite of the fact that they were all experts on the Bonnet transformation.
On the other hand, I realized that it had been pure dumb luck for me to stumble onto
M4 and M6, the precursors of the gyroid.
I was able to calculate the angle of associativity (cf. Fig. E1.23) quickly,
because I had recently made a detailed study of the geometrical calculations Schwarz carried out
in his analysis of the P and D surfaces.
Several months earlier, because I was curious about the precise shape and arc length of the
quasi-circular edges of a {6,4|4} hexagon of P
(cf. Fig. E1.2c), I had sketched the outline of a computer program for getting answers to these questions.
My colleague Jim Wixson coded the program in FORTRAN and ran it on ERC's PDP-11 minicomputer.
The output of Jim's program, combined with Schwarz's analysis, provided the required clues to the value of the angle of associativity of G
(cf. Fig. E1.23) .
They also showed that the departure from perfect circularity of the quasi-circular holes in a pipejoint unit cell of P
is in the range of approximately ± 0.5% of the hole's mean radius, implying a comparably small departure
from perfect helicity of their image curves in the gyroid.
Not only did the quasi-helical curves in the pseudo-gyroid (cf. Fig. E1.17) turn out to be
very close approximations to the corresponding curves in the true gyroid,
but — according to calculations I carried out a few years ago — these curves
are also very close to, but not quite the same as, the corresponding
curves in the 'level-set' gyroid (cf. Fig. E1.31.)
In late 1968, I decided that I must somehow force a nodal polyhedron for BCC6 into being,
and by trial-and-error I produced the
space-filling saddle polyhedron shown In Figs. E2.70a, b, and c.
It is described in
Infinite Periodic Minimal Surfaces Without Self-Intersections.
Fig. E2.70a
BCC6 Pinwheel polyhedron:
the 6-faced nodal polyhedron
of
the defective symmetric graph BCC6 of degree 6
(stereo pair)
The vertices of the graph are the complete set of vertices of the b.c.c. lattice.
BCC6 is described on p. 82 of
Infinite Periodic Minimal Surfaces Without Self-Intersections.
Fig. E2.70b
An oblique view of the BCC6 Pinwheel polyhedron
(stereo pair)
Fig. E2.70c
The BCC6 graph (orange) and its dual graph (black)
(stereo pair)
(What is not quite kosher about this black graph
is that it intersects the orange graph.
Perhaps a better design can be found for
an improvised nodal polyhedron for the BCC6 graph.)
Fig. E2.7
The six components of the brass tool — before assembly and further machining — that I used
for vacuum-forming the plastic hexagons of the 1968 model of the pseudo-gyroid shown in Fig. E1.4c.
Two years earlier,
I discovered an arrangement of two sets of the eight
solid tetrominoes
in the enantiomorphic trigonal packings of a half-cube shown in Figs. E2.8a and b.
These two arrangements of the tetrominoes pack the cube.
The shapes of these packings inspired the design of the tool parts shown above.
Incidentally, I conjecture that the packing of the assembly
of eight tetrominoes shown in Figs. E2.8a and b is unique.
If you have evidence to the contrary, please let me know!
Fig. E2.8a
Mirror-symmetric triskelion packings by the eight solid tetrominoes
Fig. E2.8b
Mirror-symmetric triskelion packings by the eight solid tetrominoes
My addiction to recreational mathematics,
which worsened considerably once I started playing with these solid tetrominoes in 1965,
was not helped by exposure to the writings of
Martin Gardner and
Solomon Golomb.
I am extremely grateful to both of them.
Fig. E2.9
Solomon Golomb
Fig. E2.10
Abstract 658-30 submitted to the American Mathematical Society in 1968
announcing the discovery of the gyroid
I referred to the gyroid here as 'L' (for 'Laves').
A few weeks later I renamed it 'gyroid'.
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This announcement was slightly premature!
I mailed in the abstract in Fig. E2.10 a month or so before the Madison summer meeting of the AMS,
even though I had not yet succeeded in proving that the
surface represented by the pseudo-gyroid (cf. Fig. E1.4c) is a single continuous minimal surface.
I had naively assumed that any expert on minimal surfaces
would be able to construct such a proof.
A few months earlier, I had sent a plastic model of the pseudo-gyroid to
Bob Osserman,
who passed it along to his PhD student
Blaine Lawson,
Blaine promised to think about the problem in his spare time, even though he was already fully occupied with his dissertation research.
It wasn't until about ten days after the Madison meeting that at last I
understood — thanks to a fruitful phone conversation with Blaine —
that the gyroid is an associate surface in the
Schwarz P−D family.
(The details are described just after Fig. E2.68c.9.)
By means of drawings based on hand calculations, I confirmed that there are no other intersection-free
associate surfaces between P and D.
A few months later, the differential geometer
Tom Banchoff
introduced me to
Charlie Strauss,
the mathematician/computer graphics expert who is his friend and collaborator.
I hired Charlie to write a computer graphics program
for producing stereoscopic perspective animations, and
I used one such animation sequence (cf. Fig E1.21) to strengthen the evidence that every other associate surface is self-intersecting.
(Twenty-eight years later, this claim was at last proved rigorously by
Karsten Grosse-Brauckmann and Meinhard Wohlgemuth, in their article,
'The gyroid is embedded and has constant mean curvature companions', Calc. Var. Partial Differential Equations 4 (1996), no. 6, 499-523.)
The precise version of the surface that I had in mind at the time of the Madison meeting had a fatal flaw:
the boundary of each of its hexagonal faces is a chain of one-quarter pitches of circular helices,
alternately right-handed and left-handed (cf. Fig. E2.7).
Even now, in 2011, it is not known how to derive an analytic solution for a minimal surface bounded by a circuit ('Schwarz chain') composed of such arcs.
Although an assembly of these hexagons looks like a single infinitely-connected minimal surface, it is not one.
To explain why I had chosen these helical curves for the surface patch boundary, I offer the following more or less chronological summary of
the tangled sequence of events
that culminated in the construction of the model of L.
In January 1968 I found my first promising lead in the hunt for the gyroid — a pair of related surfaces I named M4 and M6.
One might call them wrinkled versions of the gyroid. The original models of these surfaces are shown in Figs. E1.16a and E1.16b.
A variety of stereoscopic images of more recent models of M4 and M6 are shown in Figs. E1.19 and E1.20.
Fig. 1.16a
Fig. 1.16b
M4
M6
view: [111] direction
As the result of a lucky guess about how to remove the 'wrinkles' in M6, I produced the pseudo-gyroid,
which is shown in Figs. E1.4c and E1.17. The details of how I got from the
pseudo-gyroid to the gyroid are described in the text that precedes Fig. E1.33a.
By September 1968 I had concluded that the gyroid is the only embedded surface among the
countably many surfaces associate to P and D,
but my 'proof' was based on (a) physically bending assemblies of plastic replicas of surface modules and on
(b) computer graphics animation of that bending (cf. Fig. E1.21b).
In 1996, Karsten Grosse-Brauckmann and Meinhard Wohlgemuth published a rigorous proof that
the gyroid is embedded (free of self-intersections) and contains neither straight lines nor reflection symmetries, in
The gyroid is embedded and has constant mean curvature companions, Calc. Var. 4, 1996, 499-523.
It seems likely that before 1968, no one had ever bothered to look at any of the
countably many intermediate surfaces associate to Schwarz's P and D
surfaces to determine whether any of them were embedded.
I confess that before 1968 it had never occurred to me to look there (even though it should have!).
The gyroid has received an unusual amount of attention from physicists, chemists, and biologists since the early nineties,
because it has been found to be — at least approximately — a kind of geometrical template for
a great variety of self-assembled bicontinuous structures, both natural and synthetic.
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Fig. E1.17
The first physical model (1968) of the pseudo-gyroid
I first announced its existence at an AMS meeting
in August, 1968 (cf. abstract in Fig. E2.10),
in the mistaken belief that the pseudo-gyroid is a bona fide minimal surface.
(It is remarkably close to one!)
The actual gyroid G is described on pp. 48-54 of
Infinite Periodic Minimal Surfaces Without Self-Intersections,
NASA TN D-5541 (May 1970),
and also here in Fig. E1.23.
This model was composed of vacuum-formed plastic hexagons.
view: [100] direction
Weierstrass parametrization of G
Fig. E1.2m
The rectangular coordinates of G,
defined by Schwarz's solution (Weierstrass-Enneper parametrization)
for the entire family of surfaces associate to P and D
If θG in the term eiθG is replaced by zero,
the coordinates are those of D.
If θG is replaced by π/2, the coordinates are those of P — the adjoint of D.
(The value given above for θG agrees up to eight significant figures
with the value derived from a more recent analysis
based on a Schwarz-Christoffel mapping,
in
Adam Weyhaupt's 2006 PhD thesis (pp. 115-116).
Fig. E1.2n
Bonnet bending of one skew hexagonal face of D (blue)
into its image in G (violet) and P (red)
During the bending, the trajectory of each point on the surface is an ellipse.
.jpg)
.jpg)
Fig. E1.2o
Sequence of Bonnet bending stages from D to G to P (stereo)
One lattice fundamental domain (plus one additional hexagonal face)
The surfaces at all other angles between 0º and 90º are self-intersecting.
The above images are eleven frames selected from a movie I made in 1969
using a FORTRAN program written by the computer scientist
Charles Strauss.
The cinematographer, Bob Davis of the
MIT Lincoln Lab,
used a Bell and Howell 35mm movie camera
modified to accept input data from a PDP-11 computer.
Ken Paciulan
and Jay Epstein peformed the data input.
The enthusiastic help of all of these great guys is gratefully acknowledged.
*****************************************************************************
If you are using INTERNET EXPLORER or MOZILLA FIREFOX as your browser, you can
ANIMATE THIS TIME-LAPSE BENDING SEQUENCE BY CLICKING HERE.
Hold down the Page Down/Page Up keys to see the animation.
******************************************************************************
Stereo views of translation fundamental domain of P, D, and G
.jpg)
Fig. E1.2p
The D surface
.jpg)
Fig. E1.2q
The G surface
.jpg)
Fig. E1.2r
The P surface
For higher-resolution [pdf] versions of these three images, look
here (D),
here (G),
and
here (P).
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These pdf images will probably load slowly, because they are large (7 to 10 Mb).
For maximum image clarity, zoom in to make the width of the image almost equal to the screen width.
The associate surfaces D and P are called adjoint, because the
angle of associativity (cf. Fig. E1.23)
by which they are related via Bonnet's bending transformation is π ⁄ 2.
Like all of the other intermediate surfaces associate to the helicoid and the catenoid,
G contains neither straight lines nor plane lines of curvature.
The round tunnels centered on [100], [010], and [001] axes, which are arranged in square checkerboard arrays,
are bounded by approximately helical curves of opposite handedness in the two intertwined labyrinths of the surface.
The outermost curved edges of the eight congruent hexagonal faces in Fig. E1.3a
correspond to the edges of the regular map with holes {6,6|3}.
If you examine these edges closely, you will see that they are not quite plane.
Each of them lies half inside and half outside the enclosing cube.
Let us call a geodesic curve on a triply-periodic minimal surface a trivial geodesic
if it is either a straight line or a mirror-symmetric plane line of curvature, and a non-trivial geodesic otherwise.
Since there are neither straight lines nor mirror-symmetric plane lines of curvature in the gyroid,
all of its geodesic curves are non-trivial.
If you apply rubber bands to physical models of the three surfaces P, G, and D (as I have done),
you will easily discover a variety of periodic geodesic curves, both closed and unbounded.
I will eventually show here images of several non-trivial geodesics,
both closed and unbounded, on these three surfaces.
For a discussion of geodesics on multiply-connected surfaces, see
Steven Strogatz's 2010 NYTimes essay on geodesics,
with links to Konrad Polthier's videos on this topic.
The four images in Figs. E1.15d to E1.15g were made by
Ken Brakke,
period killer extraordinaire.
The embedded surfaces I-WP (cf. also Fig. 2.5) and F-RD (cf. also Fig. 2.6) are the
adjoints of two of the four self-intersecting Schoenflies surfaces (cf. Fig. E2.1).
In 1975, I used the laser-soap film technique mentioned just below Figs. E1.15a and b to kill
periods in order to obtain the approximate shapes of the curved edges of the octagonal surface patch of F-RD(r).
Several years ago, Ken used his Surface Evolver program to obtain the vastly improved modeling of F-RD(r) shown below.
Soon afterward I asked Ken to use Surface Evolver to obtain the hexagonal patch of I-WP(r). In order to emphasize
how the tubular structure of each cubic surface cell on the right is related to its companion
at its left by a one-eighth-turn (45º) about a vertical axis, Ken included '(r)' in its name.
Fig. E1.15d
Fig. E1.15e
F-RD
F-RD(r)
Fig. E1.15f
Fig. E1.15g
I-WP
I-WP(r)
A spectacular large model of G is shown in the
YouTube video,
Gyroid playground climbing structure at the San Francisco Exploratorium
and in Figs. E1.18a-d.
This structure was designed and built by a team that included
Thomas Rockwell, Paul Stepahin, and Eric Dimond.
Fig. E1.18a
Exploratorium gyroid
photo courtesy of Amy Snyder
Additional photos of the Exploratorium gyroid
(also courtesy of Amy Snyder) are at:
E1.18a
E1.18b
E1.18c
E1.18d
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Figs. E1.19a-e show M6, which was the immediate precursor of the gyroid (cf.
abstract in Fig. E2.10). The vertices coincide with the vertices of the
Coxeter-Petrie map {6,4|4}
on the gyroid, but the
edges of the hexagonal faces are line segments, not quasi-helical arcs as in the gyroid.
Figs. E1.20a-e show M4, the immediate precursor of M6.
_reduced.jpg)
Fig. E1.19a
65 hexagons of M6
view: [111] direction, silhouetted by bright summer sky backlighting
Fig. E1.19b
65 hexagons of M6 (stereo)
view: [100] direction
_reduced.jpg)
Fig. E1.19c
65 hexagons of M6
view: [100] direction, silhouetted by bright summer sky backlighting
_reduced.jpg)
_reduced.jpg)
Fig. E1.19d
65 hexagons of M6
view: [111] direction
_reduced.jpg)
_reduced.jpg)
Fig. E1.19e
65 hexagons of M6
view: [100] direction
_reduced.jpg)
_reduced.jpg)
Fig. E1.20a
30 quadrangles of M4 (stereo)
view: [111] direction
.jpg)
.jpg)
Fig. E1.20b
30 quadrangles of M4 (stereo)
view: [110] direction
Fig. E1.20c
30 quadrangles of M4 (stereo)
view: [110] direction
Fig. E1.20d
The edges of the model in Fig. E1.20c,
which define a portion of the 'defective' graph BCC6
view: close to [100] direction
_silhouette_[111](1)_reduced.jpg)
Fig. E1.20e
30 quadrangles of M4
view: [111] direction, backlit by summer sky
_reduced.jpg)
Fig. E1.20f
30 quadrangles of M4
view: [100] direction, backlit by summer sky
Cubic unit cell of G
Fig. E1.22
Cross-eyed stereogram of a gauzy cubic unit cell of G
G is associate to P and D, i.e.,
its rectangular coordinates
are a linear combination of those of P and D
(cf. Fig. E1.23).
This figure shows twice as much of G as is contained in one translation fundamental domain.
The lattice for G is body-centered cubic (b.c.c.).
Among the countable infinity of surfaces associate to P and D,
G is the only embedded surface.
A translation fundamental domain has genus 3.
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Cubic unit cell of G
Fig. E1.24
A view from a different corner (the lower right rear corner) of the unit cell of G shown in Fig. E1.22
The skeletal graphs of the two enantiomorphic labyrinths of G are enantiomorphic Laves graphs of girth ten (and degree 3).
Since the lattice for G is b.c.c.,
a truncated octahedron is a reasonable choice for a translation fundamental domain.
This cube unit cell has volume equal to that of two translation fundamental domains.
The sequence of three stereo images immediately below illustrates the application of
Ossian Bonnet's 1853 bending transformation, without stretching or tearing,
to the 'morphing' of Schwarz's D surface into Schwarz's P surface.
The orientation of the tangent plane at each point of the surface remains fixed throughout the bending.
The G surface shows up a little more than one-third of the way (cf. Fig. E1.23) through the transformation.
A simply-connected translation fundamental domain of genus 3 is shown in each figure.
Each such domain is composed of eight congruent regular skew hexagonal faces,
which are defined by Coxeter's regular map with holes {6,4|4} (cf. Fig. E1.4).
The lattices for the three [oriented] surfaces are
face-centered cubic (f.c.c) for D,
body-centered cubic (b.c.c) for G, and
simple cubic s.c.) for P.
|
Stereo view of G
Fig. E1.28
Cross-eyed stereogram of a larger piece of G
A hexagonal face of G
Fig. E1.29
Cross-eyed stereogram of a hexagonal face of G that corresponds to
a face of the regular map with holes {6,4|4}
{cf. Fig. E1.5).
The six vertices of the face coincide with vertices of a truncated octahedron.
The edges of the face appear to be plane curves, but they are not.
One half of each edge lies inside this truncated octahedron, and one half outside.
view: [100] direction
Two hexagonal faces of G
Fig. E1.30
Cross-eyed stereogram of two hexagonal faces of G
|
David Hoffman and Jim Hoffman (not relatives!) have animated
the associate surface transformation
P-> G -> D
for a single translation fundamental domain.
Gerd Schröder-Turk has also animated the
P -> G -> D Bonnet transformation
.
In addition, he has created a spectacular
'FlyThrough' animation of a large chunk of G.
His movie shows the enantiomorphic skeletal graphs of the two labyrinths of G.
Don't be surprised if Gerd's movie takes a while to download this file. It's not small (~91 Mb!).
The "level-set" surface
G*,
which very closely resembles G, is defined by the equation
cos x sin y + cos y sin z + cos z sin x = 0.
David Hoffman and Jim Hoffman have illustrated
the striking resemblance between G and G*.
Below are images of G* made with Mathematica.
The level surface G*
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Fig. E1.31
Various views of the level surface G*
-
(a) skew hexagonal face
-
(b) assembly of eight of the skew hexagonal faces, which defines a translation fundamental domain bounded by a cube
-
(c) the assembly shown in (b) viewed from the (1,1,1) direction
-
(d) the assembly shown in (b) viewed from the (− 1,− 1,− 1) direction
-
(e) view of the assembly in (b) from the (1,0,0) direction, showing the square array of quasi-helical tunnels.
Adjacent tunnels spiral alternately CW and CCW.
-
(f) view of the assembly in (b) from the (1,1,1) direction, showing the kagomé array of quasi-helical tunnels.
Adjacent tunnels spiral alternately CW and CCW.
-
(g) oblique view of the assembly shown in (f)
|
Flat-faced polyhedral models of minimal surfaces
Voronoi G
"Voronoi G", which is illustrated in Figs. E1.34c to E1.34g, is a triply-periodic surface composed of plane polygons
that roughly approximates the gyroid.
It is constructed by removing the three decagons from every cell
in a combined packing of the two enantiomorphic types of Voronoi cells (cf.
Fig. II-2a, p. 89 in NASA TN D-5541) that
enclose the vertices of the pair of Laves graphs, related by inversion, that are the skeletal graphs of the gyroid.
Fig. E1.34a
Voronoi cell of a vertex of one of the pair of enantiomorphic Laves graphs
that are skeletal graphs of the gyroid (stereoscopic pair)
Fig. E1.34b
Voronoi cell of a vertex of the other of the pair of enantiomorphic Laves graphs
that are skeletal graphs of the gyroid (stereoscopic pair)
Fig. E1.34c
Voronoi G, viewed in the [100] direction (stereoscopic pair)
(For a higher-resolution [pdf] version of the above image, look
here.)
Fig. E1.34d
Orthogonal projection of Voronoi G onto [100] plane (cf. Fig. E1.34c)
Fig. E1.34e
Voronoi G, viewed in the [110] direction (stereoscopic pair)
(For a higher-resolution version of the above image, look
here.)
This image demonstrates that the gyroid is symmetrical by a half-turn about a [110] axis.
Fig. E1.34f
Orthogonal projection of Voronoi G onto [110] plane
Fig. E1.34g
Orthogonal projection of Voronoi G onto [111] plane
(For a higher-resolution [pdf] version of the above image, look
here.)
Compare the shapes of the tunnel holes with the
corresponding holes in G shown in Fig. E1.19.
Fig. E1.34h
One chamber of Voronoi G
All but six of its thirty vertices lie on the sphere of radius r1 shown here.
The six outliers — shown as red dots — lie on
a slightly larger sphere of radius r2 (not shown).
The ratio r2/r1 ≅ 1.029.
Uniform polyhedra based on G
Fig. E1.35a
The uniform gyroid {6.3.4.32}, an infinite uniform polyhedron
All of its faces are regular plane polygons,
and the symmetry group of the surface
is transitive on the faces and on the vertices.
(For a pdf version of the upper image, look
here.)
This polyhedron, which was independently discovered a few years ago by John Horton Conway and named
(by him) 'mu-snub cube', was first discovered in 1969 by the mathematician Norman Johnson,
shortly after I showed him my AMS abstract (cf. Fig. E1.35c) summarizing the results
of an incomplete study of the infinite uniform polyhedra based on P, G, and D.
(For decades, I have tried — without success — to persuade Norman
to publish an account of his enumeration.)
I failed to catch {6.3.4.32} and several other examples.
Norman's list (proved by him to be complete) of the
tessellations of the {4,6} family is shown in Fig. E1.35b.
Norman is well known for his 1966 enumeration — later proved by Zalgaller to be complete — of
the 92 Johnson solids.
Fig. E1.35b
Norman Johnson's 1969 enumeration of
the uniform tessellations of the {4,6} family
(unpublished)
Fig. E1.35c
A summary of my incomplete enumeration of the
quasi-regular tessellations of the {6,4} family
(abstract submitted to the American Mathematical Society in 1969)
Uniform polyhedron models of P
Fig. E1.36
(6.43), an infinite uniform polyhedron
model constructed by my Cal Arts student Bob Fuller in 1971
Fig. E1.37
(6.43), an infinite uniform polyhedron
model constructed by Bob Fuller in 1971
Fig. E1.38
(6.43), an infinite uniform polyhedron
model constructed by Bob Fuller in 1971
Isaac Van Houten's bronze sculpture of G
Fig. E1.39
Isaac Van Houten's 2006 bronze sculpture of the gyroid G
Bathsheba Grossman's rapid prototype sculpture of G
Fig. E1.40
Two views of one of
Bathsheba Grossman's printed models of G
E2. Cubic lattice surfaces not in the Schwarz P-D family
In 1934, the mathematician
Berthold Stessmann published an article in Mathematische Zeitschrift 38, 1934 (414-442) entitled
"Periodische Minimalflächen".
* * * *
(I have been unable to learn what became of Stessmann in the WWII period. If you have any information about him, please email me.)
* * * *
On the first page of Stessmann's article there are drawings of the six skew quadrilaterals that
Arthur Moritz Schoenflies
proved in 1890 are the only skew quadrilaterals, spanned by minimal surfaces,
that generate TPMS by half-turn rotations about their edges, i.e., by repeated applications of
Schwarz's reflection principle.
These six Schoenflies quadrilaterals are reproduced here in Fig. E2.1.
The six Schoenflies quadrilaterals
Fig. E2.1
The six Schoenflies quadrilaterals
Of the six Schoenflies quadrilaterals, only I and III are patch boundaries for embedded
surfaces — Schwarz's D and P, respectively (cf. Figs. E1.2a and E1.3a).
By applying Schwarz's reflection principle, it is easy to demonstrate that the other four quadrilaterals
— II, IV, V, and VI — define surfaces with transverse self-intersections.
But what about the adjoints (cf. text below Fig. E1.23) of II, IV, V, and VI?
It is a fundamental property of any two adjoint minimal surfaces S1 and S2
that if a boundary edge in S1 (say) is a straight line segment E1, then its image in S2
is a segment of a plane line of curvature C1, lying in a plane perpendicular to E1.
Let us call this property 'Property A'.
The adjoint of II is an embedded surface of genus 9 and is called Neovius's surface.
It was first analyzed in 1883 by Edwin Rudolf Neovius — an 1869 doctoral student of Schwarz's — in his dissertation,
Zweier Speciellen Periodischen Minimalflächen auf welchen
unendlich viel gerade linien und unendlich viele ebene geodätischen linien liegen.
I sometimes identify Neovius's surface by the alternative name C9(P) —
or just C(P) —
because it has exactly the same embedded straight lines as P and has genus 9.
(The 'C' in C9(P) stands for 'complement'.)
C9(P) is illustrated in Figs. E2.2a,b,c, and
its self-intersecting adjoint surface C9(P)† is illustrated in Fig. E2.2d.
Neovius, who became Professor of Mathematics at the University of Helsinki, was a cousin of
Ernst Lindelöf
and uncle of
Rolf Nevanlinna.
(For further information, see this
short article about the history of the Finnish Mathematical Society.)
The embedded Neovius surface C9(P)
and its self-intersecting adjoint surface C9(P)†
_shaded pencil drawing.jpg)
(a)
(b)
Figs. E2.2a and b
One lattice fundamental region of
the embedded Neovius surface C9(P) (genus 9)
The images are copied from Tafeln II and III of Neovius's 1883 dissertation.
On June 27, 2012, the day after my wife Reiko and I visited the magnificent
National Library of Finland,
Mr. Jari Tolvanen, a reference librarian there, kindly informed me by email that Neovius's doctoral dissertation,
a copy of which is in the library's collection, is also available online
here.
_photo.jpg)
_adjoint_photo.jpg)
(c)
(d)
Figs. E2.2c and d
(c) An assembly of eight lattice fundamental regions of
the embedded Neovius surface C9(P)
and
(d) an assembly of seven lattice fundamental regions of its self-intersecting adjoint surface
C9(P)†
The images are copied from Tafeln IV and III of
Neovius's 1883 dissertation.
Fig. E2.3
A vacuum-formed plastic model of C9(P)
_abstract_2-11-69(2).jpg)
Fig. E2.4a
Announcement in a February, 1969 abstract published by the American Mathematical Society
describing C(D), a TPMS that is complementary to Schwarz's D surface
"Abstract 658-30" mentioned in the third line
and shown here as Fig. E2.10, refers to the gyroid,
which I originally named "L" (for Laves) in 1968.
The possibility that there exists an embedded counterpart of C(P),
which it would be reasonable to call 'C(D)',
occurred to me as I was crossing the street while returning to my NASA office from the MIT library.
I was staring at the illustrations in the photocopy of Neovius's 1883 PhD dissertation
made for me a few minutes earlier by the MIT science librarian.
When I saw the drawing shown above in Fig. E2.2b,
I was startled to see that the set of straight lines in the surface
is identical to the set of straight lines in Schwarz's P surface.
I imagine that Neovius and Schwarz (his teacher) also must have noticed this matching of lines!
I suddenly decided that what is sauce for the goose may also be sauce for the gander.
I reasoned that since both P and D can be regarded as [infinite] regular polyhedra,
it is plausible that if one of them (P)
has an embedded companion surface that contains the same set of straight lines,
then the other (D) probably does too.
I concluded that if I could confirm the existence of such a surface for D, I would
name Neovius's surface 'C(P)' and the new surface 'C(D)' —
where 'C( )' means 'complement of ( )'.
As soon as I reached my office a few minutes later, I needed only to glance at my straw model of the straight lines in D
to recognize instantly that the embedded surface C(D) exists.
A few months later, I realized that higher order complements of both P and D probably exist too
(cf. discussion of 'Notched adjoints' following Fig. E2.82).
The MIT librarian told me that no one had borrowed Neovius's thesis during the fifty years
since the library had acquired it. It was printed in large folio format,
and as a result she had to cut all the pages into quarters (which took her a good while!) before she could make a photocopy for me.
Perhaps that's a good indication of how interested mathematicians were in periodic minimal surfaces in those bygone days.
(My account of how the MIT library copy of Neovius's thesis came to see the light of day for the first time in fifty years
would hardly be complete if I failed to mention that I initially asked the librarian to obtain a copy for me via inter-library loan,
because I naively assumed that it was much too obscure to be included in the MIT collection. I even mentioned that I was aware that she might have
to send to Finland for it, and I was prepared to wait. She did not deign to reply but simply marched off in the
direction of the stacks, pausing only to look back once over her shoulder, wearing an expression of polite disdain
as if to say, "What kind of institution do you think we are?" I can't imagine that she actually knew that she would find the thesis
on the shelves, but she located it in much less than a minute!)
|
C(D)
_twopieces_thumbnail_reduced.jpg)
Fig. E2.4b
C19(D) (genus 19)
Right: One translation fundamental domain
Left: One-fourth of a translation fundamental domain
C19(D) is called the [first-order] complement of D,
because it has exactly the same embedded straight lines as D.
Ken Brakke's images of a sequence of
higher-order complements of C19(D)
of genus 35, 51, 67, ..., which belong to two families, A and B,
are shown
here.
_in_clearboxes(2)_thumbnail_reduced.jpg)
Fig. E2.4c
C19(D)
Two translation fundamental domains
_in_clearboxes_thumbnail_reduced.jpg)
Fig. E2.4d
C19(D)
One-fourth of a translation fundamental domain,
cut from two different parts of the surface
_[111].jpg)
Fig. E2.4e
What's left in 2011 of the brittle 1970 model of C19(D) shown in Fig. E2.4b
view: [111] direction
_[111]_reversed.jpg)
Fig. E2.4f
C(D)
view: [-1-1-1] direction
_[110].jpg)
Fig. E2.4g
C(D)
view: [110] direction
|
What about the adjoints of Schoenflies IV, V, and VI?
When I first examined these three quadrilaterals in 1968 with Property A (cf. discussion following Fig. E2.1) in mind, I was startled to discover that although
the infinite surfaces derived from Schoenflies IV and from the adjoint of Schoenflies IV are both self-intersecting (they each have a 120º corner and therefore a branch-point),
the adjoints of both Schoenflies V and Schoenflies VI are patches for embedded surfaces!
I chose the name F-RD for the adjoint of V (which has genus 6) and the name I-WP for the adjoint of VI (which has genus 4).
My naming conventions are explained in
Infinite Periodic Minimal Surfaces Without Self-Intersections.
In every known example of an embedded triply periodic minimal surface that contains no straight lines, the two labyrinths are not congruent.
Elke Koch and Werner Fischer
have classified such surfaces as non-balanced and surfaces that contain
straight lines — like P and D — as balanced.
I-WP and F-RD contain no straight lines and are therefore called non-balanced.
They are illustrated in Figs. E2.5 and E2.6.
I-WP
Fig. E2.5a
I-WP (genus 4)
oblique view
Fig. E2.5b
I-WP
Modeled by
Ken Brakke, using his Surface Evolver program
oblique view
Fig. E2.5c
I-WP
view: [111] direction (approximately)
Fig. E2.5d
I-WP
view: [100] direction
Portions of two non-trivial geodesics (cf. discussion below Fig. E1.24) are shown.
The one is at the left is closed, and the triply-periodic one at the bottom is unbounded.
Fig. E2.5e
I-WP
view: [100] direction
Fig. E2.5f
I-WP
view: [100] direction
Fig. E2.5g
I-WP
oblique view
F-RD
Fig. E2.6a
F-RD (genus 6)
view: [111] direction
(See my 1997 F-RD
poster essay)
I am indebted to John Brennan and Robert Fuller,
two of my outstanding students at California Institute of the Arts in 1971,
for volunteering to complete this large model of F-RD.
Unfortunately, in 1983 it — as well as most of my other models — was
destroyed by an unknown marauder.
Fig. E2.6b
F-RD
view: [111] direction
Fig. E2.6c
F-RD
view: [110] direction
Fig. E2.6d
F-RD
view: [100] direction
Fig. E2.6e
F-RD translation fundamental domain
view: [111] direction
Fig. E2.6f
F-RD
One-half of a translation fundamental domain
Fig. E2.6g
F-RD
One translation fundamental domain (stereo view)
Fig. E2.6h
F-RD
One-and-a-half translation fundamental domains
|
MY MEANDERING ODYSSEY
A more or less chronological account
of some of my physics research
before I became interested
in minimal surfaces
I realize that except in books on the history of science or mathematics, it is not customary to describe
the development of mathematical or scientific results in strictly chronological fashion, and
only an unusually dedicated reader will have the stamina required to reach the end of this story.
So much of what I did depended on chance events that it may sometimes seem to the reader like a sort of random
walk. It was more than a decade after my journey began that I first had an inkling of where it might lead. For those readers who make it all the way to the end,
I can only say, "Mazel Tov!"
1953-1957
At the University of Illinois in Champaign/Urbana, where Frederick Seitz was chairman,
research in condensed matter physics was heavily weighted toward the study of point defects in metals, semiconductors, and alkali halides.
For my PhD research in David Lazarus's group, I made radioactive tracer measurements of atomic
diffusion coefficients as a function of temperature and alloy composition
in single crystals of Ag-Cd and Ag-In, using experimental techniques developed by Dave, his post-docs, and the students who preceded me.
Although I enjoyed this work at first, my progress was slow, and as I looked in awe at the accomplishments of some of my classmates,
I gradually began to question whether I was temperamentally suited for a career as an experimental physicist.
Besides, it seemed to me that my thesis topic did not have much scientific significance,
and I saw little prospect of making any fundamental advance in the field of diffusion.
I was fascinated, however, by the mathematics of
random walks on lattices
— a famous example of
Brownian motion.
This fascination eventually led me to my first significant discovery — that by measuring
the isotope effect for self-diffusion in an elemental crystal, one could distinguish between
'substitutional' diffusion and 'interstitial' diffusion. This had not previously been possible.
How this came about is described below.
One day in the spring of 1957, I read a 1952 paper by
John Bardeen
and
Conyers Hering
entitled, "Diffusion in Alloys and the Kirkendall Effect".
Appendix A of that paper is an analysis by Hering of the difference between
the diffusion coefficient of a vacancy (vacant atomic site) and that of an atom.
It had long been widely accepted that the mechanism for self-diffusion in
noble metals, for example, is the exchange of an atom with an isolated vacancy.
The concentration of vacancies was known to relatively dilute, even at the elevated temperatures
required for observing self-diffusion.
As a consequence, after an atom has exchanged positions with a particular vacancy,
it is somewhat more than randomly likely that the next jump of that atom
will be an exchange with the same vacancy. When that happens, the two consecutive jumps
of the atom will have either partially or wholly canceled each other,
and the atom is described by Hering as undergoing a correlated random walk.
An atom in an interstitial position (e.g. a lithium atom in a germanium crystal), on the other hand,
is believed to hop from one interstitial site to another with no correlation between the directions of consecutive jumps.
Its diffusion is characterized as a strictly random walk.
Hering proved that if atoms diffuse by the vacancy mechanism
('substitutional' diffusion) and the vacancies are relatively dilute,
then the diffusion coefficient for an atom is
smaller than the diffusion coefficient for a vacancy by a fractional amount
that depends on the coordination number (number of nearest neighbors of an atom) of the host crystal.
The smaller the coordination number, the larger this fraction.
For the uncorrelated random walk of a vacancy,
which jumps a distance a with frequency Γ, the diffusion coefficient for the vacancy is given by
Dvacancy = a2 Γ / 6. (E2.11)
Hering showed that for the correlated random walk of an atom in a homogeneous crystal
in which the atom-vacancy jump vector has at least two-fold rotational symmetry,
the diffusion coefficient for the atom is given by
Datom = f a2 Γ / 6, (E2.12)
where the correlation factor f is given by
f = (1 + < cos θ >Av ) /
(1 − < cos θ >Av ); (E2.13)
< cos θ>Av is the average value of the cosine of the angle between two consecutive jumps of an atom.
Since the diffusion of an atom in an interstitial position does not involve an exchange with a
vacancy, the directions of its consecutive jumps are uncorrelated.
I had a hunch that Bardeen-Hering correlation would reduce both the self-diffusion
coefficient and the isotope effect for self-diffusion by the same fractional amount.
This turned out to be the case.
Let Dα and Dβ be the self-diffusion coefficients of isotopes α and
β of mass mα and mβ and correlation factors fα and fβ,
respectively. I defined the
isotope effect =
((Dβ / Dα) − 1)
/ ((mα / mβ)1/2 − 1). (E2.14)
In the absence of correlation effects, the isotope effect would be equal to unity. I conjectured that correlation effects would cause it to be equal instead to fβ.
If this conjecture were correct, one could distinguish between interstitial self-diffusion and self-diffusion by the vacancy mechanism
simply by measuring the isotope effect for self-diffusion.
I first estimated the influence of Bardeen-Hering correlation on the isotope effect by using an approximate model of correlation published
by Alan LeClaire and Alan Lidiard in Phil. Mag., 1, 518 (1956).
This rough estimate appeared to confirm my hunch that in crystals with the required symmetry, the isotope effect is equal to the correlation factor.
1958-1964
In 1958, in order to refine my calculation of the influence of correlation on the isotope effect,
I designed a Fortran program for extending it to a higher order of approximation.
In this program, I modeled the infinite crystal by a sequence of
four cubically symmetrical sub-crystals of successively larger volumes, each
centered at the initial site of the diffusing atom.
.jpg)
Fig. E2.14
The four cubically symmetrical sub-crystals in my
vacancy random walk Fortran program
The number of atomic sites in each sub-crystal is shown in the top row.
The vacancy is located at the center of each sub-crystal.
The four sub-crystal boundaries contain
nearest neighbors,
2nd nearest neighbors,
3rd nearest neighbors, and
4th nearest neighbors
of the vacancy,
respectively.
A vacancy, starting from a site adjacent to the diffusing atom, was allowed
to execute an infinite random walk, during which it had a finite probability of escaping through the boundary of the sub-crystal.
Program runs on an IBM 704 computer for each of five successively larger subcrystals
confirmed that correlation does indeed reduce both the isotope effect
and the self-diffusion coefficient by exactly the same fractional amount (up to at least eight significant figures),
in agreement with the calculation I had made earlier using the Lidiard-LeClaire model.
Soon afterward K. Tharmalingam and A. B. Lidiard published an algebraic proof that my results were exact,
in an article entitled 'Isotope Effect in Vacancy Diffusion'
(Phil. Mag., 4, Issue 44, 1959, pp. 899-906).
Junjiro Kanamori
In 1960, I attempted to write the the Bardeen-Hering correlation factor
as a combinatorial expression. After I received some critical help from the theoretical physicist Junjiro Kanamori,
Robert W. Lowen, Jr. and I evaluated the correlation factors for seven structures
— four 2-dimensional and three 3-dimensional.
We reduced the correlation factors for the three 3-dimensional cases to triple elliptic integral expressions and
published our results in the Bulletin of the American Physical Society, April 1960, 4, No. 5, p. 280.
They are listed in Figs. E2.15a,b. I don't recall whether we ever completed our calculations for the face-centered cubic structure,
for which the value of the correlation factor was already known to be approximately 0.78.
Fig. E2.15a
Computed values of Bardeen-Hering correlation factors (APS abstract)
|
STRUCTURE
|
Z
|
− < cos θ >Av
|
f
|
|
linear chain
|
2
|
1
|
0
|
|
honeycomb layer
|
3
|
1/2
|
.333333
|
|
square layer
|
4
|
1 − 2/π
|
.466942
|
|
triangle layer
|
6
|
5/6 − √ 3/π
|
.566057
|
|
diamond
|
4
|
1/3
|
.500000
|
|
simple cubic
|
6
|
.209841
|
.653120
|
|
body-centered cubic
|
8
|
1 − Γ4(1/4)/8π3 − 8π/Γ4(1/4)
|
.727194
|
Fig. E2.15b
Correlation factors for seven crystal structures
After 1959, I tried — with limited success — to invent a systematic duality rule
('partitioning algorithm') for associating infinite periodic graphs in pairs that represent the sets of all possible geometrical pathways for
diffusing atoms in (a) 'vacancy-exchange' diffusion and (b) 'interstitial' diffusion.
It didn't occur to me to imagine any kind of surface separating such infinite graphs until 1964, when I first learned about the
Coxeter-Petrie [infinite] regular skew polyhedra,
(Only in 1966 did I realize that these three infinite polyhedra were nothing more than
'flattened and folded' incarnations of Schwarz's P and D surfaces.
When I met Donald Coxeter for the first time at a 1966 geometry conference in Santa Barbara, he told me
that he had never heard of the Schwarz surfaces.
His face expressed something close to shock when I showed him my plastic models of P and D.
He examined them closely for a couple of minutes or more, before saying a word.)
I cataloged a variety of examples of crystal structures that could be neatly partitioned into two
disjoint substructures, and I computed the shapes of the Voronoi cells for many examples of unary, binary, and ternary crystal structures.
From time to time I made wooden models of many of these polyhedra and used some of them as nodes for ball-and-stick network models
of crystal structures.
An especially useful resource for me in those days was 'Third Dimension in Chemistry', by Alexander F. ('Jumbo') Wells.
It was there that I learned (in 1958) of the existence of the Laves graph
(cf.
John Tanaka's oral history interview of Wells).
A graph is called symmetric if
all of its vertices are symmetrically equivalent and all its edges are symmetrically equivalent.
Another way of saying this is: A symmetric graph is one that is both edge-transitive and vertex-transitive.
A regular graph, on the other hand, is one in which every vertex has the same degree.
Hence every symmetric graph is regular, but not every regular graph is symmetric.
Among all symmetric graphs on cubic lattices known to me — either then or now —
the Laves graph is the unique graph of smallest degree (three). I imagined in fantasy an elemental crystal
whose atomic sites correspond to the vertices of the Laves graph, with self-diffusion occurring by means of atom-vacancy exchanges.
As an additional part of the fantasy, I imagined measuring the isotope effect for self-diffusion in this hypothetical crystal.
I believe that the Bardeen-Hering correlation factor (cf. Eq. E1.3) has never been computed for the Laves graph,
but it is safe to predict that it would have a value of
less than one-half, since the coordination number of the Laves graph is smaller than that of the diamond graph.
(The data in Fig. E2.15b suggest the possibility that it is exactly 1/3. I may get around to calculating it some day, just for fun!)
Consequently the isotope effect, which is predicted to be equal to the correlation factor, would also be less than one-half.
But I realized that the fantasy was far-fetched, because the interstices in such a crystal would be so large that
self-diffusion would be unlikely to occur by a simple vacancy mechanism.
UPDATE:
Fig. E2.16
Toshikazu Sunada
In a 2007 article in the Notices of the American Mathematical Society,
Crystals that Nature Might Miss Creating,
the mathematican
Toshikazu_Sunada —
who was unaware of the history of the Laves graph — independently predicted its existence,
making use of results of his research on random walk on crystal lattices.
In this remarkable analysis, Sunada's 'K4 crystal' (i.e., the Laves graph) emerges as the unique mathematical twin of the diamond crystal.
He proves that diamond and K4 are the only three-dimensional crystals with the property he calls strong isotropy,
and also that the honeycomb (cf. graphene) is the only two-dimensional crystal with this property.
(I confess that I understand only the easy parts of Sunada's article!)
1964 − April 1966
In July 1964, after spending a few stimulating months consulting for a new division of Beckman Instruments
on the design of apparatus for measurements of the Mössbauer effect,
I joined the Physics Research Laboratory of Space Technology Laboratories (STL) in Los Angeles.
Within a few months, STL was acquired by TRW and changed its name to TRW Systems.
I continued — from time to time — to ponder the question of how to develop a 'partitioning algorithm' for interpenetrating pairs of triply-periodic graphs.
April 1966 −April 1967
One day in April, 1966, in a hallway of the TRW Physics Research Laboratory, I noticed an engineer who was
drawing polygons on a large plastic sphere. When I [politely] asked him what he was doing,
he replied with some impatience that he was trying to model a fly's-eye lens by arranging a few hundred hexagons on a sphere but was having some difficulties.
(In 'Ernst Haeckel (1843 − 1919) is still a problem', Eclectica (2009),
Alan Mackay
describes a similar error made — and subsequently corrected — by Ernst Haeckel.)
Trying not to sound patronizing, I suggested to the engineer that he might consider including some pentagons, and I explained why the patterm he was searching for didn't exist.
I told him the famous story about Euler and the bridges of Königsberg, and I explained that Euler had
derived a simple equation that accounts for every possible combination of polygons that tile the sphere.
Because it was obvious that he was somewhat less than pleased by my butting in, I decided not to pursue the matter further. However, I did casually mention the incident to my supervisor.
A few days later, I was invited by the research vice-president of TRW Systems to spend one or two days every month as a kind of informal consultant to
a group of company engineers who were designing a manned space station.
(I knew very little about structural engineering, but in 1965 — four years before the first lunar landing — I had submitted an invention disclosure to TRW
describing a modular building system designed for use on the moon.
It employed hollow columnar space-frames, based on the geometry of space-filling tetragonal disphenoids.
Each column, which was clad in aluminum and stored in a flat collapsed configuration, was designed to be self-deployed after delivery to the moon.
Columns could be filled with lunar sand, so that a shelter constructed from an assembly of columns would provide effective shielding from dangerous radiation.)
In order to catch up on gossip about the current state of the art in modular building systems, I paid a visit to the distinguished architect
Konrad Wachsmann,
chairman of the architecture department at nearby University of Southern California.
Wachsmann in turn referred me to the North Hollywood architect/designer
Peter Pearce, who
was studying polyhedra, crystal structures, periodic three-dimensional networks,
and the design of a modeling kit for both polyhedra and networks.
Peter had received a grant from the Graham Foundation to study natural and man-made periodic structures.
He showed me many ball-and-stck models of crystal structures he had constructed with the help of his assistant, Bob Brooks.
Illustrations of these models appeared twelve years later in Peter's book,
'Structure in Nature
is a Strategy for Design', MIT Press (1978). Peter told me that he had been inspired especially
by R. Buckminster Fuller, Alexander F. Wells,
D'Arcy Thompson,
and
Charles Eames, his former employer.
Two of Peter's models each contained a specimen of what he called saddle polyhedra and made a profound impression on me.
Peter had seen a museum exhibit designed by Charles and Ray Eames in collaboration with the mathematician
Ray Redheffer, in which
a motor-driven quadrangular wire frame emerged repeatedly from a beaker of soap solution with a physical
approximation to a minimal surface spanning its boundary.
Peter recognized that by spanning appropriate circuits of edges in triply-periodic graphs with plastic polygons
that approximated minimal surfaces,
he could fill the interstitial cavities in those graphs with saddle polyhedra.
Peter's concept of saddle polyhedron struck me instantly as
the critical ingredient required to complete the duality rule ('partitioning algorithm') I had been mulling over
in my struggle to develop a systematic relation between substitutional and interstitial sites in crystal structures.
Although I never expected to find a rule applicable to every possible triply-periodic graph,
I did hope to find one that would work at least for every symmetric graph — a graph
which is both edge-transitive and vertex transitive, i.e., a graph in which all vertices are
symmetrically equivalent and all edges are symmetrically equivalent.
As explained in pp. 76-85 of
Infinite Periodic Minimal Surfaces Without Self-Intersections),
however, I discovered that although it is not necessary for the graph to be symmetric,
it is apparently necessary to add the stipulation that for symmetric graphs,
- each vertex of the graph g is joined by an edge to every one of the Z nearest neighbor vertices
(the graph g is described as being of maximum degree with respect to the vertices.
- each vertex lies at the centroid of the positions of the Z nearest neighbor vertices
(the graph g is described as being locally centered).
(The two saddle polyhedra shown below in Figs. E2.50-E2.53 and in Figs. E2.55a,b,c demonstrate that these conditions are too restrictive.)
One (cf. Fig. E2.20) of Peter's two saddle polyhedra filled an interstitial cavity of the diamond graph, a
symmetric graph of degree four on the vertices of the diamond crystal structure, while
the other (cf. Fig. E2.25) filled an interstitial cavity of the body-centered cubic (b.c.c.) graph,
a symmetric graph of degree eight on the vertices of the b.c.c. lattice.
Each of these saddle polyhedra is called the interstitial polyhedron of the graph g
and has the following properties:
- (a) the interstitial polyhedron and the graph g have the same point group symmetry with respect to the center of the cavity;
- (b) the number of faces of the interstitial polyhedron is equal to the degree Z
(number of edges incident at each vertex) of a second [dual] graph ginterstitial, in which there is a vertex v at each
cavity center and Z edges — incident at v — that protrude through the faces of the interstitial polyhedron.
Each of these Z edges is incident also at a vertex v of one of the Z adjacent interstitial polyhedra.
Because the diamond graph happens to be self-dual, if every vertex of the graph is enclosed by a replica of the interstitial polyhedron,
the assembly of such polyhedra — just like the assembly of interstitial polyhedra that occupy the interstitial cavities —
define a packing of R3. In this role, these saddle polyhedra are called nodal polyhedra.
The nodal polyhedron of the b.c.c. graph is shown in Fig. E2.27.
For some infinite symmetric graphs — depending on the proximity of vertices in coordination shells beyond the first —
the number F of faces of the space-filling Voronoi polyhedron that encloses each vertex is greater than Z.
The simplest example of a pair of symmetric graphs that illustrate the duality expressed by properties (a) and (b)
is a pair of simple cubic (s.c.) graphs (cf. Fig. E2.58). My goal was to
incorporate the concept of saddle polyhedron in a procedure that defines this duality in a systematic way.
Among the many triply-periodic graphs that exhibit both properties (a) and (b) defined above are the seven symmetric graphs listed in the table below.
The f.c.c. graph and the FCC6(I) graph are the only examples among these seven for which the dual graphs are not also symmetric.
For the f.c.c. and s.c. graphs, both the nodal and interstitial polyhedra happen to be convex.
|
GRAPH
|
Z
|
F
|
RELEVANT FIGS.
|
|
Laves
|
3
|
17
|
E2.41, E2.42
|
|
WP
|
4
|
12
|
E2.36
|
|
diamond
|
4
|
16
|
E2.19c
|
|
s.c.
|
6
|
6
|
E2.58 − E2.60
|
|
FCC6(I)
|
6
|
12
|
E2.46 − E2.49
|
|
b.c.c.
|
8
|
10
|
E2.24b, E2.36
|
|
f.c.c.
|
12
|
12
|
E2.56d − E2.56f
|
Fig. E2.17
Examples of symmetric graphs on cubic lattices
Z = degree of the graph (coordination number).
F = number of faces of the Voronoi polyhedron
associated with each vertex.
|
dual graph
|
diamond graph
|
|
nodal polyhedron
|
expanded regular tetrahedron ERT
|
|
interstitial polyhedron
|
expanded regular tetrahedron ERT
|
Fig. E2.18
.gif)
Fig. E2.19a
Fig.2.19b
Schwarz's D surface
(images courtesy of Ken Brakke)
The diamond graph is the skeletal graph of both labyrinths.
.jpg)
Fig. E2.19c
A portion of the diamond graph (Z=4)
The edges of the expanded regular tetrahedron ERT (cf. Fig. E2.20),
interstitial polyhedron of the diamond graph,
are shown in blue.
Fig. E2.20
The expanded regular tetrahedron ERT,
interstitial polyhedron of the diamond graph
It is also the nodal polyhedron of the diamond graph.
(The diamond graph is self-dual.)
ERT is the saddle polyhedron Peter Pearce constructed in
an interstitial cavity of the diamond graph (cf. Fig. E2.19).
Each face is a regular skew hexagon
with face angle θ = cos-1(− 1/3) =~109.47°.
.JPG)
Fig. E2.21
The regular tetrahedron and the edges (black lines) of ERT
Fig. E2.22
The 16-face Voronoi cell for the vertices of the diamond graph
For a sharper [pdf] version of this image, look here.
|
dual graph
|
WP graph
|
|
nodal polyhedron
|
expanded regular octahedron ERO
|
|
interstitial polyhedron
|
tetragonal tetrahedron TT
|
Fig. E2.23
Fig.2.24a
A cubic unit cell of I-WP
(image courtesy of Ken Brakke)
The skeletal graphs are
the b.c.c. graph − aka the I graph −and the WP graph.
Fig. E2.24b
The b.c.c. graph (Z=8)
The edges of the tetragonal tetrahedron TT (cf. Fig. E2.25)
are shown in blue.
Fig. E2.24c
The b.c.c. graph (green vertices) and its dual,
the WP graph (orange vertices)
.jpg)
Fig. E2.25
The tetragonal tetrahedron TT,
interstitial polyhedron of the b.c.c. graph
TT is the saddle polyhedron Peter Pearce constructed in
an interstitial cavity of the b.c.c. graph (cf. Fig. E2.24).
Each face is a regular skew quadrangle
with face angle θ = cos-1(1/3) =~70.13°.
.jpg)
Fig. E2.26
The eight edges (purple) of TT
For a sharper [pdf] version of this image, look here.
Fig. E2.27
The expanded regular octahedron ERO,
nodal polyhedron of the b.c.c. graph
Fig. E2.28
It appears that ERO is identical to the asymptotic limit surface suggested by this image from
Ken Brakke's Surface Evolver sequence of surfaces
of successively higher genus in the Neovius C(P) family.
(I first observed this curiosity in 1975 and described it
in response to some questions from the physicist Tullio Regge
about minimal surface soap film experiments.
I had met Tullio a few weeks earlier at a Providence conference hosted by Tom Banchoff.
Tullio, it turned out, was an expert in differential geometry,
having studied the work of the pioneering 19th century
Italian masters (Bianchi, etc.) of the subject
when he was a student.)
Fig. E2.29
The twenty-four edges (black) of ERO
For a sharper [pdf] version of this image, look here.
Fig. E2.30
The 14-face Voronoi cell for the vertices of the b.c.c graph
For a sharper [pdf] version of this image, look here.
Fig. E2.31
Fig. E2.32
Fig. E2.33
Fig. E2.34
|
dual graph
|
b.c.c. graph − aka the I graph
|
|
nodal polyhedron
|
tetragonal tetrahedron TT
|
|
interstitial polyhedron
|
expanded regular octahedron ERO
|
Fig. E2.35
Fig. E2.36
The WP graph (Z=4)
The edges of ERO (cf. Figs. E2.27, E2.37) ,
interstitial polyhedron of this graph, are shown in blue.
Fig. E2.37
The expanded regular octahedron ERO,
interstitial polyhedron of the WP graph
In 1966, while assembling the faces of this model,
I discovered that if adjacent hexagons are related
by rotation instead of reflection,
the result is an infinite smooth surface — Schwarz's D surface.
(That was my introduction to triply-periodic minimal surfaces.
A few minutes later, I replaced the 90° hexagons by 60° hexagons
and obtained Schwarz's P surface.)
_reduced.jpg)
Fig. E2.38
The tetragonal tetrahedron TT,
nodal polyhedron of the WP graph (cf. Fig. E2.36)
Fig. E2.39
The expanded octahedron EO,
the Voronoi cell for the vertices of WP
For a sharper [pdf] version of this image, look here.
|
dual graph
|
enantiomorphic Laves graph
|
|
nodal polyhedron
|
trigonal trihedron T
|
|
interstitial polyhedron
|
trigonal trihedron T' (enantiomorph of T)
|
Fig. E2.40
Fig. E2.41
A portion of the Laves graph (Z=3)
The edges of the trigonal trihedron TT ,
interstitial polyhedron of the Laves graph,
are shown in blue
(cf. model in Figs. E2.43 and E2.44).
Fig. E2.42
A view of the edges of TT
from a direction different from that in Fig. E2.41
_reduced.jpg)
_reduced.jpg)
Fig. E2.43
The trigonal trihedron TT,
nodal polyhedron for the Laves graph (cf. Figs. E2.41 and E2.42)
is a skew decagon with 120° face angles.
_reduced.jpg)
_reduced.jpg)
Fig. E2.44
Another view of TT
|
dual graph
|
a non-symmetric graph of degree 10
|
|
nodal polyhedron
|
pinwheel polyhedron PP
|
|
interstitial polyhedron
|
doubly expanded tetrahedron DET
|
Fig. E2.45
Fig. E2.46
Four vertices' worth of the FCC6(I) graph (Z=6),
a locally-centered defective symmetric graph (LCDSG)
The vertices of the FCC6(I) graph are those of the f.c.c. lattice.
At each vertex, six of the twelve edges
of the standard f.c.c. graph are omitted.
FCC6(I) is described on pp. 47-48 of
Infinite Periodic Minimal Surfaces Without Self-Intersections.
Fig. E2.47
Four vertices' worth of the FCC6(I) graph
inscribed on the surface of the polyhedron VPdiamond
(Voronoi polyhedron for the diamond crystal structure)
Fig. E2.48
Five VPdiamonds' worth of the FCC6(I) graph
Fig. E2.49
Five VPdiamonds' worth of the FCC6(I) graph
Fig. E2.50
Doubly expanded tetrahedron ('DET'),
the interstitial polyhedron of the FCC6(I) graph
Six faces are regular skew quadrangles,
and four faces are regular skew hexagons.
_reduced.jpg)
_reduced.jpg)
Fig. E2.51
Two views of DET
(stereo)
Fig. E2.52
The Doubly Expanded Tetrahedron is so named because
its twenty-four edges are produced by reflecting the edges of each face
of a regular tetrahedron in each of the three edges of the face.
(Every edge of the regular tetrahedron is reflected twice,
once for each of its two incident faces.)
Fig. E2.53
When an infinite set of DETs is assembled by gluing hexagonal faces together in pairs,
the quadrangular faces remain exposed and define Schwarz's D surface.
Fig. E2.54
When an infinite set of DETs is assembled by gluing quadrangular faces together in pairs,
the hexagonal faces remain exposed and define Schwarz's P surface.
Fig. E2.55a
Pinwheel polyhedron PP,
the nodal polyhedron of the FCC6(I) graph
Fig. E2.55b
Pinwheel polyhedron PP,
rendered in Mathematica
Fig. E2.55c
The red edges are the edges of
the nodal polyhedron PP of the FCC6(I) graph.
It has the same volume as the
Voronoi polyhedron (rhombic dodecahedron).
Fig. E2.55d
Partial packings of PP, rendered in Mathematica
(The quadrangular surface patch module is rendered here as a
doubly-ruled surface.
Although it resembles the actual minimal surface, it is only an approximation.)
Below are four views of a portion of the compound [self-intersecting] surface
composed of replicas of just one of the two enantiomorphous quadrangular surface
patches that make up the pinwheel polyhedron PP (above). Three patches are incident at
edges of type [111], but along edges of type [100], two patches are smoothly related by a half turn.
I learned of the concept of compound surfaces of this type from the mathematician
Dennis Johnson, who explained that intersections of three patches along directions
of type [111] are analogous to the 120º intersections in froths of soap bubbles.
Fig. E2.55e
Viewed in [100] direction
Click
here for higher resolution image
Fig. E2.55f
Viewed in [110] direction
Click
here for higher resolution image
Fig. E2.55g
Viewed in [111] direction
Click
here for higher resolution image
Fig. E2.55h
Orthogonal projection on [111] plane
Click
here for higher resolution image
|
dual graph
|
fluorite graph
|
|
nodal polyhedron
|
rhombic dodecahedron
|
|
interstitial polyhedra
|
regular tetrahedron, octahedron
|
Fig. E2.56a
Fig. E2.56b
Fig. E2.56c
Two views of F-RD
Image at left courtesy of Ken Brakke
Fig. E2.56d
The nodal polyhedron of the FCC graph
is the Voronoi polyhedron (rhombic dodecahedron).
Fig. E2.57
The
fluorite
graph is the dual of the FCC graph.
Its nodal polyhedra are
the regular tetrahedron and
the regular octahedron.
Fig. E2.58
The FCC graph and the fluorite graph
|
dual graph
|
s.c. graph
|
|
nodal polyhedron
|
cube
|
|
interstitial polyhedron
|
cube
|
Fig. E2.59
Fig. E2.60
The s.c. graph,
skeletal graph of one labyrinth of Schwarz's P surface
Fig. E2.61
The s.c. graph, which is also the
skeletal graph of the other labyrinth of Schwarz's P surface
Fig. E2.62
The congruent skeletal graphs of the two disjoint labyrinths of Schwarz's P surface
(stereo pair)
|
A few days after I met Peter Pearce, I observed with astonishment that
for certain shapes of saddle polygons spanned by a minimal surface, e.g.,
the 90° regular skew hexagon (a module for Schwarz's D surface)
or the 60° regular skew hexagon (a module for Schwarz's P surface),
if two specimens of the saddle polygon are related
by a half-turn about a common edge, instead of by mirror reflection in a plane containing that edge
(which is the arrangement in most, although not all, of the saddle polyhedra I had explored by then),
not only does the junction between the two polygons appear to be perfectly smooth,
but an endless sequence of these half-turns produces
a single smooth, embedded infinite labyrinthine surface with the global topology and symmetry of a
Coxeter-Petrie regular skew polyhedron!
I had accidentally stumbled onto two examples of the application of Schwarz's
reflection principle, his
P and D surfaces − two objects that I had never heard of.
I was unable to locate a reference to either of these surfaces in my books on geometry or differential geometry.
Because I realized that some contemporary mathematicians must be familiar with these two surfaces, I paid
a visit to the UCLA math department, where I showed my plastic models to the two faculty specialists
in differential geometry. But neither of them recognized the two surfaces!
Next: a visit to the UCLA science library, where I learned that the late
Johannes C. C. Nitsche,
a prolific mathematician on the faculty of the University of Minnesota,
was a noted authority on minimal surfaces (cf. his summary of the field,
"A Course in Minimal Surfaces").
When I telephoned him and described what I had been doing,
he kindly explained that I had probably made models of the two TPMS for
which H. A. Schwarz (and also Riemann and Weierstrass, as I was to learn later) had developed solutions in 1866.
He referred me to Vol. 1 of Schwarz's Collected Works.
From a quick perusal of this tome,
I learned that Schwarz had also discovered two other examples of TPMS —
H and CLP, both also of genus 3 —
and I made plastic models of them too.
I soon noticed that on p. 271 of
Hilbert and
Cohn-Vossen's
'Geometry and the Imagination';
the authors write that
In this way, Neovius4 succeeded in constructing a minimal surface that extends over the entire space
without singularity or self-intersection and has the same symmetry as the diamond lattice (italics added).
4E. R. Neovus, Bestimmung zweier speziellen periodischen Minimälflachen, Akad. Abhandlung, Helsingfors, 1883
|
|
I foolishly assumed that the authors had simply become confused here and were actually referring to Schwarz's D surface.
Eighteen months later, I learned that Neovius had treated an entirely different surface of genus 9.
I invented a naive scheme for identifying and labeling these surfaces, each of which I regarded as lying
between the two triply-periodic graphs of a dual pair.
I named these pairs of graphs 'skeletal graphs', because I thought of them as the skeletons of their respective hollow labyrinths.
I found it helpful to regard the skeletal graph edges as thin hollow tubes that could be enlarged by inflating them
until the whole graph was transformed into the TPMS. Then if the tubes were overinflated, the graph would eventually shrink down
into the dual graph! I imagined that for at least a portion of this inflation cycle, the surface of the graph would
define a triply-periodic surface of non-zero constant mean curvature.
As I began my admittedly superficial study of the mathematical underpinnings of these surfaces, beginning with the two
Schwarz reflection principles,
I couldn't help wondering what other examples of embedded 'TPMS' might exist.
In particular, I wondered whether there was a TPMS whose skeletal graphs were the enantiomorphous pair of Laves graphs I had learned about
seven years earlier in
'Third Dimension in Chemistry',
by Alexander F. Wells.
A pair of enantiomorphic Laves graphs that are related by inversion has b.c.c. translation symmetry.
It struck me as curious that of the three different cubic lattice symmetries —
simple cubic (s.c.), face-centered cubic (f.c.c.), and body-centered cubic (b.c.c.) —
b.c.c. was missing from the inventory of cubic lattice symmetries for known examples of TPMS of ultimately simple topology (genus three).
A second reason for my focus on the Laves graph was that it was apparently the only other example of a
triply-periodic graph — besides the simple cubic and diamond graphs,
which are the skeletal graphs of P and D, respectively — in which congruent regular polygons
are incident at each edge.
In the Laves graph, there are precisely two regular polygons incident at each edge. They happen to be infinite helical polygons,
centered on lines parallel to two of the three coordinate axes.
(I had been strongly influenced by Coxeter's 'Regular Polytopes', and I believed one should take regularity very seriously!)
The Laves graph is not a reflexive regular polyhedron, however, and its lack
of reflection symmetries made it impossible for me to imagine just how it could serve as the skeletal graph
of the labyrinth of an embedded TPMS.
But the most compelling reason for my conviction that there must exist an embedded
TPMS whose skeletal graphs are enantiomorphic Laves graphs was that
the simple cubic graph, the diamond graph, and the Laves graph were the only
examples I could identify of symmetric triply-periodic graphs of cubic symmetry that are self-dual.
Even though I knew of no theoretical justification for claiming that such graphs — regarded as skeletal graphs
of embedded TPMS — play a unique role in defining embedded TPMS of cubic symmetry,
I nevertheless believed that they must play such a role!
I was aware of the fact that the concept of skeletal graph was itself somewhat ill-defined.
It seemed to me to be a very 'natural'
construct when applied to the then known examples of embedded TPMS, but I had no idea how to prove
that for every possible example of an embedded TPMS
there is a unique pair of skeletal graphs.
All of these considerations at times seemed to me to smack more of theology than of mathematics. (I am reminded that
the young Riemann, who was probably the first to solve the equations for what we now call Schwarz's P and D surfaces,
as a young man abandoned the study of theology (his pastor father's choice) for a career in mathematics!
I resolved to learn more about TPMS, which I recognized as far more interesting objects than saddle polyhedra,
but in the meantime I was determined to continue exploring the relation between
triply-periodic graphs and saddle polyhedra.
On evenings and weekends throughout the spring and summer of 1966, I used a toy
vacuum-forming machine and home-made moulds cast from polyester resin
poured against a thin stretched rubber membrane to make dozens of saddle polyhedra
of different shapes, all of which I shared with Peter Pearce.
He preferred to make his saddle polygons by draw-forming— pushing a tool
in the shape of a skew polygon outline against a transparent vinyl sheet
that had been softened by heating. I preferred vacuum-forming with solid moulds, but it was clear that Peter's method also worked well.
It has the advantage of not requiring the extra labor involved in making a mould, but the disadvantage is that it cannot replicate the shape
of a minimal surface as well as a carefully crafted mould can.
During these months of experimenting, I found no counterexample to my improvised
duality rule, even for triply-periodic graphs that are not symmetric.
In May, I hit on the idea of what I rather lamely called a 'defective' symmetric graph
(I decided later that 'deficient' might be a more appropriate name) — a symmetric graph
A derived from a second symmetric graph B
by omitting some of the edges but none of the vertices of B.
In A, not every pair of nearest neighbor vertices is joined by an edge.
I required that every defective symmetric graph be locally-centered, i.e., that every vertex
lie at the centroid of the vertices with which it shares an edge.
For the simple cubic lattice, it's easy to prove — simply by enumerating
each of the possible locally-centered subsets of edges that contains at least three edges — that it is
impossible to construct a locally-centered defective symmetric graph (LCDSG) on the vertices.
I have no idea why I failed to ask myself in those days whether
there exists a LCDSG on the vertices of the body-centered cubic lattice.
The only example of a LCDSG that I examined in 1966 was a graph of degree six I call FCC6(I)
(cf. Fgs. E2.45 - E2.55b).
Its vertices are those of the face-centered cubic lattice.
I used the name FCC for the familiar symmetric graph of degree twelve on the vertices of the f.c.c. lattice,
in which every pair of nearest neighbor vertices is joined by an edge.
I derived the defective graph FCC6(I) by removing a symmetrical set of six out-of-plane edges from each vertex,
leaving behind a flat six-edge cluster that occurs in each of the four possible [111] orientations. Although FCC6(I)
proved not to be a counterexample to the duality rule,
I was not confident that the rule would always hold even if I were to restrict it to symmetric graphs only.
As it happened, not only did the rule not fail in the case of FCC6(I)
— it yielded an unexpectedly interesting pair of saddle polyhedra. I call the interstitial polyhedron —
shown in Fig. E2.51 — the Doubly Expanded Tetrahedron ('DET'), and the corresponding nodal polyhedron — shown in Fig. E2.55a
— the Pinwheel Polyhedron. The Doubly Expanded Tetrahedron is the first example I had encountered of a space-filling saddle polyhedron in which the
faces are of two kinds — hexagons and quadrangles. As illustrated in Figs. E2.53 and E2.54, Schwarz's P and D surfaces
can be formed from either the quadrangular or the hexagonal faces of an infinite 'porous packing' of DETs,
according to whether neighboring DETs share quadrangular faces (P) or hexagonal faces (D).
April 1967 − July 1970
In the spring of 1967, the physicist
Lester C. Van Atta,
who was Associate Director of the NASA Electronics Research Center ('ERC') in Cambridge,
Massachusetts, came to Los Angeles for a few days to visit his physicist son Bill, a friend and colleague of mine who happened to be a whiz
at solving combinatorial puzzles.
A year earlier, stimulated by a
Martin Gardner
column in Scientific American that described Piet Hein's SOMA puzzle,
I had become hooked on investigating the possible symmetries of complementary half-cube
packings by the eight solid tetrominoes (cf. Fig. E1.13).
After Bill lent his father a set of these puzzle pieces,
his father told Bill that he 'wanted to meet the guy who had cost [him] a night's sleep'.
At the end of a very long late evening visit to my home,
Lester abruptly invited me to join the NASA/ERC research staff in Cambridge, Massachusetts.
He explained that I would be required only to 'follow my nose'.
I found it difficult to believe that he was making me a serious job offer, and I didn't make any response.
A few days later, Van Atta phoned me from Cambridge and told me in no uncertain terms that I had only three or four days left to get
the paper work (a few dozen pages of federal employment application forms) in the mail,
because it would be impossible to keep the position open longer than that.
This time I took him seriously, and in July 1967 I moved to Massachusetts.
I wanted to immerse myself immediately in the study of TPMS.
However, I believed that I should first make a better organized attack on my embryonic duality rule ('partitioning algorithm').
I knew it was unlikely that the rule would be applicable to every possible triply-periodic graph,
but I had no idea how to characterize those graphs for which it worked and those for which it didn't.
Inspired by Polya's rules for problem solving,
I continued to emphasize symmetric graphs — those graphs for which there is a symmetry group
transitive on both vertices and edges.
I wondered whether my duality rule worked for every possible symmetric graph.
If I could find a counterexample, I wanted it to be as simple as possible.
I had been hired at NASA/ERC as a mathematician
(chief of a special section Van Atta created for me, called the 'Office of Geometrical Applications'),
even though I was at best an amateur mathematican.
Curiously it was my forays into recreational mathematics that had led Van Atta to hire me.
I was never told by him or anyone else what I should work on,
but of course it was understood that if I saw possible applications of
what I was doing that might be of interest to NASA, I should not fail to pursue those applications.
From the start, I undertook to learn more about the mathematics used in the study of minimal surfaces.
I pored over three books on differential geometry and Schwarz's Collected Works.
I was especially curious to know whether there were additional examples of TPMS just waiting to be discovered,
but when I began to read the published literature in the field (some of it in German, of course), I often felt overwhelmed. I believed that
I didn't have time to become sufficiently knowledgeable about the deep foundations of
the relevant branches of mathematics — differential geometry and complex analysis — to make any
'breakthrough' advances in the field.
I had developed a special interest since 1958 in the properties of infinite periodic graphs, and this interest
had eventually led me to papers and books by Donald Coxeter. My interest in these graphs had sprung from
my research on atomic diffusion in crystalline solids and from the mathematics of
correlated random walks on discrete lattices. Because I had discovered (in 1957) that the magnitude of the isotope effect for atomic diffusion
in crystals could distinguish between interstitial diffusion and substitutional diffusion, I was curious to learn which
elements and compounds were likely to be good candidates for measuring the isotope effect for diffusion.
I tried — with only slight success, initially — to develop an algorithm ('duality rule')
for deriving the infinite periodic graph whose vertices correspond to the principal interstitial sites of a crystalline solid.
WARNING: Continue at your own risk.
Most of the following text will be integrated into earlier discussions.
I was able to confirm my hunch that the duality rule was valid
for many pairs of triply periodic graphs — one of the pair being called substitutional and the other interstitial.
In every case I tested, both nodal and interstitial polyhedra exhibited the following properties:
(a) the number of faces of the polyhedron is equal to the number of edges of the associated periodic graph;
(b) the symmetry of the polyhedron is identical to the symmetry of the associated vertex of the periodic graph.
It didn't matter which graph was called substitutional and which was called interstitial. The dual relation between them is, after all, symmetrical.
(Of course from a physical point of view, it would be absurd to call the large interstitial cavities in silicon — which can be occupied by smaller atoms like lithium,
for example — substitutional and the silicon atomic sites interstitial!)
I call the saddle polyhedron in Fig. E2.25 the interstitial polyhedron for the b.c.c. graph of degree 8,
because it is bounded by edges [of a periodic graph] that are the bars of a sort of interstitial cage.
This same polyhedron is the nodal polyhedron for another triply periodic graph, which I named WP.
The nodal polyhedron encloses a vertex of the periodic graph at its center, and the number of faces of the nodal polyhedron is equal to the number of edges
incident at that vertex.
The saddle polyhedron in Fig. E2.20 is both nodal polyhedron and interstitial polyhedron for the diamond graph.
That means that in a space-filling array of these saddle polyhedra, each polyhedron can either (a) enclose at its center a vertex of the diamond graph
or (b) occupy a single interstitial region bounded by the edges and vertices of the diamond graph.
For some periodic graphs, either the nodal polyhedron or interstitial polyhedron (or both) may turn out to be a convex polyhedron with plane faces.
For the simple cubic graph of degree six, for example, both polyhedra are cubes.
In the simplest cases, the graph is unary, i.e., all the vertices of the graph are equivalent.
But the duality rule works smoothly without requiring any ad hoc adjustments even for many non-unary graphs.
(I plan eventually to post a picture or two of the polyhedra for such a graph.)
As explained below, in early 1968 I searched for — and eventually found — a periodic graph for which the duality rule failed,
and that failure led to the discovery of the pseudo-gyroid, which is composed of hexagons with perfectly helical boundary curves.
Recall that FCC6(I) is a locally-centered defective graph of degree six
on the vertices of the f.c.c.lattice (cf. Fig. E2.48).
In February, 1968, I realized that I had never searched for the obvious b.c.c. counterpart to FCC6(I):
a locally-centered defective graph on the vertices of the b.c.c. lattice.
(It's easy to prove that no locally-centered
defective graph on the vertices of the s.c. lattice exists.)
Once I started looking, it didn't take me long to discover BCC6, a symmetric graph of degree six on the vertices
of the b.c.c. lattice.
A portion of this graph is shown in Fig. E1.20d.
BCC6 proved to be the long-sought counterexample to my 'duality rule'.
Its edges are those of the infinite regular warped polyhedron ('IRWP') that
I call M4 (cf. Fig. E1.16a and E1.20a to E1.20f).
In the case of BCC6, the breakdown in the duality rule occurs at the very first step — the construction of the interstitial polyhedron.
Instead of the finite interstitial polyhedron the duality rule was intended to generate,
an infinite one — M4 — appeared.
I shed no tears over this failure of the duality rule, because the two-labyrinth character of M4 suggested that something of
potentially greater interest might be in the offing: an example of a previously unknown TPMS.
I observed that the skeletal graphs of M4 were enantiomorphic Laves graphs.
This was potentially exciting, because it suggested that M4
might somehow be transformed into the minimal surface (the gyroid) whose
existence I had speculated about almost two years earlier.
Appendix II of
Infinite Periodic Minimal Surfaces Without Self-Intersections
explains why I constructed M6 (in February 1968).
M6 was the result of an attempt to improve on its predecessor, M4
(cf. Figs. E1.16a and E1.16b), which is composed of skew quadrangles spanned by minimal surfaces.
I was looking for a way to 'smooth out the wrinkles' in M4.
The hexagons in M6 are the duals of the quadrangles in M4.
In M4, the dihedral angle between adjacent faces is 60°.
In M6, it is only ~44.4°.
I hoped that this small reduction would enable
M6 to look at least a little more like a continuous minimal surface than M4 did.
As soon as I had constructed the model of M6 shown in Fig. E1.16b, I noticed that its edges define
regular helical polygons with straight edges, centered on lines parallel to the rectangular coordinate axes
(cf. Figs. E1.19b, c, e).
I speculated that if I replaced the straight edges of M6 by helical ones, the ~44.4° dihedral angle
between adjacent faces might shrink to nearly zero.
This was a wild and woolly guess,
with no theoretical justification whatsoever, but the new physical model I constructed (cf. Fig. E1.17)
was encouraging.
The shapes of these edges in the true gyroid just happen to differ so slightly from circular helices that it is virtually impossible to detect the error by eye.
The glaringly obvious hint that I had missed from the outset
lay in the identical combinatorial structure of each of the three regular tessellations of the three surfaces — P, D,
and the pseudo-gyroid surface in Fig. E1.17.
All three of these surfaces can be constructed of surface patches that correspond to faces of any of the three Coxeter-Petrie
infinite regular skew polyhedra,
with tangent planes identically oriented at corresponding vertices.
Since I was already familiar with the details of how curves transform and how the tangent plane remains invariant at each point in the Bonnet
bending of the catenoid into the helicoid,
it should have occurred to me (but didn't!) that the almost perfectly circular lines of curvature in the coordinate planes of Schwarz's P surface
would be transformed by Bonnet bending into almost perfectly helical lines of curvature before they finally became the
linear asymptotics in Schwarz's D surface parallel to the coordinate axes.
For the previous several months, I had begun to feel pressure to do something 'useful'.
Even though Dr. Van Atta himself never once hinted that he was less than satisfied about how I chose to spend my time,
there were growing signs that I could not afford to ignore indefinitely NASA's expectations that my work suggest
at least the possibility of some 'practical' offshoots.
Since I had no idea how to obtain an analytic solution for a minimal surface patch bounded by six helical arcs,
I decided to give up trying to prove that the pseudo-gyroid is a minimal surface.
Instead I sent a physical model of the pseudo-gyroid to Bob Osserman, who passed it along to his PhD student Blaine Lawson
at Stanford.
Blaine agreed to think about the problem, but he warned me that his dissertation would be keeping him extremely busy.
I delved more deeply into the analysis of the 'continuous transformation on vertices and edges' mentioned in the abstract of Fig. E2.10.
I attempted to identify every possible example of non-self-intersecting 'infinite regular warped polyhedra' (and 'infinite
quasi-regular warped polyhedra'), whose faces are regular skew polygons.
(cf. Figs. E1.35b and E1.35c.).
At the same time, I analyzed the geometry of what I called the 'graph collapse' transformation,
which is diagrammed for the 2-dimensional square graph in Fig. E2.68c.
To the exclusion of almost everything else, I concentrated for several weeks on the engineering requirements for the application of this
transformation to the design of expandable space frames.
Finally I wrote a patent application with the help of two NASA patent attorneys
who for two weeks flew up to Cambridge every morning from Washington.
Here is a
synopsis of the expandable space-frame patent,
which was issued in 1975, and here is
the complete text of the NASA patent, from which an illustration is shown in Fig. E2.68a.
I estimated that with realistically designed struts, a value of 80:1 was feasible
for the ratio of the expanded to collapsed volume of the space frame.
With Charles Strauss, Bob Davis, and Ken Paciulan as collaborators, I made an animated film of the collapse of the Laves graph
and of three other regular graphs.
Time-lapse frames showing the geometry of the underlying transformation applied to the Laves graph are illustrated on pp. 86-88 of
Infinite Periodic Minimal Surfaces Without Self-Intersections.
In the fully collapsed state, the vertices and edges
of the [infinite] Laves graph are mapped onto the four vertices and six edges, respectively, of a single regular tetrahedron
(cf. the tetrahedron AOBC in Fig. E2.68b.2).
If one vertex (vertex O in Figs. E2.68b.0, E2.68b.1, and E2.68b.2) is fixed, the collapse trajectories
of all the other vertices are ellipses centered on that vertex.
For every vertex V, the major radius of the ellipse is equal to the initial distance of V from O.
The minor radius of the ellipse is equal to the edge length of the graph if V is related to vertex a, b, or c by a translation
that is a symmetry of the Laves graph — i.e., if V is red, green, or blue.
The minor radius is equal to zero if V is related to vertex O by a translation
that is a symmetry of the Laves graph — i.e., if V is yellow.
Collapse onto tetrahedron AOBC occurs twice in each period of the transformation: at the two moments when either one-quarter
or three-quarters of each elliptical trajectory has been traversed.
Each of the vertices a, b, c rotates on a circular trajectory in one of the three orthogonal coordinate planes.
Because of the screw isometries of the Laves graph, edges collide only at the two instants of collapse in each period.
In an actual physical space-frame, however, struts are of finite thickness, and this causes
edge collisions to occur well before collapse.
(Precisely how early the collisions occur in each period depends on the thickness of the struts.)
The analogous transformations
for those regular graphs derived from Coxetrie-Petrie maps that contain reflection isometries are not physically realizable,
because edges collide early in the transformation even though they are of zero thickness.
Fig. E2.68a
A hinged joint in the expandable space frame
The collapse of the Laves graph is readily depicted by regarding the graph as initially embedded in the D surface
(cf. Fig. E2.69b.0)
and then allowing every vertex to be translated along a linear trajectory in a direction normal to the surface.
Vertices related by a translational symmetry of the graph are colored the same.
If the two sides of the D surface are labeled A and B,
with motion in the direction from A to B defined as positive and motion
in the direction from B to A defined as negative, then the two
vertices incident on each edge move along normals of opposite sense.
The computed positions of the vertices at each stage of the collapse are scaled by the requirement that edge lengths remain invariant,
thereby causing the graph to shrink continuously, with all of its edges finally collapsing
onto the six edges of a single regular tetrahedron — the tetrahedron with vertices O, A, B, C in Fig. E2.68b.2.
Fig. E2.68b.0
The Laves graph embedded in the D surface
The arrows indicate the initial directions of the curvilinear displacements of the vertices.
Green diplacements are called positive, and
red diplacements are called negative.
Fig. E2.68b.1
The yellow vertex at O is now fixed.
The arrows here indicate the initial directions of the curvilinear displacements
in a coordinate system in which the yellow vertex O is at the origin.
Fig. E2.68b.2
The circular trajectories of vertices a, b, and c
and the elliptical trajectory of vertex d, in Fig. E2.68b.1
One-fourth of a complete trajectory period is shown here for these four vertices.
Perhaps the easiest way to illustrate the collapse is to depict
the circular trajectories, in orthogonal coordinate planes,
of the three vertices that are nearest neighbors of any vertex V
if V is regarded as fixed..
The next four images illustrate these trajectories for
V = a yellow, red, green, or blue vertex.
Fig. E2.68b.3
Rotation of R, G, and B vertices around Y vertex
stereo view
Fig. E2.68b.4
Rotation of G, B, and Y vertices around R vertex
stereo view
Fig. E2.68b.5
Rotation of B, Y, and R vertices around G vertex
stereo view
Fig. E2.68b.6
Rotation of Y, R, and G vertices around B vertex
stereo view
The stereo images in Fig. E2.68c illustrate the application of the graph collapse transformation to the square graph.
In this 2-dimensional example,
all the edges of the graph coalesce into a single vertical edge. The images below are parametrized by the value of θ,
the angle of rotation of each edge of the graph out of the horizontal plane.
The vertex at A is regarded as fixed. Collapse onto a single vertical edge occurs at θ=90° and θ=270°.
Fig. E2.68c.0
Square grid graph before the start
of the collapse transformation
θ=0°
Fig. E2.68c.1
θ=18°
Fig. E2.68c.2
θ=52.2°
Fig. E2.68c.3
θ=81°
Fig. E2.68c.4
θ=90°
Fig. E2.68c.5
θ=279°
Fig. E2.68c.6
θ=307.8°
Fig. E2.68c.7
θ=345°
Fig. E2.68c.8
θ=360°
Fig. E2.68c.9
Below are six stereo views of the 20-faced Voronoi polyhedron that encloses
each vertex of a partially collapsed Laves graph. The value of the expression
[i,j,k], just below each figure caption, is proportional to the coordinates of the
station point with respect to the origin at the center of the polyhedron.
The first two images show the polyhedron from opposite directions.
In the last four images, successive station points are separated
by a quarter-turn about a vertical line through the origin.
The collapse is near 'stage 4', shown in Figure II-1d on p, 88 of
Infinite Periodic Minimal Surfaces Without Self-Intersections.
Fig. E2.68d.1
[1,1,-1]
Fig. E2.68d.2
[-1,-1,1]
Fig. E2.68d.3
[0,1,0]
Fig. E2.68d.4
[1,0,0]
Fig. E2.68d.5
[0,-1,0]
Fig. E2.68d.6
[-1,0,0]
Immediately below is an oblique stereo image of
the Voronoi cell {chamfered 'cube') for a vertex
of a certain symmetric graph of cubic symmetry
at an early stage of the graph collapse.
(Can you guess which graph it is?)
Fig. E2.69
A space-filling polyhedron with twenty faces.
Its twelve 'cube edges' are partially chamfered,
and a pair of opposite 'cube corners' are trncated.
Some additional history
One day in early spring 1969, it occurred to me that there must be examples of moderately low-genus embedded TPMS that are hybrids
of topologically simpler surfaces. 'Hybrid' here means the curved-edge adjoint of a linear combination of two straight-edged polygons
that are adjoints of known embedded surfaces.
(A linear combination of two polygons P1 and P2 is
equivalent to a polygon that is interpolated between P1 and P2.
The idea of constructing a linear combination of two polygons
occurred to me after I recalled a description of linear combinations of convex polyhedra
that I had once read in a book on polyhedra. I no longer recall the name of the author of the book,
but it may have been Aleksandr Aleksandroff. I have been unable to trace it.)
Fig. E2.71 shows the page of notes from 1969 that describes a scheme for constructing an example of a hybrid surface.
_hybrid_reduced.jpg)
Fig. E2.71
1969 notes on a possible embedded hybrid of P and C(P) (genus 14)
(The scribbled captions should say "Schwarz's P", not "Schwarz's D".)
The relative weights assigned to the surfaces P and C(P) need to be
adjusted by trial and error to make arcs 23 and 15 coplanar.
Cognoscenti call this process 'killing periods'.
The parents of the hypothetical surface suggested in Fig. E2.71 are Schwarz's P and Neovius's C(P).
After I sketched those 1969 notes, I phoned Blaine Lawson, an expert on minimal surfaces.
He was then approaching the last stages of his PhD dissertation research at Stanford under Bob Osserman.
I asked Blaine if he thought it was plausible that a hybrid of these two straight-edged polygons
would define an embedded adjoint surface. He replied that it was not an unreasonable idea,
because the intermediate value theorem guarantees successful 'period killing' — or words to that effect.
(I'm sure 'period killing' wasn't the exact phrase he used. I recall first hearing
those words several years later in a telephone conversation with David Hoffman,
who had been working with Bill Meeks on the Costa surface.)
I mailed (faxed?) Blaine a copy of my Fig. E2.71 notes,
but then I abandoned the P—C(P) hybrid and turned instead to a
topologically simpler case — the hybrid of P and
I-WP that I call O,C-TO (cf. Figs. E2.79 to E2.81). Its genus is only 10.
I was determined to build a physical model of some hybrid surface, and I knew that
vacuum forming a plastic surface patch with a severe undercut — like
the one shown at left center in Fig. E2.71 — would be difficult or impossible.
Then for the next forty-two years, I forgot all about the P—C(P) hybrid!
This process of hybridization is equivalent to attaching a handle to a minimal surface.
Many other people subsequently discovered handle attachment and have applied it to a variety of minimal surfaces — not just periodic ones.
(I have frequently been chided by mathematicians for not having taken the trouble to publish my work in refereed journals,
since that would have made it more readily accessible to others.)
A few days ago (it's now May, 2011), I found the Fig. E2.71 notes in my files and emailed a copy to Ken Brakke.
He quickly confirmed (with his Surface Evolver)
that the P—C(P) hybrid is embedded.
Ken's pictures of this surface are shown in Figs. E2.72, E2.73a, and E2.73b and also at his
Triply Periodic Minimal Surfaces web site,
where it is called "N14".
Fig. E2.72
Ken Brakke's 2011 Surface Evolver solution for an elementary patch of the embedded hybrid of P and C(P)
(cf. Fig. E2.71)
(image courtesy of Ken Brakke)
Fig. E2.73a
E2.73b
A cubic unit cell of the hybrid of P and C(P) (genus 14)
The unit cells in the two images are displaced with respect to
each other by one-half of the body diagonal of an enclosing cube.
(cf. Fig. E2.72)
(images courtesy of Ken Brakke)
Fig. E2.74 shows Ken's picture of the
manta surface (genus 19),
which is one of many examples of hypothetical minimal surfaces whose existence I conjectured in 1971. Some of them were inspired
by experiments with soap films blown inside a kaledioscopic cell.
Others were inspired by considering the structure of various highly symmetrical inorganic crystals.
Manta is a balanced surface;
the P—C(P) hybrid is non-balanced.
If you compare Fig. E2.73 and Fig. E2.74, you will see that
the P—C(P) hybrid has a simpler topology than Manta.
Manta has [100] tunnels, while the P—C(P) hybrid does not.
A few days after Ken sent me his image of the P—C(P) hybrid, which is shown in Fig. E2.73,
he sent me images of two slightly more complicated surfaces he said he had
obtained by 'poking holes' in the P—C(P) hybrid.
Images of this new pair — N26 and N38 — can be seen at his website
Triply Periodic Minimal Surfaces,
together with several other hybrids.
Of course, purists rightly claim that the existence of all of these hypothetical minimal surfaces is somewhat suspect,
since it has not been established by rigorous mathematical proof.
A diagram of the unit cell of the cubic phase of the compound BaTiO3 is shown in Fig. E2.75 for comparison with manta.
One can try to match the sizes and positions of the ions in BaTiO3 to symmetrical cavities in the labyrinths of the surface.
At my request, Ken produced the three orthogonal projections of manta shown in Figs. E2.76, E2.77, and E2.78, together with the following numerical data
on the radii of spheres that fit snugly against the surface in three classes of symmetrical cavities:
Radii of tangent spheres in 1x1x1 unit cell of manta genus 19 surface:
corner sphere radius: 0.18560130
center sphere radius: 0.18560130
midedge sphere radius: 0.23553163
(Here's
a useful Wikipedia article about ionic radii.)
|
Fig. E2.74
manta (genus 19)
(image courtesy of Ken Brakke)
Fig. E2.75
Unit cell of cubic phase of barium titanate (BaTiO3)
Ba2+ red
Ti4+ green
O2− blue
Fig. E2.76
Orthogonal projection of manta unit cell on [100] plane
(image courtesy of Ken Brakke)
Fig. E2.77
Orthogonal projection of manta unit cell on [110] plane
(image courtesy of Ken Brakke)
Fig. E2.78
Orthogonal projection of manta unit cell on [111] plane
(image courtesy of Ken Brakke)
Fig. E2.79
O,C-TO, a hybrid of I-WP and P
(genus 10)
cubic unit cell
view: [100] direction
Fig. E2.80
O,C-TO
unit cell
view: [111] direction
.jpg)
Fig. E2.81
O,C-TO
1.5 unit cells
oblique view
|
'Notched adjoints': grafting handles onto embedded surfaces
.jpg)
Fig. E2.82a
A note by Ernst Eduard Kummer (H.A. Schwarz's father-in-law)
that is included in Schwarz's Gesammelte Werke
The illustration shows eight triangular Flächenstücke of
Schwarz's diamond surface D inside a tetragonal disphenoid.
Fig. E2.82b
Free translation of the note by Kummer in Fig. E2.82a
One weekend in 1968, while I was reading p. 150 of Schwarz's Collected Works (cf. Fig. E2.82a),
it occurred to me that one might be able to model a small portion of a triply-periodic minimal surface, like the portion
of Schwarz's D surface shown in Fig. E2.82, by means of
a soap film in the interior of the appropriate polyhedral cell.
(Schwarz and
Plateau
had a very active correspondence for many years about soap films and minimal surfaces.
I'm surprised that they seem not to have performed experiments of this type.)
I quickly constructed a transparent model of the tetragonal disphenoid from four vinyl triangles,
stretching cotton threads along the two internal symmetry axes, which are clearly visible in Kummer's Fig. E2.82 sketch.
I cut a hole in one of the faces of the cell large enough to provide access to the interior with a soda straw.
Using a soap solution containing some glycerine, I discovered that the modeling of the minimal surface is quite easy!
One of its elegant features is that by blowing through the straw on one face or the other of the soap film,
you can toggle back and forth between two stationary states:
a triangular patch of D and a quadrangular patch of C19(D) (cf. Figs. E2.4b, c, d).
You stretch the soap film by blowing one corner of it right up to a corner of the cell, and then
with a light puff of air through the straw, you push that part of the film just beyond the cell corner.
At that point it automatically slides down into its other equilibrium position.
My model of the tetragonal disphenoid was a flimsy one.
When I arrived at NASA on the following Monday morning, I phoned Hal Robinson,
the sculptor and model-maker who had recently started to work for me
part-time and asked him to make a more physically rugged tetragonal disphenoid.
Within a couple of days or so, he produced a lucite model with highly accurate proportions,
using monofilament nylon instead of cotton threads for the internal symmetry axes.
Next I asked Hal to make me a lucite model of another Coxeter cell relevant for cubic TPMS — the quadrirectangular tetrahedron,
which is one-quarter of the tetragonal disphenoid. It contains only one internal axis of two-fold rotational syrmmetry,
not two. With this cell, you can toggle back and forth between a patch of P and a
patch of the Neovius surface C9(P) (cf. Figs. E2.2a, b),
again by blowing on the soap film to stretch it over one corner of the cell.
Just as with the tetragonal disphenoid, from there
the soap film slides into its other stable stationary state automatically.
Not until the spring of 1970 did it occur to me that perhaps Schwarz's P surface and Neovius's surface
C9(P) are merely the topologically simplest members of a
countably infinite sequence of embedded surfaces
of progressively higher genus:
P, C9(P), C15(P), C21(P), ...
I obtained experimental evidence for the existence of C21(P) by
producing the four-sided Flächenstück of
C21(P) as a soap film in a stationary — but unstable — state inside
the quadrirectangular tetrahedron. This was a more difficult soap film experiment than toggling back and forth between
P and C9(P),
which are in stable equilibrium.
Although the C21(P) soap film corresponds to a [mathematical] stationary state,
its area is larger than that of nearby lying [non-minimal-surface] soap films,
and I had to struggle to maneuver the film into its unstable equilibrium position
long enough for a camera to capture it.
In 1992 I made an impromptu video about minimal surfaces in which I attempted to demonstrate the art of blowing these
unstable soap films inside Coxeter cells, but I had run out of glycerine that day. After many tries, I succeeded for a fleeting moment
in capturing the gracefully curved Flächenstück of C21(P).
I plan to post here a snapshot or two from these videos, but the images of the soap films are somewhat obscured by the transparent tape I used to join the faces of the vinyl tetrahedra.
I plan to obtain clearer photos of soap films inside glass tetrahedra I have recently made.
Fig. E2.83a
Twelve elementary triangular Flächenstücke
of Neovius's embedded surface C9(P)
Fig. E2.83b
Twelve elementary triangular Flächenstücke
of Neovius's self-intersecting surface C9(P)†
In both C9(P) and C9(P)†, the triangular Flächenstück abc
is analytically continued by reflection in its edges.
The normal vectors (red arrows) at corresponding points of the
two adjoint surfaces C9(P) and C9(P)†
have the same directions.
In 1971, with the assistance of my Cal Arts students John Brennan and Bob Fuller,
I performed additional soap film experiments aimed at modeling the elementary Flächenstücke
for higher-genus variants of the P and D surfaces and a variety of non-cubic TPMS. All of these experiments involved
modifying the shapes of stationary-state soap films inside transparent plastic models of Coxeter cells by blowing on them.
Fig. E2.84 shows some of Ken Brakke's Surface Evolver
solutions
for some of these higher-genus variant surfaces.
I was unable to produce the genus-15 soap film, but occasionally I succeeded with the genus-21 case.
genus=9
genus=15
genus=21
genus=27
genus=33
Fig. E2.84
Ken Brakke's Surface Evolver solutions for the first few
high-genus variants of Neovius's C9(P)
'NOTCHED' ADJOINT SURFACES
genus=3
genus=9
genus=15
.jpg)
_surface.gif)
genus=21
genus=27
genus=33
Fig. E2.85
Stereo images of the sequence of 'notched' variants of
the adjoints of P, C9(P), ... (left)
and Ken Brakke's images of
the corresponding embedded surfaces (right)
(The relative lengths of the edges in the notched adjoint outlines
do not have the true values, which were obtained by Ken Brakke
when he killed periods to obtain the
embedded surfaces shown at the right.)
The genus pk of the kth surface
Mk in the family {Mk}
is defined as pk = p0 + k gap (k = 0, 1, 2, ...);
the values of p0 and the positive integer gap are characteristic of the family.
These families include — but are not limited to — high-genus complements of P and D.
For what I will call the
P
and
D families,
for example,
p0=3 and gap=6. The surfaces in these two families are of genus 3, 9, 15, 21, ... .
On that day in 1971, John and Bob and I blew a large variety of "finely filigreed" soap films
in a variety of Coxeter cells, and we made detailed drawings of our results.
Most — but not all — of these soap films included one or two nylon threads
stretched along 2-fold symmetry axes of the enclosing polyhedral cell.
The curved soap film boundary edges lying in face planes of the cell are 'mirror-symmetric plane lines of curvature'.
Every face plane is a plane of reflection symmetry for both the assembly of cells and the soap films in their interiors.
The soap films meet the enclosing face planes orthogonally.
Each soap film is an approximate model of the stationary state of the adjoint of
a minimal surface bounded both by straight line segments and by either one or two curved edges
— according to whether the number of rotational symmetry axes through the cell is one or two.
Films with k=0 or 1 are in stable quilibrium.
If k=2, the film is in a delicate state of unstable equilibrium. I found it impossible to produce films
for k>2. Some skill is required to arrest a film for k=2 in the neighborhood of its equilibrium position long enough to
confirm the existence of the equilibrium. (The films were composed of a mixture of distilled water, detergent, and glycerine and were thick and viscous enough for
both gravity and capillarity effects to impose some limits on the accuracy of the modeling.)
In 1999, I began sending Ken Brakke data from these 1971 experiments as well as some additional data for surfaces whose existence I conjectured
during the following three years,
for authentication with his Surface Evolver computer program.
Many of these authenticated surfaces are illustrated on his
Triply Periodic Minimal Surfaces web site.
In this work, Ken uses Surface Evolver to 'kill periods' — i.e., to derive the unique values for relative edge lengths that allow the surface to be embedded
(cf. the brief description of this problem on pp. 45-46 of
Infinite Periodic Minimal Surfaces Without Self-Intersections).
In Infinite Periodic Minimal Surfaces Without Self-Intersections,
I included only two examples of hybrid surfaces — C(H) (genus 7) and O,C-TO (genus 10),
because at the time of writing, these were the only examples of such surfaces for which I had already constructed and photographed vacuum-formed plastic models.
(In a footnote on p. 46, I mentioned a third example, of genus 5, that I called g-g'.
I soon renamed that surface g-W, after I confirmed that its dual skeletal graphs are related to the
structures of
hexagonal graphite
and of
wurtzite.) Figs. E3.7, E3.8, and E3.9 show three views of g-W.
E3. Triangle lattice surfaces
All of the minimal surfaces described in this section are named according to conventions introduced in
Infinite Periodic Minimal Surfaces Without Self-Intersections.
Fig. E3.1
Schwarz's H surface (genus 3)
oblique view
_unit_cell.jpg)
Fig. E3.2
C(H) (genus 7)
The [first-order] complement of Schwarz's H surface
unit cell
view: c-axis
_[111].jpg)
Fig. E3.3
C(H)
view along c-axis
_basal_plane.jpg)
Fig. E3.4
C(H)
view: c2 axis (intersection of horizontal and vertical mirror planes) in basal plane
_basal_1.jpg)
Fig. E3.5
C(H)
view: line in basal plane that is below and parallel to a linear asymptotic (2-fold axis embedded in the surface)
Note the infinitely long straight tunnels with pointy oval cross-section
_silhouette(1).jpg)
Fig. E3.6
C(H)
view: c-axis, silhouetted by bright summer sky backlighting
Note the infinitely long straight tunnels.
(The trigonal symmetry of the surface would be slightly more apparent
if the image had been rotated 60º in the image plane,
as in Fig. 3.3!)
Fig. E3.7
g-W ("graphite-wurtzite") (genus 5)
oblique view
.jpg)
Fig. E3.8
g-W
oblique view
.jpg)
Fig. E3.9
g-W and C(H)
oblique view
Fig. E3.10
H''-R (genus 5)
view: c-axis
.jpg)
Fig. E3.11
H''-R (genus 5)
view: c-axis, silhouetted by bright summer sky backlighting
.jpg)
Fig. E3.12
H''-R
oblique view
Fig. E3.13
H''-R
view: c2 axis (intersection of horizontal and vertical mirror planes) in basal plane
Fig. E3.14
H''-R
view: line in basal plane that is below and parallel to a linear asymptotic (2-fold axis embedded in the surface)
Note the infinitely long straight tunnels with pointy oval cross-section
Fig. E3.15
H'-T (genus 4)
view (stereo): c-axis
E4. Surfaces on other lattices
Fig. E4.1
S'-S''
genus 4
view: oblique
E5. Background
E6. Bibliography
For online minimal surface videos, discussion, analysis, and images, including
— but not restricted to — examples of embedded triply periodic surfaces, see
- Ken Brakke's
Triply Periodic Minimal Surfaces
- EPINET's
Triply Periodic Minimal Surfaces
- Shoichi Fujimori and Matthias Weber's
A Construction Method for Triply Periodic Minimal Surfaces
- Paul J. F. Gandy, Sonny Bardhan, Alan L. Mackay, and Jacek Klinowski's
Nodal surface approximations to the P, G, D, and I-WP triply periodic minmal surfaces,
- Darren Garbus's
Isoperimetric Properties of Some Genus 3 Triply Periodic Minimal Surfaces Embedded in Euclidean Space,
M.S. Thesis, May 2010
- Chaim Goodman-Strauss and John Sullivan's
Cubic Polyhedra
- Wojciech Gòzdz and Robert Holyst's
High Genus Periodic Gyroid Surfaces of Non-Positive Gaussian Curvature
- David Hoffman and Jim Hoffman's
Geometry: Minimal Surfaces
- David Hoffman and Jim Hoffman's
The Lidinoid Surface
- Stephen Hyde, Christophe Oguey, and Stuart Ramsden's
Triply connected graph embeddings
- Stephen T. Hyde, Michael O'Keeffe, and Davide M. Proserpio's
A Short History of an Elusive yet Ubiquitous Structure in Chemistry, Materials and Mathematics
- "Touching Soap Films, An Introduction to Minimal Surfaces",
by Hermann Karcher and Konrad Polthier.
Besides providing a wealth of information in text form, it includes some beautiful videos.
- Hermann Karcher and Konrad Polthier's
list of references
to materials that are not online.
- Hermann Karcher's
The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions,
Manuscripta Math. 64, 291-357 (1989)
- Hermann Karcher and Konrad Polthier's
Exhibition of Historical Minimal Surfaces
(video)
- Jacek Klinowski's
Periodic Minimal Surfaces Gallery
- Elke Koch and Werner Fischer's
3-periodic minimal surfaces at the '[Details]' hyperlink in their Mathematical Crystallography
- Xah Lee's
Gallery of Famous Surfaces
- Eric Lord and Alan Mackay's
Slide Show of Triply Periodic Surfaces
- Eric Lord and Alan Mackay's
Periodic Minimal Surfaces of Cubic Symmetry
- William H. Meeks III's
Introduction to Minimal Surfaces
- Alan Schoen's
Infinite Periodic Minimal Surfaces Without Self-Intersections,
NASA TN D-5541 (May 1970)
- Gerd Schröder-Turk's
The bicontinuous fish tank
- Gerd Schröder-Turk's
Bonnet transformation between the D, G and P minimal surfaces
(video)
- Gerd Schröder-Turk, Stuart Ramsden, Andrew Christy, and Stephen Hyde's
Medial Surfaces of Hyperbolic Structures
-
Gerd Schröder-Turk, Andrew Fogden, and Stephen Hyde's
Local v/a variations as a measure of
structural packing frustration in bicontinuous mesophases,
and geometric arguments for an alternating Im3m (I-WP) phase
in block-copolymers with polydispersity
- Gerd Schröder-Turk, Andrew Fogden, and Stephen Hyde's
Bicontinuous geometries and molecular self-assembly:
comparison of local curvature and global packing variations in genus three cubic, tetragonal, and rhombohedral surfaces
- Martin Steffens and Christian Teitzel's
Grape Minimal Surface Library
- Toshikazu Sunada's
Crystals That Nature Might Miss Creating
- Matthias Weber's
Gallery of Minimal Surfaces
Be sure not to miss Matthias's 'Archive' animations!
- Adam Weyhaupt's
New Families of Embedded Triply Periodic Minmal Surfaces of Genus Three in Euclidean Space
- Adam Weyhaupt's
Meet the Gyroid
- Adam Weyhaupt's
Deformations of the gyroid and Lidinoid minimal surfaces
-
3D-XplorMath-J Applets
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E7. Minimal surface people
Fig. E7.1
Shoichi Fujimori at Bloomington, Indiana (2008)
Fig. E7.2
Wojciech Góźdź
Fig. E7.3
Bathsheba Grossman at Santa Cruz
Fig. E7.4
Stefan Hildebrandt at Berkeley (1979)
photo by George M. Bergman
©George M. Bergman
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH
Fig. E7.5
Robert Holyst
Fig. E7.6
Stephen Hyde at Canberra
Fig. E7.7
Hermann Karcher
(left),
David Hoffman (center), and Manfredo Perdigão do Carmo (right)
at Granada (1991)
photo by Dirk Ferus
©Dirk Ferus
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH
.png)
Fig. E7.8
Katsuei Kenmotsu at Oberwohlfach (2009)
photo by Renate Schmid
Source: Mathematisches Forschungsinstitut Oberwohlfach gGmbH
Fig. E7.9
Blaine Lawson
(left) and
Bill Meeks (right) at Rio (1980)
photo by Dirk Ferus
©Dirk Ferus
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH
Fig. E7.10
Johannes C. C. Nitsche (1925-2006)
photo by Ludwig Danzer
©Ludwig Danzer
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH
Fig. E7.11
Robert Osserman (1926-2011) at Berkeley in 1979
photo by George M. Bergman
©Mathematisches Forschungsinstitut Oberwolfach gGmbH
Fig. E7.12
Raymond Redheffer (1921-2005)
Fig. E7.13
Gerd Schröder-Turk at Erlangen-Nürnberg
Fig. E7.14
Isaac Van Houten at Carbondale (2008)
photo by the author
.jpg)
Fig. E7.15
Matthias Weber at Oberwohlfach (2009)
photo by Renate Schmid
©Mathematisches Forschungsinstitut Oberwohlfach gGmbH
Fig. E7.16
Adam Weyhaupt (right) at Edwardsville
with two of his students — Darren Garbuz and Caroline Coggeshall
photo by the author
Return to GEOMETRY GARRET
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