Welcome to the
GEOMETRY GARRET!
A Pot-Pourri of
People, Penrose Patterns, Polyhedra, Polyominoes, Posters, and Puzzles
Alan H. Schoen
alan_schoen@verizon.net
Comments are welcome.
Outline of topics:
- A. Polyhedra
- 1. Roundest Polyhedra
- B. Finite tilings by rhombs
- 1. Rosettes (and pseudo-rosettes)
- 2. Rombix
- C. Infinite tilings by rhombs
- 1. Penrose tilings
- 2. d7-symmetric generalized Penrose tilings derived from de Bruijn heptagrids
- 3. RP7 tilings (recursive pseudo-Penrose tilings of d7 symmetry)
- 4. Rhombic wallpaper (periodic tilings derived from a variant form of the de Bruijn multigrid)
- 5. Others
A1. Roundest Polyhedra
The theory of roundest polyhedra (sometimes called best polyhedra) was first treated by Simon Lhuilier (1750-1840). The subject is conventionally referred to as
In the 19th century this problem was investigated by Jakob Steiner, E. Kötter, Ernst Lindelöf, and Hermann Minkowski, and in the 20th century by Ernst Steinitz, Michael Goldberg, and Victor Klee. Klee greatly deepened the analysis and also extended it to dimensions beyond three.
Ernst Steiner (1796-1863)
Hermann Minkowski (1864-1909)
Ernst Steinitz (1871-1928)
Gyorgy Voronoy (1868-1908)
Ernst Lindelöf (1870-1946)
Victor Klee (1925-2007)
My interest in roundest polyhedra was kindled in 1972 when Victor Klee gave me copies of both Parts I and II of his filmscript booklets, 'Shapes of the Future — Some Unsolved Problems in Geometry '. With the permission of the Mathematical Association of America, I am making 'Part II: Three Dimensions' available here. I intend to post 'Part I: Two Dimensions' as well.
At AMS meetings in the late sixties, I met Michael Goldberg several times. We talked mostly about minimal surfaces and Voronoi polyhedra. Unfortunately he never mentioned his 1934 article, 'The Isoperimetric Problem for Polyhedra'. I regret that by the time (1972) I had become interested in this subject, he had already 'shuffled off this mortal coil'.
The roundest polyhedron question for R3 is:
(S = surface area and V = volume.)
In 1897, Minkowski proved that a roundest polyhedron exists
for every n ≥ 4.
It was proved by Lindelöf in 1899 that a necessary condition
for a polyhedron P to be roundest is that
( l) P circumscribes a sphere, and
(2) the inscribed sphere is tangent to all the faces of
P at their respective centroids.
For any polyhedron circumscribed about the unit sphere,
S/V = 27V = 9S.
Hence minimizing S 3/V 2
is equivalent to
minimizing S (or V).
It has been conjectured — but never proved — that the
roundest polyhedron is always simple,
i.e., that its vertices are all of degree three.
My computational geometry 1986 conference paper,
'A Defect-Correction Algorithm for Minimizing the Volume
of a Simple Polyhedron Which Circumscribes a Sphere',
summarizes a 1986 experimental investigation of roundest polyhedra.
Conjectured solutions for 4 ≤ n ≤ 35 and for n = 42 are described
and illustrated.
A supplement added in May, 1986
includes conjectured solutions for 36 ≤ n ≤ 41
and n = 43. Included in the supplement is a summary of William Tutte's analysis
of the number of simple polyhedra as a function of the number n of faces.
I will soon post images of face
templates for four examples of 'roundest' polyhedra (n=8, 25, 33, 44),
so that you can construct physical models of them.
[Probably] the 'roundest' of all polyhedra with 42 faces
IBM Computational Geometry Conference, 1986
(If you can identify this participant, please tell me who she is!)
In 1989, my former student David M. Aubertin wrote a computer science masters thesis in which he designed and implemented a program for obtaining and displaying solutions for the 4-dimensional version of the isoperimetric problem for polyhedra. His thesis, which is entitled 'Optimization of Four-Dimensional Polytopes', contains stereoscopic perspective drawings of projections of the solution 4-polytopes into R3. It also includes a listing of the elegantly structured FORTRAN program Dave wrote to obtain his results, together with a short BASIC program for displaying the perspective images. He has told me that you are welcome to copy and use both of these programs.
The lowly rhomb — a parallelogram with sides of equal
length — has acquired special notoriety during the past few decades
as the shape of the two prototiles of Penrose patterns.
But there are several other kinds of tiling
patterns by rhombs in which the rules that govern tile placement, though
less subtle than the rules for Penrose
tilings, are nevertheless sufficiently non-obvious to require some analysis.
Some of these other kinds of tilings, both finite and infinite, are discussed below.
But first— a frivolous design:
COMPOSED OF SIX NESTED RINGS OF RHOMBS
This arrangement of nested eccentric rings is not an edge-to-edge tiling (cf. p.18 of Tilings and Patterns, by Grünbaum and Shephard). From now on, all the tilings I consider are of edge-to-edge type.
B1. ROSETTES (and PSEUDO-ROSETTES)
A rosette is an example of a finite tiling by rhombs. In § C1, C2, and C3, I consider a different kind of finite tiling by rhombs—the finite core of an infinite recursive tiling. One of the major differences between these two types of tiling is that while the number of rhombs of each shape in a rosette is fixed by the choice of the positive integer n, the number of rhombs of each shape in the core of an infinite recursive tiling — where there is no upper bound on the size of the core — is not restricted.
I define a simple rosette as any edge-to-edge tiling by rhombs of a regular convex polygon with an even number 2n of sides; n is called the order of the rosette. The sides are defined to be of unit length. A square is a rosette of order two. The number of rhombs in a rosette increases quadratically with n (no. of rhombs = n(n-1)/2), and this increase is accompanied by a combinatorial explosion in the number of ways the rhombs can be arranged. Of particular interest are symmetrical rosettes. When n >> 2, it is not immediately obvious what rules govern their structure. I explore this question below.
First, however, let's examine how to use straight edge and compass to draw a particular class of tilings I'll call compound rosettes. My seventh-grade classmates and I learned this trick in 1936 from Miss Beatrice Wild, our art teacher in Room 314 of William Wilson Jr. School in Mount Vernon, N.Y.
The first step is to inscribe 2n (n = 2,3,4,...) unit circles inside — and tangent to — a circle of radius 2. The unit circles are spaced uniformly along the circle boundary.
The second step is to join by line segments certain pairs of points where circles — whether small or large — intersect in non-overlapping four-point cycles that define rhombs. The rhombs inside each unit circle define a simple rosette I call the STRAWBERRY. Since the unit circles overlap, the STRAWBERRIES do too (except in the case n = 2). Examples of compound rosettes for n = 2, 3, and 4 are shown below.
Fig. B1.1
Compound rosettes for n = 2, 3, and 4
From now on, unless stated otherwise, every rosette will be assumed to be simple.
Some vital statistics:
For odd n:
There are (n − 1)/2 different shapes of rhomb, with smaller face angles equal to
kπ/n
(k =1, 2, ..., (n − 1)/2), respectively.
For n = 2:
There is a single rhomb — the square.
For even n ≥ 4:
In addition to n/2 specimens of the square, there are n specimens of each non-square rhomb,
with smaller face angles equal to
kπ/n
(k =1, 2, ..., n/2), respectively.
There are altogether n(n − 1)/2 rhombs in every rosette.
They comprise what I call the Standard Rhombic Inventory of order n, or SRIn.
For rosettes of order n > 4, the n(n − 1)/2 rhombs in SRIn can be arranged to form a variety of different rosettes. For n = 5, for example, there are six different arrangements:
The six rosettes for n = 5
Using a 'rhomb-shuffling' computer program, I once found forty-nine different rosettes for n = 6, but my program may have missed some.
If we perform a variation on Miss Wild's tangent-circles construction
for the half-integer n-values
3/2, 5/2, and 7/2, we obtain the pseudo-compound rosettes shown below.
The 2n quadrangles incident at the center of the large circle are kite-shaped;
all the other quadrangles are rhombs.
In a pseudo-compound rosette of order n, there are n − 1/2 different
shapes of quadrangle, and there are 2n
specimens of each shape.
In every kite, the face angle at the vertex opposite the central vertex
is equal to π/2n.
If each kite were elongated by translating its central vertex
to the nearest vertex on the kite's symmetry axis, it would be transformed into a rhomb.
Fig. B1.3
Pseudo-compound rosettes for n = 3/2, 5/2, and 7/2
It seems reasonable to call the arrangement of polygons inside each small circle in a
pseudo-compound rosette a pseudo-STRAWBERRY.
Prove that neither the tiles in a pseudo-strawberry nor those in a pseudo-compound rosette can be arranged in any other pattern with convex boundary.
I don't recall ever hearing Miss Wild mention that the rhombs in a compound rosette can be rearranged to form other patterns. I first learned about rosettes as mathematical objects from Coxeter's description of them as projections of polar zonohedra, in his Regular Polytopes, Third Edition, Dover Publications (1973) pp. 27-30. Coxeter also describes them briefly in Mathematical Recreations and Essays, Thirteenth Edition, Dover Publications (1987) pp. 141-142. (In a footnote on p. 141, Coxeter asks in how many ways the regular 2n-gon can be dissected into n(n − 1)/2 rhombs. It is easy to prove that for n = 2, 3, 4, and 5, the numbers are 1, 1, 1, and 6, respectively.)
When you design a symmetrical rosette 'by hand', after deciding on the order n and the symmetry class it's natural to wonder whether there are alternatives to simply shuffling paper rhombs in hit-or-miss fashion. In 1978, while examining dozens of examples, I succeeded in sniffing out a few simple numerical rules that significantly reduce the labor. In 1979 I wrote an unpublished summary of some of those rules, but I included only a few proofs. At that time, I was able to prove only a few of the main results.
One of these rules is useful for designing rosettes of dihedral symmetry. It generates a list of all the partitions of a diametral pendant into two radial pendants of equal projected length. A diametral pendant is a string of rhombs joined tip-to-tip, lying along a line of reflection symmetry and extending from one boundary vertex to the opposite boundary vertex. Diametral pendants are the 'structural ribs' of dihedral rosettes.
A second rule, which is is useful for designing rosettes with cyclic symmetry, generates a list of all the partitions of a ladder into two half-ladders of the same projected length. A ladder is a strip of rhombs in a rosette that extends from one edge of the {2n} to the opposite edge. Every rhomb in a rosette belongs to two intersecting ladders.
You can learn about part of what I discovered in 1979 about both cyclic and dihedral rosettes here, in a condensed version of the 1979 report. I plan to expand this note soon. I will describe complete proofs of all of the claimed results. (It wasn't until 1991 that I returned to the subject of rosettes and succeeded both in proving what until then had been only conjectures and also in expanding some of the results. I was pleasantly surprised to find that only elementary methods were required for all the proofs.)
In 1979 I mailed copies of my report on rosette design rules to
several mathematicians, hoping that someone would provide the missing proofs.
Ira Gessel,
who had been given a copy of the report by Gian-Carlo Rota (one of the experts
on my mailing list), sent me an extremely helpful reply.
Fig. B1.4
Ira Gessel
With Ira's permission, I include his reply here. Ira's letter explains that the rhombs in each ladder of a rosette are the duals of points of intersection in an arrangement of n chords of a circle, indexed in order of their angular orientation. The sides of each rhomb are perpendicular to the two chords whose intersection is the dual of the rhomb. This means that one can generate rosettes by a simple 'automatic' method, without having to shuffle paper rhombs. After receiving Ira's letter, I wrote a Fortran program to implement his algorithm, and my first-semester computer graphics students were soon creating examples by the dozen of rosettes of various orders n. Rosettes generated by this method typically differ in appearance from the ones my students had been creating 'from the ground up' by applying the diametral pendant and ladder partitioning rules I had taught them. The 'automatic' rosettes tend to be somewhat bland in appearance. Later, after I have defined the total curvature of a rosette, the reason for this difference will become clear. (The strawberry appears to have the smallest possible value of total curvature. I conjecture that this is true for every n.)
At the January meeting of the AMS in San Antonio a few months after I had mailed out copies of my rules report, I showed a copy of it to Ron Graham. Paul Erdös, Ron's friend and collaborator, was standing a few feet away. After briefly examining the report, Ron introduced me to Paul and asked him if he would mind looking it over. Paul scrutinized it for a few minutes and then said,
"Zeese is veddy in-ter-est-ing. Zee next time I am in Car-bon-dale, vee vill prove zeeze sings too-ged-der."
Although I was intrigued by the possibility of someday having an Erdös number of one, my Erdös number will forever remain zero, because there was no 'next time'. Later that year, Mel Nathanson, Paul's regular host in Carbondale, left SIU for Rutgers, and Paul never returned to Carbondale.
In 2007, the algebraist Bob Fitzgerald compared the 'vanishing sum of sines' equations predicted for n = 15 and n = 18 by the ladder partitioning rule (described here) with a set of equations, known to be complete, that he obtained by invoking cyclotomic field theory. Here's the email Bob sent me. He had confirmed that for n = 15 and n = 18, every possible equation of this vanishing-sum-of-sines type is at most a linear combination of the equations derived from my rules for ladder partitioning.
Below are scanned images of examples of rosettes, almost all of which were drawn (long ago) with TrueBASIC.
Fig. B1.5
n = 39
c13
pdf version
Fig. B1.6
n = 36
d3
pdf version
The image in Fig. B1.4\5 is the result of a search for rosettes that contain the maximum possible number of convex hexagons (call them ovals) tiled by three rhombs. In 1980 I examined hundreds of rosettes without finding any examples containing fewer than n − 2 ovals. I widely circulated—but didn't publish—the conjecture that in every rosette of order n, there are at least n − 2 ovals.
When I mentioned the n − 2 ovals conjecture to
Jim Propp
in 1995, he immediately
circulated it, with my permission, by email.
(I hadn't yet opened an email account.)
Fig. B1.7
Jim Propp
Fig. B1.8
Alexander Postnikov
(A simplex is bounded, in Postnikov's words, if its interior is not intersected by any hyperplane.) Belov's paper, On a problem of combinatorial geometry, was published in Russian Math Surveys, Vol. 47 (1992), pp. 167-168. Alex said that he and Igor Pak had developed another proof 'in a more general context, based on the fact that every (n − k)-dimensional polyhedron has at least n − k edges going out of every vertex'.
In two dimensions, the simplices are the triangular interstices defined by the pairwise intersections of lines. The lines are not required to be straight, but any pair of lines is required to have only one intersection. If they are not straight, they are called pseudo-lines. I suspect that it may be unknown whether there exists a dual configuration of straight lines for every possible rosette.
(I think I'll take a short break before doing any more wild conjecturing.)
February 29, 2009
Well, that 'suspicion' of mine was dead wrong!
When I wrote to Alex and asked him about it a few hours ago,
he replied that it is false, and he even described how to
construct counterexamples. He wrote,
"There are rhombus tilings that cannot be represented by straight
lines. Equivalently, there are pseudo-line configurations that cannot
be straightened. ... Moreover, for sufficiently large n almost all pseudo-line
configurations cannot be straightened."
Fig. B1.9
n = 45
pdf version
.jpg)
Fig. B1.10
n = 45
c15
pdf version
.jpg)
Fig. B1.11
n = 45
c15
(The colors are reversed with respect to the arrangement in Fig. B1.6.)
pdf version
Fig. B1.12
n = 39
c13
pdf version
Fig. B1.13
n = 27
c9
pdf version
Fig. B1.14
n = 49
c7
pdf version
Fig. B1.15
n = 50
c5
pdf version
Fig. B1.16
n = 50
c5
pdf version
Fig. B1.17
n = 27
d1
(redrawn with Mathematica)
pdf version
Fig. B1.18
n = 27
d1
(redrawn with Mathematica)
pdf version
Fig. B1.19
n = 30
d5
pdf version
Fig. B1.20
n = 45
d1
pdf version
Fig. B1.21
n = 45
d9
pdf version
Fig. B1.22
Rosette images redrawn with Mathematica
You'll have noticed that the symmetry of all of these rosettes — whether cyclic or dihedral — is of odd order. That's no accident. It's not hard to prove that symmetries of even order are forbidden. One can prove further that the order of the allowed cyclic or dihedral symmetries is restricted to odd divisors of n (including n itself). Because a rosette that appears to have no symmetry is in fact invariant under rotation by a full turn, it is classified as having c1 symmetry (cyclic symmetry of order one).
You can make physical rosettes from magnetized plastic rhombs
of SRI7,
marketed under the name
'FRACTILES'.
To learn about KADON's sets of translucent lucite rhombs from SRI7, look
here.
B2. ROMBIX (a dissection puzzle)
Beginning in 1991, when ROMBIX first appeared in the marketplace,
I wrote several different versions of a manual for it.
(Look at KADON's
ROMBIX sets.)
C. INFINITE TILINGS BY RHOMBS
C1. Penrose tilings
In 1977,
Martin Gardner
introduced Roger Penrose's quasiperiodic tilings in his legendary
Scientific American column, Mathematical Recreations.
He explained how either the kite and dart or the thin and thick rhombs (the so-called "golden" rhombs)
serve as prototiles for these tilings. Figs. C1.1 and C1.2 show these tiles with markings that conform to Penrose's
'matching rules'. We'll call a tiling by replicas of the prototiles of either Fig. C1.1 or C1.2 matched if the tiling is
edge-to-edge and the tile markings are continuous across every edge.
Penrose proved that
(a) there is no periodic matched tiling of the plane by the prototiles of either Fig. C1.1 or C1.2, and
(b) there is an uncountable infinity of aperiodic matched tilings by these prototiles.
Such tilings are called quasiperiodic.
Fig. C1.1
Fig. C1.2
Periodicity of a tiling is easily confirmed by demonstrating that the tiling is composed of copies of a finite region replicated by translation in two independent directions in the plane. Quasiperiodicity, by contrast, is not identified simply by inspecting the tiling visually. It is a subtle kind of long-range order that is best defined in terms of the symmetry of the diffraction pattern of the tiling.
Two indispensable books for the study of Penrose tilings and related topics are
Tilings and Patterns (W. H. Freeman and Company, 1987), by Grünbaum and Shephard, and
Quasicrystals and geometry (Cambridge University Press, 1995),
by Marjorie Senechal.
Quasiperiodicity is discussed in Wikipedia's Penrose tiling.
There are several different methods of generating Penrose tilings, although they are equivalent (cf. de Bruijn). One of these is Penrose's remarkable inflation/deflation mechanism (so named by John Conway). Chapter 7 of Martin Gardner's The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems, W. W. Norton & Co. (2001), contains a description of inflation/deflation and how it leads to a proof that the number of different Penrose tilings is uncountable. Two effective tools for learning about inflation/deflation are David Austin's java applet, 'The Inflation Hierarchy', in his Penrose Tiles Talk Across Miles, and Stephen Collins's interactive program, Bob - Penrose Tiling Generator and Explorer.
Websites devoted to Penrose tilings, quasicrystals, and related matters include:
David Austin's
Penrose Tiles Talk Across Miles
Martin Gardner's
Penrose tiles,
pp. 73-89, excerpted from Chapter 7 of his The Colossal Book of Mathematics
Eric Hwang's
Penrose Tilings and Quasicrystals
Ron Lifshitz's
Quasicrystals
Michael S. Longuet-Higgins'
Nested icosahedral shells, or How to grow a quasi-crystal,
Math. Intelligencer 25, 25-43 (2002).
Paul J. Steinhardt's
A New Paradigm for the Structure of Quasicrytals
Eric Weisstein's
Penrose tiles
Michael Widom's
Quasicrystals.
Penrose is said to have remarked that before 1984 almost nobody but Alan Mackay took seriously his notion that his quasiperiodic tilings might have a physical counterpart in 3-dimensional space. The astonishing discovery of quasicrystals in 1984 by Shectman and his co-workers changed all that. Besides the extensive experiments on growing solid quasicrystals from the molten state, there has been much theoretical research — especially by Steinhardt and his collaborators, by Michael C. Longuet-Higgins, and by others too numerous to mention here — aimed at explaining the growth of quasicrystals. Fragments excerpted from the book, In Our Own Image: Personal Symmetry in Discovery, by Istvan and Magdolna Hargittai, describe some of the turbulent events that accompanied the discovery of quasicrystals, including Alan Mackay's independent prediction of their existence.
One might almost say now (in 2010) that the long-sought single prototile ('einstein') for two-dimensional quasi-periodic patterns
has at last been found, and it is the regular decagon. But of course it is inaccurate to call this decagon a
prototile, since it is the overlapping unit of a covering rather than a tiling.
An article by Petra Gummelt, inspired by an idea originally suggested
by Sergei E. Burkov, provides some background.
Steinhardt and Hyeong-Chai Jeong have developed Gummelt's model significantly, demonstrating that
Penrose coverings by decagons with matching overlaps are isomorphic to Penrose tilings by rhombs
with matching edges. Their analysis is supported by HAADF electron microscopy images of decagonal AlNiCo
obtained by Koh Saitoh et al. The agreement between the predictions of the overlapping decagons
model and the experimental results is striking.
In a landmark article in 1981, Nicolaas G. de Bruijn described an algebraic theory of Penrose tilings. He introduced the concept of pentagrids, which are composed of five superimposed grids of parallel lines unit distance apart. Below is an example of a pentagrid.
Fig. C1.3
An example of a de Bruijn pentagrid
Every pentagrid is identified by a set of five
shifts γi (i = 0,1,2,3,4) — radial displacements
of the five grids from the origin at the center. De Bruijn proved that the tiling
by rhombs that is dual to a given pentagrid is a Penrose tiling — i.e., satisfies Penrose's
matching rules — if and only if Γ, the sum of the five shifts,
is equal to an integer.
Among Penrose tilings for which the five shifts are equal, the one for which Γ = 1 is called SUN and
the one for which Γ = 2 is called STAR. They are shown below.
If you toggle up and down using the Page Up and Page Down keys,
you will observe that
any pair a and b of these six pentagrids for which γa + γb = 1
are related by inversion in the origin at the center
and are therefore equivalent.
Hence it's unnecessary to consider shifts > 1/2.
Since SUN and STAR are the only two equal-shift tilings for which Γ = 1 and γi ≤ 1/2, they are the only two Penrose tilings with d5 symmetry.
The remarkable CARTWHEEL tiling, for which all five shifts
are zero, has only d1 symmetry.
Each of its ten infinite triangular sectors, which — except for the Conway 'worms'
that line their sides — are congruent, contains an alternating sequence
of successively larger central regions of SUN and STAR.
The ratio of the distances from the origin to the centers of any two consecutive such regions
is found to be equal to the golden ratio φ (≅ 1.618). CARTWHEEL is shown below.
In a 1978 AMS abstract, I conjectured that both SUN and STAR have a recursive structure that allows them to be generated from a small central core without regard for tile-matching rules. By comparing tilings, I verified that the conjecture is correct at least up to the fourth stage of recursion. The abstract describes the recursion for kite/dart tilings, not tilings by rhombs. Here I consider the tilings by rhombs. A brief summary of the recursion conjecture follows.
The SUN and STAR Penrose tilings arise from a small central core
via a recursive sequence of
(a) radially outward reflections in the enclosing necklaces
(pentagonal rings of Conway mirror worms)
of successively larger central regions,
followed by
(b) lateral reflections of the images produced in (a)
to fill the empty triangular gaps left by (a).
Penrose STAR
three stages
d5
pdf version: Toggle back and forth between images,
using Page Up and Page Down keys,
to see how each triangular gap in stage three
(and in all subsequent stages, according to the Recursion Conjecture)
is filled by an image of the nearby triangular tiled region, reflected in an adjacent mirror edge (red).
Penrose SUN
three stages
d5
pdf version: Toggle back and forth between images,
using Page Up and Page Down keys,
to see how each triangular gap in stage three
(and in all subsequent stages, according to the Recursion Conjecture)
is filled by an image of the nearby triangular tiled region, reflected in an adjacent mirror edge (red).
Here I will assume the truth of the Recursion Conjecture.
At each stage of recursion, the area in the interior of the pentagonal boundary
mirror of the SUN or STAR increases by the factor
Since 2 cos π/5 = (1+√ 5)/2 = φ, the golden ratio (≅ 1.618),
the total tiled area in successive steps is proportional to
2+3φ
13+21φ
89+144φ
...
Let S1 = {1, 2, 13, 89,...} and S2 = {0, 3, 21, 144,...}.
Since the Fibonacci sequence is {.., 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...},
where
F1 = 1,
F2 = 0,
F3 = 1,
F4 = 1,
F5 = 2,
F6 = 3,
etc., then
S1 = {F1, F5, F9, F13, ...} and
S2 = {F2, F6, F10, F14, ....}.
De Bruijn's landmark paper
is entitled
"Algebraic theory of Penrose's non-periodic tilings of the plane",
Nederl. Akad. Wetensch. Proceedings Ser. A 84 (Indagationes Math. 43)
(1981) 38-66.
It was reprinted in The Physics of Quasicrystals, ed. P. J. Steinhardt and
S. Ostlund, World Scientific Publ. Comp., Singapore (1987), pp. 673-700.
For a concise summary of parts of the paper,
see
Laura
Effinger-Dean's undergraduate honors thesis.
The duality between a vertex (point of intersection) in a pentagrid and a thin or thick rhomb in the corresponding Penrose tiling is illustrated in the representative pentagrid shown in Fig. C1.3. For any pair of intersecting grid lines, the smaller angle of intersection is either 36º or 72º. The sides of a rhomb dual to a vertex are orthogonal to the two grid lines that intersect at the vertex.
grid.gif)
Fig. C1.6
In this representative pentagrid, the two gray rhombs
illustrate the duality between rhombs and points of intersection.
The amplitude of each vector in the central shift star for this pentagrid is equal to 1/4.
CARTWHEEL (Γ = 0)
Next — two differently shaded images of each of the d5-symmetric Penrose tilings, SUN and STAR. In the second image of each pair, concentric pentagonal rings of Conway mirror-worms are distinguished by color. The role of these mirrors in the conjectured recursive structure is explained below.
STAR (de Bruijn shift = 4/10). (TWO images: Scroll down for the second image.)
Here are the pentagrids for SUN and STAR.
pentagrid for SUN
Below are images of
- the three special Penrose tilings —SUN, STAR, and CARTWHEEL—and their dual de Bruijn pentagrids
- three generalized Penrose tilings and their dual de Bruijn pentagrids.
de Bruijn (3/10) (de Bruijn shift = 3/10)
de Bruijn (5/10) (de Bruijn shift = 5/10)
(Toggle with Page Down and Page Up keys)
C2. d7-symmetric generalized Penrose tilings from de Bruijn's heptagrids
Fig. C2.1.1
Left to right:
David Klarner, George Polya, and Nicolaas de Bruijn.
Stanford University, May 1973
Here are seven of the eight tilings of this generalized class.
(I haven't yet constructed the eighth one — de Bruijn (0/14) — which is a rough analog of the Penrose CARTWHEEL.)
de Bruijn (2/14)-1
de Bruijn (2/14)-2
de Bruijn (2/14)-3
de Bruijn (3/14)-1
de Bruijn (4/14)-1
de Bruijn (4/14)-2
de Bruijn (5/14)-1
de Bruijn (6/14)-1
de Bruijn (6/14)-2
de Bruijn (7/14)-1
C3. RP7 tilings (recursive rhombic tilings of d7 symmetry)
in the Hortus deliciarum of Herrad von Landsberg
(from Wikipedia)
I call any pseudo-Penrose tiling for n ≥ 3
an RPn tiling if it
(a) is composed of all of the different shapes of rhomb in SRIn
(cf. § B1) — that's n ⁄ 2 shapes for even n and (n − 1) ⁄ 2 shapes for odd n,
(b) is recursive in the restricted sense described below in § C3.1, and
(c) has dihedral symmetry dn.
If a tiling satisfies both (a) and (b), but
has symmetry d2n instead of dn, it is called a RP2n tiling.
If a tiling satisfies (b) and has symmetry d2n, but contains only the two smallest
shapes of rhomb in SRIn, it is called a RP2n∗ tiling.
Numerous examples of RP7 tilings are shown in § C3.12.
Several RPn tilings for n ≠ 7 are shown in § C3.13 and § C3.15.
RP2n and RP2n∗ tilings are represented by examples in § C3.14 and § C3.16.
RPn tilings for n > 7 are of little interest, because — as Fig. C3.0.1 demonstrates up to n = 12 and Fig. C3.0.2 demonstrates up to n = 42 — as n increases, each stage of the tiling inherits a steadily decreasing fraction of the previous stage, because of overlap (cf. § C3.1).
Although the principal topic of § C3 is RP7 tilings, much of the discussion is applicable also to RPn tilings for odd n ≠ 7. Not much needs to be said about the 'even' tilings (RPn tilings for even n and RP2n tilings), because they are both conceptually and mathematically simpler (cf. Fig C3.1.2) than the 'odd' tilings (RPn tilings for odd n).
Fig. C3.0.1
The geometrical basis of recursion in RPn and RP2n tilings (3 ≤ n ≤ 12)
(For details, cf. § C3.1)
RP7 tilings were an afterthought of a 1978 conjecture (cf. § C1) suggesting that the d5-symmetric Penrose SUN and STAR tilings are structured in strict recursive fashion. RP7 tilings are designed to mimic this putative recursive structure, at least partially, but unlike Penrose tilings, they are not quasiperiodic and cannot be generated by de Bruijn's algorithm (cf. § C2).
Fig. C3.0.3 shows the three prototile rhombs of RP7 tilings, each with sides of unit length. Rhomb j (j=1,2,3) has upper and lower face angles 2jπ/7.
The three prototile rhombs for n = 7
Fig. C3.0.4
The regular 14-gon is the smallest convex region
for which there exists a centro-symmetric tiling by the three RP7 prototiles
RP7 tilings are created by iterating the reflection of a finite d7-symmetric core tiling generated up to a given stage of recursion in the edges of its heptagonal boundary mirror, thereby producing a heptagonal annular tiling enclosing the core and its boundary mirror. The mirrors M1, M2, M3 for the first three stages are shown in the schematic diagram of Fig. C3.1.1. The structure suggests a two-dimensional onion whose concentric layers increase exponentially in thickness from the center outward.
In the Penrose SUN and STAR, the analogous reflection of the pentagonal region generated up to a given stage leaves five empty triangular gaps, which are magically filled by lateral reflection of the newly generated tiled regions adjacent to the gaps (cf. Figs. C1.4 and C1.5). In the case of RP7 tilings there is no such magic. Instead, as illustrated below in Fig. C3.1.1, in addition to gaps (green), overlaps (red) are created in each new heptagonal annulus. Each gap-overlap pair must be replaced by an orderly ad hoc arrangement of tiles called a wedge, in order to connect the adjacent tiled regions (gray) seamlessly. Beginning with the second stage, after the gap-overlap pair is replaced, the remaining portion (gray) of the reflected tiling in the interior of each heptagonal mirror is called the remainder.
In designing a wedge, it is inevitably found necessary to include parts of the gray areas that fall outside the boundaries of the red and green regions of Fig. C3.1.1. After the second recursion stage, wedge design is somewhat challenging, because of the large number of rhombs involved.
I assume without proof that it is possible to construct a wedge at every stage of recursion.
Fig. C3.1.1
Schematic diagram of the first three stages of a RP7 tiling.
M1, M2, and M3
are the boundary mirrors of these three stages.
Fig. C3.1.2 illustrates the difference between odd and even n in the expansion of
the kth central polygonal region.
odd n even n
Rk is the circumradius and rk is the inradius of the kth central polygonal region.
The linear expansion ratio fn = Rk+1/Rk
= rk+1/rk
for the concentric polygonal regions of consecutive stages is equal to
1 + 2 cos π/n for odd n ≥ 3
and
3 for even n.
f7 ≅ 2.8019
Although I conjecture that RPn tilings for odd n can in principle be extended indefinitely from the center outward, in practice they are restricted to only a few stages, because the time required to design a wedge eventually becomes prohibitive. At each stage, the tiled area increases by the factor g7 = f72 ≅ 7.85. Note that g72 ≅ 62, g73 ≅ 484, g74 ≅ 3799, and g75 ≅ 29,825.
At the center of every RP7 tiling (e.g., pointed tiling (1,1,0)P) lies a d7-symmetric core enclosed by a heptagonal necklace composed of seven beadstrings (Conway worms), each of which contains one or more beads (cf. Figs. C3.3.1 and C3.3.2). A bead is a d2-symmetric polygon, not necessarily convex, tiled by prototile rhombs. If every bead — with the possible exception of a bead at the center of the beadstring (central bead) — is a single rhomb, the beadstring is called minimal. (Central beads are discussed in § C3.7.) The regular heptagon whose edges bisect the beadstrings is called a mirror (e.g., the green heptagons in pointed tiling (1,1,0)P). In many of the examples shown below, beadstrings are distinguished by color from other components of the tiling.
Any radial line through the center of the tiling is called a central ray. Beads related by reflection in a central ray are called conjugate beads. (Every central bead is self-conjugate.)
The boundary curve of every beadstring necessarily has d2 symmetry.
It is symmetrical by reflection in both
(a) the mirror edge that bisects it, and
(b) the perpendicular bisector of that mirror edge.
The tiling of a non-central bead need not be symmetrical, so long as it is related by reflection to the tiling of its conjugates. But it is found that most non-central beads can be tiled in an arrangement with d1 symmetry (symmetry by reflection in a line perpendicular to the mirror edge associated with the bead). Only in central beads is the tiling forced — by the d7 symmetry of the tiling — to have d1 symmetry. Central beads are found only in beadstrings whose signatures (cf. § C3.4) have at least one odd component (cf. § C3.7). If the signature has exactly one odd component, the central bead may be either simple, i.e., tiled by a single rhomb, or compound, i.e., tiled by more than one rhomb (cf. § C3.7). If the signature has more than one odd component, the central bead must be compound. Every simple bead has d2 symmetry. A central compound bead has d1 symmetry, but a non-central compound bead need not be symmetrical.
As an example, consider the beadstrings of the three necklaces in the three-stage blunt tiling (0,0,2)B-2. Although there is no central bead in the beadstrings of the innermost necklace, the beadstrings of each of the next two necklaces have a central bead. The signatures (cf § 3.4) of these first three beadstrings are (0,0,2)B, (1,2,4)B, and (4,7,10)B, respectively.
In some of the tiling examples in § C3.12, every beadstring is minimal, i.e., contains at most one compound bead. In other examples in § C3.12 (e.g., Fig. C3.12.1), some beadstrings contain compound beads that could equally well be replaced by two or more simple beads.
If the boundary of every bead were not symmetrical by reflection in the mirror edge e associated with it, the tiling on opposite sides of the beadstring would not be related by reflection in e. But it is not necessary for the boundary of individual beads to have d2 symmetry, so long as the entire beadstring is symmetrical by reflection in e. Fig. C3.13.3 shows an example of a RP5 tiling in which there is a bead whose tiling is unavoidably asymmetric.
I consider here principally two types of RPn and RP2n tilings — pointed and blunt.
I call a tiling homogeneous if it is either pointed or blunt, and
mixed if some of its necklaces are pointed and some are blunt (cf.
the mixed RP2n tiling for n = 5 in Fig. C3.14.2).
In pointed RP7 tilings (cf. Fig. C3.3.1), beadstrings are terminated at each end by a bead vertex (call it a vertex of type V) that coincides with a vertex of the heptagonal mirror (red). Adjacent beadstrings are joined at type V bead vertices. In some cases, depending on the shape of the terminal beads of the beadstring, they may also be joined along a common edge.
In blunt RP7 tilings (cf. Fig. C3.3.2), a corner rhomb that is congruent to rhomb 1 (cf. Fig. C3.0.3) is inserted at each end of every beadstring. The pair of corner rhombs lengthens the mirror edge associated with the beadstring by tan π/n (cf. Fig. C3.3.3).
Fig. C3.3.1
for the pointed tiling (1,1,0)P
Fig. C3.3.2
The core (orange and blue), mirror (red heptagon), and necklace
(closed chain of seven 3-rhomb convex beadstrings)
for the blunt tiling (0,0,1)B
A corner rhomb (blue) is inserted at each end of every blunt beadstring.
The corner rhomb ρ0 incident at each end of every blunt beadstring
(cf. Fig. C3.3.2)
If a tiling is pointed, its signature, beads, beadstrings, and mirrors are also called pointed (with a similar convention for blunt tilings). The vertices of pointed mirrors coincide with bead vertices, but each vertex of a blunt mirror lies in the interior of a corner rhomb.
As shown in Fig. C3.3.2, blunt beadstrings are not incident at mirror vertices. Every blunt beadstring is terminated by a bead edge (call it an edge of type E) that is orthogonally bisected by the associated mirror edge. Adjacent blunt beadstrings share a vertex (call it a vertex of type V*) that does not coincide with a vertex of the heptagonal mirror. In some blunt tilings, depending on the shape of the bead at the ends of the beadstring, adjacent beadstrings may also share a common edge. Every corner rhomb shares an edge and a vertex with each of its two adjacent beadstrings. A corner rhomb is regarded as part of the necklace but not as part of a beadstring.
It is clear from Fig. C1.4 that the Penrose STAR can be classified as a blunt tiling.
The Penrose SUN (cf. Fig. C1.3), which is neither blunt nor pointed, could perhaps be called overlapping blunt.
Both tilings are homogeneous.
A RP7 tiling grows recursively by outward expansion from its core via successive reflections in the edges of concentric heptagonal mirrors of exponentially increasing size. At each reflection, the entire tiling in the interior of the outermost mirror is reflected, except for rhombs in the necklace beadstrings that are bisected by edges of that mirror.
The composition of each beadstring in the outermost necklace at the kth stage of recursion is specified by the kth signature
where σk(1), σk(2), and σk(3) are non-negative integers. σ1 is called the initial signature. If σ1 = (0,0,0), the tiling is called the null tiling.
For both pointed and blunt beadstrings, the three components of the signature σk are defined as follows:
σk(j) is the number of boundary edges with projected length sin jπ/7 (j=1,2,3)
on the mirror edge that bisects the beadstring
(cf. Fig. C3.4.1).
In a beadstring, rhomb j contributes 2sin jπ/7 to the total length of a mirror edge.
In either the left or right half of each horizontal beadstring in Fig. C3.3.1, for example, one of the two contributing boundary edges has projected length sin π /7 and the other has projected length sin 2π /7. Hence the signature is (1,1,0)P.
Similarly, in Fig. C3.3.2, in either half of each horizontal blunt beadstring, the single contributing boundary edge has projected length sin 3π /7, and the signature is therefore (0,0,1)B.
It is convenient to assign signatures to beads according to the same convention as for beadstrings.
Now define the vector
In pointed tilings, (λk)pointed — half the length of an edge of the kth mirror — is equal to the scalar product of σk and S:
In blunt tilings, (λk)blunt — half the length of an edge of the kth mirror — is equal to the scalar product of σk and S plus the term (1/2) tan π/7 (≅ 0.2403) contributed by one corner rhomb (cf. Fig. C3.5.2).
The contribution (1/2) tan π/7 of a corner rhomb to
the half-edge length (λk)blunt of a blunt mirror (cf. Fig. C3.3.2)
I discovered and proved the following trigonometric identity, which makes it possible to treat both pointed and blunt tilings in a unified way:
= (1,1,− 1) ⋅ S
= δ ⋅ S,
Hence
= τk ⋅ S,
The expansion of the tangent in sines for n = 7 led me to conjecture that for all odd n ≥ 3,
where
For n = 3, 5, 7, 9, 11, …, this expansion takes the forms
(1/2) tan π/3 = sin π/3
(1/2) tan π/5 = − sin π/5 + sin 2π/5
(1/2) tan π/7 = sin π/7 + sin 2π/7
− sin 3π/7
(1/2) tan π/9 = − sin π/9
+ sin 2π/9 + sin 3π/9 − sin 4π/9
(1/2) tan π/11 = sin π/11 + sin 2π/11 −
sin 3π/11 − sin 4π/11 + sin 5π/11
(1/2) tan π/13 = − sin π/13 + sin 2π/13 +
sin 3π/13 − sin 4π/13 − sin 5π/13 + sin 6π/13
…
The pattern of signs in this set of equations is
+
− +
+ + −
− + + −
+ + − − +
− + + − − +
+ + − − + + −
− + + − − + + −
…
When I described to Bob Fitzgerald the conjecture that the tangent expansion in sines is valid for all odd n ≥ 3, he promptly proved it. Here's Bob's tidy proof.
Although it's not relevant to the present discussion, it's also true that for odd m,
(Bob's tidy proof requires only a trivial addition to prove this result.)
The recursive relation between the signatures σk and
σk+1 of consecutive beadstrings
is expressed by the symmetric tri-diagonal linear expansion matrix E3 shown in Fig. C3.4.3.
For odd n, let m = (n − 1)/2.
E3 is the n = 7 version
of the m x m linear expansion matrix Em
that is applicable to all RPn tilings for odd n.
All of the non-zero elements eij of Em are equal to 1, except for
emm = 2.
Fig. C3.4.3
The linear expansion matrix E3 for n = 7
For pointed tilings,
and for blunt tilings,
Here is proof that for linear recursion via the matrix E3 between signatures of consecutive pointed n = 7 beadstrings, the ratio of the edgelengths of the associated mirrors is equal to the linear expansion ratio fn (cf. § C3.1).
And here is proof that for linear recursion between extended signatures of consecutive n = 7 blunt beadstrings via the matrix E3, the ratio of the edgelengths of the associated mirrors is also equal to the linear expansion ratio fn (cf. § C3.1).
Let
where ρk(j) = the number of specimens of rhomb j (j = 1,2,3) in the interior of the mirror Mk of generation k (cf. Fig. C3.1.1),
and let Ak = the area in the interior of Mk.
Here is a proof that if
which is equal to
,
Fig C3.5.1
then
It might appear that the appropriate first step in the construction of a RP7 tiling would be to choose a core. As it happens, however, only one-third of all possible initial signatures allow the construction of recursively related necklaces. Consequently, a more prudent course is to choose the initial signature first, then construct the first necklace, and finally tile the core.
An initial signature σ1 = (σ1(1), σ1(2), σ1(3)) is called allowed if and only if
the beadstring with signature σ2 = (σ2(1), σ2(2), σ3(3)) contains two replicas of the beadstring with signature σ1 — one at either end.
Hence σ2(1) ≥ 2σ1(1),
σ2(2) ≥ 2σ1(2),
and σ2(3) ≥ 2σ1(3).
An initial signature that is not allowed is called forbidden.
I have proved that
for pointed tilings, the initial signature σ1 = (σ1(1), σ1(2), σ1(3)) is allowed if and only if
σ1(1) + σ1(3) ≥ σ1(2) ≥ σ1(1),
σ1(1) + σ1(3) ≥ σ1(2) ≥ σ1(1) − 1.
For both pointed and blunt tilings, it is readily proved that if a given signature
σk is allowed,
its successor σk+1
is allowed. Hence the signature of every descendant of an allowed initial signature is allowed.
No example is known of an initial signature — whether allowed or forbidden — for which no core can be designed, but it has not been proved that such signatures do not exist.
I have proved that for pointed tilings, the number of allowed initial signatures for which 0 ≤ σ1(i) ≤ n (i=1,2,3) is
Since the total number of pointed signatures for which
0 ≤ σ1(i) ≤ n (i=1,2,3) is (n+1)3,
it follows that only one-third of all possible initial pointed signatures are allowed.
A similar result holds for initial blunt signatures.
The parity Π of a beadstring signature σk = (σk(1), σk(2), σk(3)) is defined as
Signature parity determines whether or not a beadstring incorporates a central bead (cf. § C3.2) and — if it does — whether that bead is simple or compound.
Π(σk) has eight possible values. Parity (0,0,0) is called even; the seven other parities are called odd.
(1,0,0)
(0,1,0)
(0,0,1)
(1,1,0)
(1,0,1)
(0,1,1)
(1,1,1)
The catalog of initial signatures shows examples of both pointed and blunt beadstrings of each of the eight parity classes.
In beadstrings with even parity (0,0,0), there is no central bead.
In beadstrings with odd parity (1,0,0), (0,1,0), or (0,0,1), there is a central bead, and a simple bead will suffice.
In beadstrings with odd parity (1,1,0), (1,0,1), (0,1,1), or (1,1,1), there is a central bead, and it must be compound.
The catalog of initial signatures includes cores, beadstrings, and mirrors for every initial signature σ1 — pointed or blunt, allowed or forbidden — for which 0 ≤ σ1[i] ≤ 2 (i = 1,2,3). Each signature σ1 is followed by the signature of its daughter σ2. If σ1 is forbidden, σ2 is marked with an asterisk. Inside the necklace defined by each initial signature is an example of a tiled core. The order in which the rhombs are placed in each beadstring and also the arrangement of rhombs in each core are uniquely determined only for the smallest beadstrings.
In each of the catalog illustrations from Fig. 1 to Fig. 26, the figure labelled "a" shows a pointed beadstring, and the figure labelled "b" shows a blunt beadstring, but the cores for the two figures are identical. For the signature (2,2,2) in catalog Fig. 27, beadstrings and cores for each of the six possible bead sequences are shown for both pointed and blunt tilings.
Fig. C3.7.1 shows four pointed compound beads of minimum area. Every blunt bead, whether simple or compound, can be constructed simply by attaching a string of parallel rhombs to the boundary of the bottom half of the corresponding pointed bead.
Fig. C3.7.1
The smallest possible pointed compound beads for beadstrings of odd parity
For pointed signatures
(a) if the initial signature has even parity (0,0,0),
all subsequent signatures have parity (0,0,0).
(b) if the initial signature has odd parity,
subsequent signatures cycle repeatedly through the seven odd parities
in counter-clockwise order as listed in Fig. C3.7.2a.
Fig. C3.7.2a
One period of the [counter-clockwise] odd-parity sequence for pointed beadstrings
For blunt signatures,
(a) if the initial signature has odd parity (1,1,1),
all subsequent signatures have parity (1,1,1).
(b) if the initial signature has parity ≠ (1,1,1),
subsequent signatures cycle repeatedly through the seven parities
listed in Fig. C3.7.2b, in counter-clockwise order.
Fig. C3.7.2b
One period of the [counter-clockwise] seven-term parity sequence for blunt beadstrings
I conjectured as a result of numerical experiments that iff n is prime, the sum
where c1, c1, … c(n-1)/2 are rational numbers, is equal to zero iff
I was unable to prove the conjecture, but Bob Fitzgerald made light work of this one too! Here's Bob's proof.
The conclusion, then, is that the composition of prime n beadstrings that are recursively related via
the expansion matrix E3 (cf. § C3.4)
is unique. But because of the peculiar coincidence that not only the edge length but also
the area of each polygonal mirror is expressed as a linear combination of sines
with rational coefficients, it follows from
Bob's proof that the populations of the three rhombs
contained inside each mirror are also unique.
After constructing the innermost necklace, the first task is to design the core of the tiling (cf. § C3.6). So long as the beadstrings of the innermost necklace are moderately short, designing a core tiling is quite easy. But it may be helpful in any case to know how the composition of the radial pendants of the core depends on the composition of the beadstring.
… (to be continued)
The first step in the construction of a wedge is the design of a beadstring for the necklace that will enclose the entire enlarged tiling. It is not obvious without proof that in every generation there exists a string of beads of precisely the right length to bridge the gap (base of green isosceles triangle in Fig. C3.1.1) between the two beadstring replicas inherited from the previous generation. That such a beadstring always exists (and is of unique composition) is proved in § C3.4 and § C3.8.
Although a mathematical description of the recursive structure of beadstring composition is slightly simpler for pointed tilings than for blunt tilings (cf. § C3.4), it is no more difficult to design wedges for blunt tilings than for pointed tilings.
In the three-generation pointed tiling (1,1,0)P, the arrangement of rhombs in the wedge just below the bottom vertex of the innermost mirror is identical to the arrangement of rhombs just beyond each vertex of the next larger mirror. Although it is not essential to design each wedge so that it incorporates a portion of the wedge of the previous stage, it does save labor to do so. The three-generation blunt tiling (2,2,1)B-1 provides an additional example.
The concentric regular heptagons M1, M2, and M3 in the diagram in the middle of Fig. C3.11.1 represent the three innermost mirrors of the tiling. (The colored tilings at the left and right sides of the central image should have been rotated by π/7 about their centers.)
Fig. C3.11.1
'Proof [almost] without words'
produced by reflection in the mirror edges at each stage
define a ring of fourteen stars
with centers at the vertices of a regular 14-gon.
C3.12 Gallery of RP7 tilings
When you examine some of the images listed below, if you are able to freeze the display mid-course by right-clicking with the mouse, you may be able to see what overlaps look like before they are replaced by wedges. (Pointed tiling (1,1,0)P-8 is a good example. After the image is displayed, toggle between zoom up and zoom down.)
In the tiling directly below in Fig. C3.13.1, mirror edges are shown as faint line segments superimposed on necklace beadstrings. For those images in which mirror edges are not drawn (e.g., Fig. C3.13.2), it may be difficult to identify the necklaces of the inner stages. It will probably help if you work your way recursively backwards from the outermost necklace, which is always easy to recognize because it is incident at the pattern boundary.
Fig. C3.13.1
n = 3
(1)P
f3 = 2 (cf. § C3.1).
four stages
p3m1
pdf version
Fig. C3.13.2
n = 5
(1,2)P
f5 = 1 + φ (≅ 2.618)
three stages
d5
pdf version
The signatures of the first few beadstrings are
(1,2)
(3,5)
(8,13)
(21,34)
(55,89)
…
In the schematic diagram at the right
(and in all similar diagrams),
gaps (cf. § C3.1) are green.
.gif)
Fig. C3.13.3
n = 5
(1,2)B
f5 = 1 + φ (≅ 2.618)
three stages
d5
pdf version
Fig. C3.13.4
n = 9
(0,0,1,2)B
f9 ≅ 2.879
(two stages)
d9
pdf version
In the schematic diagram at the right,
overlaps are red and gaps are green.
The linear expansion ratio fn is equal to three for RP2n tilings for both odd and even n (cf. Fig C3.1.2). A necessary condition for the existence of blunt RP2n tilings is that tan π/n can be expressed as a linear combination, with integer coefficients, of sines of integer multiples of π/n (cf. § C3.4). This requirement is met by odd n but not by even n. Hence there are no blunt RP2n tilings for even n. The anomalous 'mixed' RP2n tiling for n = 5 in Fig. C3.14.3 can be regarded as an example of a blunt tiling if the pointed necklace of stage 1 is ignored.
For the 'crystallographic integers' n = 2, 3, 4, and 6, RPn tilings are found to have translation symmetry. The RP2n tiling in Fig. C3.14.1 immediately below has translation symmetry if the coloring is ignored.
Fig. C3.14.1
RP2n tiling (1)P for n = 3
three stages
f2(3) = 3
p6m
pdf version
RP2n tiling (1,0)P for n = 5
three stages
f2(4) = 3
d10
pdf version, showing mirrors
Fig. C3.14.3
RP2n tiling (0,1)mixed for n = 5
three stages
f2(5) = 3
d10
pdf version, showing mirrors
RP2n tiling (1,0,0)P for n = 7
three stages
f2(7) = 3
d14
pdf version
RP2n tiling (0,1,0)P for n = 7
three stages
f2(7) = 3
d14
pdf version
Fig. C3.15.1
RP4 tiling (1,1)P for n = 4
three stages
f4 = 3
p4m
pdf version
Fig. C3.15.2
RP6 tiling (0,1,0)P for n = 6
three stages
f6 = 3
p6m
pdf version
P.gif)
Fig. C3.15.3
(1,1,0)P for n = 6
three stages
f6 = 3
p6m
pdf version
P.gif)
Fig. C3.15.4
RP6 tiling (0,1,1)P for n = 6
three stages
f6 = 3
p6m
pdf version
Unsuccessful attempt to produce second stage of RP8 tiling (1,1,0,0)P for n = 8
f8 = 3
(a) upper left
Diagram of overlaps and gaps for two-stage tiling
(b) upper right
Tried to extend the tiling from the (1,1,0,0)P core, but was unable to generate
second-stage necklace with the required symmetry.
Constructed a proper second-stage necklace, but that forced the replacement
of most of the tiling outside the core, with the result that
the first-stage necklace no longer had the required symmetry.
The tiling in (b) is visible underneath this one.
(d) lower left
Tried to use the entire region inside the outermost necklace of (d)
as a core for the tiling (3,3,0,0)P.
Failed, because after this region was reflected in its outermost mirror, the scenario in (c) was repeated:
most of the reflected tiling (not shown), including the current outermost necklace, had to be replaced.
The tiling was therefore abandoned.
When an experiment similar to this one was carried out for n = 12,
the results were similarly unsuccessful, suggesting (without proof!) that
for even n > 6, RPn tilings may be impossible.
Fig. C3.16.1
Three-stage RP2n tiling (1)P for n = 2
f2(2) = 3
p4m
The [degenerate] beadstrings and their associated mirror edges are identical in this example.
pdf version
Fig. C3.16.2
Three-stage RP2n tiling (10)P for n = 4
f2(4) = 3
d8
pdf version
Fig. C3.16.3
Three-stage RP2n tiling (1,0)P-1 for n = 4
f2(4) = 3
d8
Three-stage RP2n tiling (1,0)P-2 for n = 4
f2(4) = 3
d8
Fig. C3.16.5
Three-stage RP2n tiling (1,0)P-3 n = 4
f2(4) = 3
d8
Aside from color differences, this tiling is the same as the one in Fig. C3.16.2.
Three-stage RP2n tiling (0,1)P for n = 4
f2(4) = 3
d8
pdf version
This tiling resembles — but is not identical to — the 1977-1982 'Ammann-Beenker tiling'
illustrated in the Tiling Encyclopedia of Dirk Frettlöh and Edmund Harriss.
Fig. C3.16.7
Three-stage RP2n tiling (0,1,0)P for n = 6
f2(6) = 3
d12
pdf version
Fig. C3.16.8
Three-stage RP2n tiling (0,1,0,0)P for n = 8
f2(8) = 3
d16
pdf version
In his legendary 1977 essay, Martin Gardner introduced
Penrose tilings in his Scientific American column,
Mathematical Recreations.
SUN and STAR
show the central regions of the two d5-symmetric Penrose tilings.
(These images were derived as duals of
de Bruijn pentagrids.)
In 1981, Nicolaas G. de Bruijn proved that the dual of
every regular pentagrid that satisfies a simple
restriction on the translational shifts γj of the five grids of
the pentagrid is identical to a Penrose tiling
created by arranging rhombs constrained by Penrose's matching rules.
In 1977, after studying Gardner's article, I persuaded a few students
to join me in the study of Penrose tilings.
I ordered steel-rule dies for cutting kites, darts,
and long and short bow-ties from
color-printed cardstock, and we soon amassed a large supply of die-cut tiles.
I decided to focus on the three special Penrose tilings:
SUN (d5 symmetry),
STAR (d5 symmetry),
and
CARTWHEEL (d1 symmetry).
When I examined the image of the CARTWHEEL tiling,
I noticed that each of its ten triangular sectors contains
an infinite alternating sequence of successively larger replicas of the central portion
of the SUN or the STAR.
I also observed, but did not prove, that for any two consecutive replicas in this sequence,
the distance from the center of the tiling to the center of the
replica increases by a multiplicative factor equal to the golden ratio.
Looking at the growth outward from the center of the SUN and STAR, I noticed an apparently
recursive structure, which is described below.
I tried unsuccessfully to prove that this recursion continues
beyond its first few iterations.
In the April 1978 Notices of the American Mathematical Society
I published a preliminary report on this topic, although I failed to state that
I had not actually proved anything!
In late 1978,
I welcomed two surprise visitors at my home:
Hank Saxe and Cynthia Patterson, ceramic artists extraordinaire
from Taos, New Mexico.
Hank had been my student and close friend at CalArts several years earlier.
I insisted on holding my two guests (and their golden retriever)
hostage until they agreed to
consider designing and making Penrose patterns from ceramic tiles.
I knew that Hank was expert in the production of colored ceramic glazes,
and I was convinced that together Hank and Cynthia would produce spectacular
works of mathematical art. They didn't disappoint me.
They quickly obtained permission from Roger Penrose to make ceramic versions of his tilings,
and you can see samples of their work on their
website .
In 1979, I encouraged my students to make a 24' x 24' CARTWHEEL tiling by gluing colored paper
rhombs (2 cm. edgelength) onto thirty-six 4' x 4' sheets of masonite. We planned to use the panels
for a traveling Penrose exhibit at Illinois high schools.
When the project was still unfinished at the end of the semester,
we stored the panels in our department building.
Unfortunately, the panels — and other property
stored in the building — mysteriously disappeared during the end-of-semester break,
and work on the project was never resumed.
In those days, I hadn't yet heard of Ammann's work on
aperiodic tilings. It wasn't until 1981 that Nicolaas de Bruijn published his
pair of ground-breaking monographs explaining the
connection between multigrids and Penrose tilings.
(At a Cincinnati AMS meeting soon afterward, David Klarner invited me to spend the evening
with him and his friend de Bruijn, who graciously gave me
copies of his multigrid monographs.)
Now for the first time it became possible to construct Penrose tilings
without having to proceed step-by-step, following matching rules
and hoping you wouldn't end up at an impasse and have to backtrack.
Until de Bruijn's breakthrough, matching rules were the principal device
for forcing aperiodicity.
I also benefited from reading the early articles by the physicist Paul Steinhardt and his students.
This was still several years
before Shectman and his colleagues at NIST astonished the world by discovering three-dimensional quasicrystals,
confirming the earlier expectations of Roger Penrose and Alan Mackay (and perhaps nobody else!).
I naively wondered whether one could find a set of matching rules
for tilings by rhombs for n = 7 that would
also force aperiodicity, i.e., allow only non-periodic tilings.
I tried everything I could think of,
but nothing worked. It was only later that I learned that such matching rules
had been proved impossible for n = 7. I was intrigued by
tilings by the three rhombs of SRI7, but
I recognized that tilings by those rhombs — or
any other non-Penrose rhombs — lack the special
charm of authentic Penrose tilings based on the magic number five.
One day in December 1979 I suddenly decided to invent a
new tiling puzzle, by somehow marrying the idea of a polyomino
(the brainchild of Sol Golomb) with the idea of tiling
a regular 2n-gon by
a set of n(n − 1)/2 rhombs, as described by Donald Coxeter.
I decided to experiment by constructing a twin,
in every possible edge-to-edge configuration, from
every possible pair of rhombs in Coxeter's set
of n(n − 1)/2 rhombs.
When I tested this idea for every n ≤ 10, I
was startled to discover that so long as one adds exactly
one specimen of each single rhomb ('keystone') to the collection of twins,
the combined area of all the pieces is precisely equal to that of the regular 2n-gon.
I quickly proved that this holds for all n ≥ 3.
I found that I was able to tile the interior of the regular 2n-gon
with some arrangement of the pieces of every set of order n ≤ 12.
Of course it took a lot longer for the larger values of n.
At first I called the puzzle 'CYCLOTOME', but later I shortened the name to 'ROMBIX'.
Several years and a couple of patents later, injection-molded ROMBIX sets
composed of the sixteen pieces for n = 8
were manufactured in China and marketed in the U.S.
The manufacturer was unwilling to spend any money to advertise them, however,
and production stopped after a couple of years.
But in 1991 Kate Jones of KADON
made a laser-cut acrylic version of ROMBIX that is still sold today.
Back to RP7 tilings
Immediately below is a 2005 photo of a paper RP7 tiling I started to paste in 1981.
It's the same tiling as
(0,0,1)B-1,
(0,0,1)B-2,
(0,0,1)B-3, and
(0,0,1)B-4.
After a couple of weeks of cutting and pasting, it was still barely half finished.
But I became bored with it and left it in this unfinished state for the next twenty-four years.
When I examined the unfinished panel,
it suggested some
mathematical questions that I probably hadn't thought much about in 1981 but
that now seemed to demand answers.
I've since found answers for most — but not all — of these questions.
Many of the questions and answers are discussed in § C3.1-C3.8.
The most awkward of these questions is how to prove that as the pattern grows radially outward, it can retain the
d7 symmetry of the central nucleus (core) at every stage of recursion,
i.e., that a proper wedge exists at every stage (cf.§ C3.1 and § C3.10).
I still can't answer this question,
but at least it can now be said (cf.§ C3.4 and § C3.8) that for both pointed and blunt tilings,
(a) the d7-symmetric
necklaces that decorate the heptagonal boundary mirrors
can be constructed at every stage of recursion, and
In practice it hardly matters
whether one can prove that a tiling can be extended symmetrically outward forever,
since the tiling expands so rapidly at each stage that
it is unrealistic to consider physical tilings (or even mere computer images!)
that exceed four or five stages.The precise number
of stages depends, of course, on whether
you're planning to cover just the floor of a room or
an area the size of Rhode Island.
RP7 tilings are in a very rough sense self-similar centro-symmetric tilings
of the Euclidean plane by rhombs.
The three prototile rhombs of RP7 tilings are unmarked, and there are
no matching rules.
The two prototile rhombs of Penrose tilings,
by contrast, are marked
in such a way as to prevent periodic tilings
and allow only aperiodic tilings,
when Penrose's matching rules are imposed on the marked tiles.
Although there is a superficial connection between RP7 tilings and
the two most symmetrical examples of Penrose tilings,
the d5-symmetric SUN and STAR,
none of the subtle features of Penrose tilings are found in RP7 tilings.
These features include:
(a) quasiperiodicity,
which implies Conway's town theorem ('local isomorphism'),
Fig 3.18.1 is a cross-eyed stereoscopic view of (1,0,0)B
transformed into a
stepped pyramid.
Using the n(n − 1)/2 rhombs of SRIn as prototiles, it's easy to
construct examples of periodic tilings in which there are
no embedded rosettes (aside from the rosette
composed of a single square when n is even).
Here is an example of such a tiling for n = 7 :
For n = 2, 3, 4, and 6, there exist RW tilings that I call
the trivial examples. They are based on
two of the three regular tilings:
This question is related to a slightly more restricted one:
Imagine an infinite orchard of identical trees arranged on a square lattice.
Let ρtree = the tree density (number of trees per unit area).
A straight cable is stretched from each tree to 2n other trees,
which are called its 'connected neighbors'.
Each cable is incident at no tree other than the two at its ends.
The arrangement of the 2n connected neighbors ('connected-neighbor
configuration') is identical, up to rotation about a vertical axis, for all trees.
The number of distinct crossed pairs of cables ('nodes') at each point
at which m cables cross is equal to m(m − 1)/2.
Let ρnode = the node density (total number of
crossed cable pairs per unit area of the rhombic tiling dual to the set of nodes),
and call the ratio ρnode/ρtree
the tree-normalized node density.
This orchard model is a paraphrase of the 'star grid' scheme for
generating RW tilings that is described below.
In the first three of the four examples shown above (Figs. C4.0.1b-C4.0.3b), the rosettes of order n
have maximal density,
i.e., they occupy the largest possible fraction of the tiling area
among all RW tilings of the same order.
But the density of the tiling for n = 6 in Fig. C4.0.4b
is almost seven percent less than that of the kagome tiling
in Fig. C4.0.5, which is conjectured to have maximal density.
For small n, there are a few empirical rules — described below — that
predict which RW tiling has maximal density.
For every n>2, there is an uncountable infinity of RW tilings in which the
density of the rosettes of order n is less than maximal.
I describe below a systematic procedure, adapted from
Gessel's algorithm. for rosettes and from
de Bruijn's multigrid algorithm
for aperiodic tilings,
for generating RW tilings of both maximal and sub-maximal density.
For n ≠ 2, the density can be made arbitrarily close to any
value less than or equal to the maximal density, by using the splitting and augmenting
operation illustrated in § 4.8.
In any aperiodic tiling for n ≥ 3 that is the
dual of a de Bruijn multigrid,
one cannot help noticing the not necessarily symmetrical rosettes of order
n that are embedded here and there.
(Here is an example for n = 7.)
The occurrence of rosettes is statistically inevitable in these aperiodic tilings.
A rosette is the dual of the k(k − 1)/2 points of
intersection of any set of k lines in which each line
intersects every other at a unique point.
In every de Bruijn pentagrid, arrangements of five lines
that satisy this condition occur infinitely often, since Conway's
town theorem guarantees that the portion P of the tiling dual to every such arrangement is
no farther from its nearest replica than slightly more than
twice the diameter of P.
Here is an example of one of the six five-line arrangements in a pentagrid:
These six line arrangements define the following six rosettes:
For n > 5, without analogs of Conway's n = 5 town theorem
we don't have an upper bound on the distance between a rosette that is
embedded in a de Bruijn aperiodic tiling and its nearest replica.
(Incidentally, for n > 5, the number of ways a rosette of order n
can be tiled by the rhombs of
SRIn is unknown. But I have found that for n = 6,
this number is at least 49.)
The scheme I have devised to explore the rosette density problem
for n ≥ 3 can be used to create an uncountable infinity of RW tilings
containing embedded rosettes of order n.
It generates a 'star grid' from one or more 'lattice stars'. Every intersection
of two lines of the star grid is the dual of a rhomb
in a RW tiling.
I'll first describe the scheme for single-lattice tilings,
in which a rosette of order n is centered at every lattice point of the tiling.
There are no other such rosettes in the lattice fundamental domain.
Then I will describe the version for poly-lattice tilings,
in which a rosette of order n is centered
at each of m points (m>1) in
the lattice fundamental domain. In this case, the rosettes are centered at the lattice points of
m congruent sub-lattices of the tiling. (The kagome tiling in Fig. C4.0.5 is an
example of a poly-lattice tiling with m = 2.)
Scheme for single-lattice tilings
Scheme for poly-lattice tilings (m congruent sub-lattices)
For the unit square lattice (integer lattice),
we define the magnitude of a lattice star — using the conventions of
taxicab geometry — as the sum of
the 'manhattan lengths' of the 2n rays of the star.
By this measure, the lattice star in Fig. C4.0.8, for example, has magnitude 44.
For RW tilings for small n on either square or rectangular lattices,
rosette density and star magnitude are found to be inversely correlated.
Examples are shown in § 4.8. When two different lattice stars for the same n
have the same magnitude, the rosette density is usually found to be
larger in the tiling for which the tree-normalized node density
in the star grid is smaller.
(If the tree-normalized node density were replaced by a weighted count of nodes,
each weight being the area of the rhomb dual to each node, then the
tree-normalized node density would be exactly inversely correlated.)
1. A tiling with the highest density of any tiling
of order n is called
superdense.
2. A tiling with the highest density of any tiling
of order n of its type is called
dense.
3. A tiling with density less than the highest density
of any tiling of order n of its type is called
sparse.
Although all of the tilings for n<9 conjectured to be superdense
have reflection symmetries, the tiling for n = 9 (cf. Figs. C5.9.1a and b)
that is conjectured to be superdense does not — it has only p3 symmetry.
Its density is almost 30% larger than that of the tiling with p3m1 symmetry in Fig. C4.10.2.
Every RW tiling treated here is characterized as belonging to either the row class or the dispersed class.
Within the row class, there are two sub-classes: straight and zig-zag.
Within the dispersed class, there are three sub-classes: square lattice,
hexagonal lattice, and other lattices.
This classification scheme is by no means exhaustive.
The kagome tiling in Fig. C4.0.5 belongs to neither dispersed nor row class.
Neither do tilings in which all the rosettes are arranged in closed rings, with every
rosette incident at each of two neighbors.
In another example, some rosettes are incident
at no other rosettes, and still other rosettes are arranged in rows.
I have not investigated examples of every one of the types listed below.
Except for a few small values of n, I have not proved
maximal density. The labels 'dense' and 'superdense' should be regarded as tentative
except where stated otherwise.
A RW tiling may also be categorized according to the residue class of
n, e.g., even, odd, congruent to 3 (mod 6), etc.
For every tiling, whether it is of the
dispersed or row class, the wallpaper group is identified.
But at every point of intersection of three or more lines in a star grid,
the orientation in the tiling plane and — for
four or more intersecting lines —
the symmetry of the arrangement of rhombs
inside the convex polygon ('oval')
dual to the point of intersection is indeterminate. An arbitrary choice
of orientation (and also of symmetry, when four or more lines are involved) must be made in each such case.
The wallpaper group of the tiling depends on precisely which choices are made.
For some families of dense tilings of a particular type that are
parametrized by the order n of the tiling,
the dependence of density on n — for small n — can be described
by an algebraic expression conjectured to hold for all n.
In some cases, it is not difficult to guess an asymptotic form for this expression
that is confirmed by numerical calculations,
but I have not proved any of these results. Some of these asymptotic expressions appear following Fig. C4.10.2.
EXAMPLES
Let's now examine examples of tilings of several types,
in increasing order of n.
The tiles are defined to have unit edge length.
In several cases, I have included the lattice star or star grid (or both).
In addition to density ρn, I record the wallpaper group and — for
row tilings — λn, the
distance between the center-lines of adjacent rows.
Some types have no representatives, because I am considering only edge-to-edge tilings,
i.e., tilings in which the corners and sides of the tiles coincide
with vertices and edges of the tiling (cf. Tilings and Patterns, by Grunbaum
and Shephard, p. 18).
C4.1 Straight row tilings, aligned: even n
C4.6 Square lattice tilings: even n
In addition to the very prominent d3 rosettes of order 12,
smaller rosettes of orders 3, 4, and 6 appear
in this pattern.
You can count a total of sixteen rosettes in each translational fundamental region
of the lattice.
(The symmetry of this tiling is reduced by errors in the
orientation of the rosettes.)
C4.7 Square lattice tilings: odd n
C4.8 Rectangular lattice tilings: even n
C4.9 Hexagonal lattice tilings: even n
Below are:
C4.11 Additional examples of RWn
Below are examples of both dense and sparse RWn tilings
for n in the interval [2,22].
In many of the the RW tilings shown here,
each shape of rhomb has a characteristic color (or shading).
Some color schemes
produce striking subliminal patterns—approximations of
circles, triangles, hexagons, etc.
The scale of such patterns is sometimes so large that
an assembly of several unit cells of the tiling may be required to reveal them.
It is characteristic of RWn tilings that
along lines of reflection of the tiling lie infinite linear strings of rhombs
that are analogs of minimal pointed beadstrings
(cf. § C3.2): each bead is composed of a single rhomb.
Some tilings in addition contain finite strings composed of blunt beads, aligned in directions
that do not coincide with lines of reflection of the tiling.
RW21(4) is an instructive example.
The axes of three pointed strings of infinite length,
intersecting at the center of the image, lie on
lines of reflection of the tiling at 0, 60, and 120 degrees from the vertical.
In addition every pair of nearest neighbor rosettes is joined by a finite string of blunt beads.
These blunt strings can only partially mimic the mirroring effects of blunt beadstrings in RPn tilings,
described in § 3.2 and § 3.3. The tiling would be perfectly symmetrical by reflection
in the longitudinal axis of each such blunt string if it were not for the fact that neither
(a) the arrangement of rhombs in the interior of the beads of each string
is symmetrical by reflection in those axes.
In RW21(4), the rosettes would require
d6 symmetry, but symmetry of even order is impossible for rosettes (cf. § B1).
RW9, illustrates the same effects.
n = 3: Regular tiling by hexagons,
each tiled by three congruent rhombs (hexagonal lattice)
n = 7: Not every rosette is centered at a lattice point (rectangular lattice).
I will soon post a picture of the 25" x 38" RHOMBBURST poster,
which includes an article at the bottom summarizing the state of knowledge in 1995 about Penrose tilings.
Unlike RW8(2) (above), the tiling of the
RHOMBBURST poster is not a wallpaper pattern—it is a centro-symmetric
tiling with a single center of symmetry.
It has 16 lines of reflection
and therefore has d16 symmetry. How many lines of reflection do you find in a unit cell
(fundamental domain under translation) of
the periodic RW8(2) (above)?
RW10(1) is a row tiling.
For other images of RW10(1), see
RW10(2) ,
RW10(3)
or
RW10(4) .
Note that exactly halfway between adjacent
rows of large rosettes in RW10(1), there is a string of strawberry rosettes of
order 5. In row tilings of odd order,
there is a medial string of convex polygons, in a regular
alternating sequence, with n-1 and n+1 sides, respectively.
I will soon add an example or two.
In RW10(1), the large rosettes in
adjacent rows are 'staggered' (offset) by a
rosette circumradius.
There also exist 'non-staggered'
row tilings, for both odd and even n.
In row tilings,
the convex 'wave fronts' of tiles that border
large rosettes in one row are transformed
into concave 'wave fronts'
by the time they reach the large rosettes in an adjacent row.
I call the pattern elements that mediate this change
ginkgo leaves.
RW16_row and
RW22_row demonstrate
the hierarchical arrangement of ginkgo leaves in row tilings.
Ginkgo leaves bear a superficial resemblance to
the arbelos (shoemaker's knife) of Archimedes, but while
the circular arcs that define the
boundary of the arbelos do not all have the same curvature,
all three boundary 'curves' of a ginkgo leaf
have the same curvature.
n=16: RW16(1) illustrates how
broken symmetries unavoidably appear when you embed a rosette—which
necessarily has symmetry of odd order—in a
RW tiling with no symmetries of odd order.
(In its present form, this tiling has no symmetries,
but a half-turn rotation of every rosette in alternate horizontal rows
would introduce horizontal lines of reflection.)
In RW16_row, a row of n=8 small rosettes lies
halfway between each pair of adjacent rows of n=16 large rosettes:
Fig. C3.17.1
A paste-up of (0,0,1)B-3, left unfinished in 1981
In 2005, my interest in (0,0,1)B-3 was revived
by a phone call from
Tom Rodgers, the Atlanta impresario who founded — and continues to host — the
now binennial Gatherings for Gardner.
Tom told me that he had seen ceramic versions of d7 rhombic tilings on the
Saxe-Patterson
website and wanted to use such a design as a logo for G4G7.
After Tom's call, I decided to take up cutting and pasting again. Here's what the tiling looks like now:
Fig. C3.17.2
The completed (0,0,1)B-3 paste-up
(b) for prime n the composition of every necklace is unique.
(b) projection from a five-dimensional cubic honeycomb,
(c) a basis for mathematical modeling of the 3-dimensional
quasicrystals
discovered by Dan Shechtman and his collaborators in 1984,
(d) the dual relationship
between the rhombs of a tiling and the vertices of a de Bruijn pentagrid,
(e) derivation by the inflation
of a finite tiling,
(f) matching rules
for kite/dart tilings and tilings by rhombs,
(g) ubiquitous role of the
golden ratio and Fibonacci sequences,
etc.
C4. Rhombic wallpaper (periodic tilings derived from a variant form of de Bruijn multigrids)
C4.0 Introduction
Fig. C4.0.1
Periodic tiling for n = 7 in which
there are no embedded rosettes
4.4.4.4, tiled by squares,
6.6.6, tiled by hexagons,
and on
two of the eleven
uniform tilings:
4.8.8, tiled by squares and octagons and
4.6.12, tiled by squares, hexagons, and 12-gons.
Figs. C4.0.1 a and b
Figs. C4.0.2 a and b
Figs. C4.0.3 a and b
Figs. C4.0.4 a and b
Four regular and uniform tesselations (left)
and the four trivial examples of RW tilings based on them (right)
Fig. C4.0.5
density = (8√ 3 − 12)/3 ≅ .618802
The original article by de Bruijn is entitled
"Algebraic theory of Penrose's non-periodic tilings of the plane",
Nederl. Akad. Wetensch. Proceedings Ser. A 84 (Indagationes Math. 43)
(1981) 38-66.
It was reprinted in The Physics of Quasicrystals, ed. P.J. Steinhardt and
S. Ostlund, World Scientific Publ. Comp., Singapore (1987), pp. 673-700.)
Fig. C4.0.6
Ten points in a de Bruijn pentagrid
that are dual to a rosette of order 5
Fig. C4.0.7
The six rosettes for n = 5
(i) Choose a lattice L.
(ii) Construct a lattice star composed of 2n distinct rays
(line segments), each of which extends from a common root-lattice point P0
to a terminal lattice point Pk (k = 1, 2, ..., 2n).
The 2n terminal lattice points are all distinct.
(iii) Generate a star grid by translating the lattice star
to each lattice point of L.
The rhombs of the tiling are dual to the vertices of the star grid.
(i) Choose a lattice L.
(ii) Construct m lattice stars—one for each of
m sub-lattices Lk (k = 1, 2, ...,
m) congruent to L, each composed of 2n distinct rays.
In each lattice star, every ray extends from a common root-lattice point P0
to a terminal lattice point Pk
(k = 1, 2, ..., 2n), which may belong either to the same
sub-lattice or to a different sub-lattice. The 2n terminal lattice
points are all distinct.
(iii) Generate a star grid by translating the lattice star for each
sub-lattice Lk
to every other point of Lk.
The rhombs of the tiling are dual to the vertices of the star grid.
Fig. C4.0.8
Star magnitude = 44
It is convenient to identify RW tilings of order n by class and type.
These terms are defined below. Tilings are ranked according to density as follows:
The tilings for n = 2 and n = 3 in Figs. C4.0.1b and C4.0.2b, respectively,
are superdense, since their density is equal to 1.
The tiling for n = 4 in Fig. C4.0.3b
is superdense. Its density is equal to 2 (√ 2 − 1) ≅ .828427,
which is maximal. (It is easily proved that there exists no tiling by regular octagons
with higher density.)
The tiling for n = 6 in Fig. C4.0.4b is
sparse,
since its density of 1/√ 3 ≅ .577350 is
less than the density (8√ 3 − 12)/3 ≅ .618802 of the kagome
tiling, shown in Fig. C4.0.5, which is conjectured to be superdense.
Within each of these two sub-classes, there are four types.
Every rosette is contiguous to exactly two other rosettes in the same row.
No rosette shares a vertex with a rosette in another row.
Fig. C4.0.9
Examples of straight row tiling lattice stars for small n.
Top row: lattice stars for the square lattice.
Bottom row: lattice stars for the hexagonal lattice.
Fig. C4.1.1
n = 2
Fig. C4.1.2
n = 4
Fig. C4.1.3
n = 6
Fig. C4.1.4a
n = 8
Straight row aligned edge-sharing
ρ8 ≅ .394591
λ8 ≅ 10.1371
pmm
(This is the same tiling as the one at the right in Fig. C4.1.6.)
Fig. C4.1.4b
n = 8
Lattice star for the tiling of Fig. C4.1.4a
Fig. C4.1.4c
n = 8
Star grid for the tiling of Fig. C4.1.4a
(This star grid is drawn on the same scale as the lattice star in Fig. C4.1.4b.)
Fig. C4.1.5a
n = 8
Straight row aligned vertex-sharing
ρ8 ≅ .253965
λ8 ≅ 15.4476
pmm
(This is the same tiling as the one in Fig. C4.1.5d
and is a slightly rearranged version of the tiling at the left in Fig. C4.1.6.)
Figs. C4.1.5b and C4.1.5d demonstrate that although the angular distribution
of the 2n rays in a lattice star is not uniform,
those rays represent 2n radial lines that are uniformly distributed:
each side of every rhomb in the tiling is perpendicular to one of the two
radial lines associated with the pair of intersecting
rays whose intersection is dual to the rhomb.
Fig. C4.1.5b
n = 8
Lattice star for the tiling of Fig. C4.1.5a
Fig. C4.1.5c
n = 8
Star grid for the tiling of Fig. C4.1.5a
(This star grid is drawn on the same scale as the lattice star in Fig. C4.1.5b.)
Fig. C4.1.5d
n = 8
Straight row aligned vertex-sharing
Ladders 0 (red), 1 (green), 2 (blue), 3 (violet)
(cf. rays 0, 1, 2, 3 in the lattice star of Fig. C4.1.5b)
Each thick black radial segment in the central rosette is
perpendicular to the rungs of its associated ladder.
Fig. C4.1.6
n = 8
Fig. C4.1.7
n = 10
C4.2 Straight row tilings, aligned: odd n
Fig. C4.2.1
n = 3
ρ3 ≅ .75000
λ3 ≅ √ 3
pmm
Fig. C4.2.2
n = 5
ρ5 ≅ .5414
ρ5 ≅ .5150
λ5 ≅ 4.9798
λ5 ≅ 4.8541
pmm
Fig. C4.2.3
n = 7
ρ7 ≅ .3466
ρ7 ≅ .3745
λ7 ≅ 9.8447
λ7 ≅ 9.3488
pmm
C4.3 Straight row tilings, staggered: even n
Fig. C4.3.1
n = 2
Fig. C4.3.2
n = 4
Fig. C4.3.3
n = 6
Fig. C4.3.4
n = 8
C4.4 Straight row tilings, staggered: odd n
Fig. C4.4.1
n = 3
ρ3 ≅ 1
λ3 ≅ 1.5
pmm
Fig. C4.4.2
n = 5
ρ5 = (1 - 8√ 5)[25 - 10√(6 - 2√ 5)] ≅ .669153
λ5 ≅ √ 5 + 1.5 ≅ 3.7361
pmm
Fig. C4.4.3
n = 7
ρ7 ≅ .460658
λ7 ≅ 7.5978
pmm
C4.5 Zig-zag row tilings, aligned: odd n
Fig. C4.5.1a
n = 5
Lattice star for sites of type A in zigzag-row tiling of Fig. C5.5.1d.
A and B are the two inequivalent types of sites in each lattice fundamental domain.
Fig. C4.5.1b
n = 5
Lattice star for sites of type B in zigzag-row tiling of Fig. C5.5.1d.
A and B are the two inequivalent types of sites in each lattice fundamental domain.
Fig. C4.5.1c
n = 5
Star grid for the zigzag-row tiling of Fig. C5.5.1d
(This star grid is drawn on the same scale as the lattice stars in Figs. C5.5.1a and b.)
Fig. C4.5.1d
n = 5
Zigzag-row tiling
ρ5 = 5 − 2√ 5 ≅ .527864
λ5 ≅ 4.9798
In Figs. C4.5.2 and C4.5.3 are two zig-zag row tilings for n = 7.
Each of them is a poly-lattice tiling, with m = 2.
Fig. C4.5.2a
n = 7
Lattice stars A and B for zigzag-row tiling no. 1 of Fig. C4.5.2c
Fig. C4.5.2b
n = 7
Star grid for zigzag-row tiling no. 1 of Fig. C4.5.2c
Fig. C4.5.2c
n = 7
Zigzag-row tiling no. 1
ρ7 ≅ .4655
λ7 ≅ 7.8948
Fig. C4.5.3a
n = 7
Lattice stars A and B for zigzag-row tiling no. 2 of Fig. C4.5.3c
Fig. C4.5.3b
n = 7
Star grid for zigzag-row tiling no. 2 of Fig. C4.5.3c
Fig. C4.5.3c
n = 7
Zigzag-row tiling no. 2
ρ7 ≅ .4547
λ7 ≅ 8.5429
Now let us look at some examples of lattice tilings — mostly, but not
all — dense tilings. We begin with
Fig. C4.6.1
n = 4
ρ4 = 2 (√ 2 − 1) ≅ .828427
Fig. C4.6.2
n = 4
ρ4 = 2 (√ 2 − 1) ≅ .828427
Fig. C4.6.3
n = 4
ρ4 = 2 (√ 2 − 1) ≅ .828427
Fig. C4.6.4a
n = 8
If you fill these 81 holes (and half-holes) ...
Fig. C4.6.4b
n = 8
with these 81 polka dots ...
Fig. C4.6.4c
n = 8
you get this superdense tiling.
ρ8 ≅ .425442
Fig. C4.6.5
n = 8
ρ8 ≅ .0143164
Compare this quite sparse tiling
to the superdense tiling in Fig. C4.6.4c.
(To see this one in brighter colors, look here.)
Fig. C4.6.6a
n = 10
ρ10 ≅ .226550
Introducing a non-convex prototile into this tiling
changes it into the one shown in Fig. C4.6.6b.
Fig. C4.6.6b
n = 10
This tiling has reflection symmetries
and a translational fundamental domain only half as large as in Fig. C4.6.6a.
Fig. C4.6.7
n = 12
ρ12 ≅ .261899
For a pdf version of this tiling, look
here.
Fig. C4.8.1a
n = 8
Lattice star for rectangular lattice tiling in Fig. C4.8.1c
Star magnitude = 52
Fig. C4.8.1b
n = 8
Star grid for rectangular lattice tiling in Fig. C4.8.1c
Fig. C4.8.1c
n = 8
Star magnitude = 52
Fig. C4.8.2a
n = 8
Lattice star for rectangular lattice tiling in Fig. C4.8.2c
Star magnitude = 44
Fig. C4.8.2b
n = 8
Star grid for rectangular lattice tiling in Fig. C4.8.2c
.gif)
Fig. C4.8.2c
n = 8
Star magnitude = 44
_split.gif)
_expanded.gif)
Fig. C4.8.3
n = 8
Splitting (left) and augmenting (right) the tiling in Fig. C4.8.2c
_tiling.gif)
Fig. C4.8.4a
n = 8
Lattice star for rectangular lattice tiling in Fig. C4.8.4c
Star magnitude = 44
_tiling.gif)
Fig. C4.8.4b
n = 8
Star grid for rectangular lattice tiling in Fig. C4.8.4c
_rect_lattice.gif)
Fig. C4.8.4c
n = 8
Star magnitude = 44
Fig. C4.8.5a
n = 8
Star magnitude = 36
Fig. C4.8.5b
n = 8
.gif)
Fig. C4.9.1
n = 6
ρ6 =1/√ 3 ≅ .577350
superdense
C4.10 Hexagonal lattice tilings: odd n
Fig. C4.10.1a
n = 9
ρ9 ≅ .341644
superdense
p3
.gif)
Fig. C4.10.1b
(Another coloring of the superdense tiling of Fig. C4.10.1a)
Fig. C4.10.2
n = 9
ρ9 ≅ .263346
sparse
p3m1
C4.11 Additional examples of RWn
C4.12 How to construct a RWn
C4.13 Origins of RWn
nor
(b) the tiling of the rosettes
n = 2: Regular tiling by squares (square lattice)
RW7(1)
RW9(1)
is a more colorful version of RW9.
Note that the image is rotated, relative to RW9, by one-sixth of a turn around its center.
In this orientation, it is symmetrical by reflection in a vertical line through the center.
Two colorings of RW10(1)
n=15: RW15 has a hexagonal lattice.
RW15
For a pdf version, look
here.
RW15(1)
is another coloring of RW15.
Fig. C4.11.4
RW16_row
Fig. C4.11.5
Adjacent n = 8 rosetttes in RW16_row are oppositely rotated.
n=21: RW21(1) is an ambitious example of a RWn tiling. To see the smallest rhomb clearly, you may have to enlarge the image.
It's apparent that d3 rosettes fit more harmoniously on a hexagonal lattice than on a square lattice. I'll explain below how the overall symmetry is affected by the parity — (even n vs. odd n) — of the SRIn.
To see how color choices for the rhombs affect RW21, see
RW21(2),
RW21(3),
RW21(4),
RW21(5), and
RW21(6).
RW21(6)
is a monstrously large piece of RW21.
It contains seven times as many rhombs as the other versions.
I didn't attempt to find color choices that would emphasize 'subliminal'
image effects, but you can see suggestions of such effects
in the roughly circular 'watermarks' embedded in the pattern.
Here is an ordered sequence of ten images of RW21. In each image only one shape of rhomb is highlighted. The sequence begins with the smallest rhomb and ends with the largest rhomb. By closely examining image sets like these, one could probably discover how to enhance the strength of particular subliminal images. (I have no plans to do that!)
n=22: RW22 is
a row tiling of order 22 that shows the 'mortar' between rosettes
but not the rosettes themselves.
C4.12 How to construct a RWn
Let's first recall that a de Bruijn multigrid is a set of n overlapping uniformly rotated grids of parallel lines. Here's an example for n = 5:
Fig. C4.12.1
A de Bruijn pentagrid
If no point of a multigrid belongs to more than two of its n grids, de Bruijn calls the multigrid regular; otherwise he calls it singular.
A star grid of order n is the periodic counterpart of a de Bruijn multigrid of order n.
DEFINITIONS:
(1) A lattice star of order n is a set of 2n rays (line segments)
associated with the lattice L.
One end of every ray is incident at the common lattice point P0;
the other end is incident at a distinct one of the
lattice points Pk (k=0, 1, 2, ..., n-1).
(2) A star grid of order n
is the union of congruent parallel lattice stars of order n,
one specimen of which is rooted at every point of the lattice L.
The 2n rays of the lattice star are labelled CCW
by their n integer indices
0, 1, 2, ..., n, n+1, ..., n+2, ..., 2n-1 (mod n).
A lattice star has the same point symmetry as its root lattice point P0.
In a star grid of order n, n lines intersect at every lattice point. Mimicking de Bruijn, we call any point of a star grid at which only two lines intersect a regular point, and any point at which more than two lines intersect a singular point.
The dual of every regular point in a star grid is a rhomb whose face angles are defined by the difference between the indices of the two rays that intersect there. The set of rhombs dual to a singular point at which m lines intersect (m=3, 4,...,n) is a convex assembly of m(m-1)/2 rhombs that tile a 2m-gon. The precise arrangement of these rhombs is indeterminate. It is appropriate, whenever possible, to arrange them in a tiling of the 2k-gon that has the point symmetry of the singular point. If the point symmetry of the singular point is dihedral of odd order, it is always possible to find an arrangement of these rhombs with the same symmetry. But if the point symmetry of the singular point is dihedral of even order, there exists no arrangement of the rhombs with the same symmetry. For these reasons, RWn tilings of odd order are somewhat more 'harmonious' in appearance than those of even order — their space group is likely to contain more symmetries.
For large separation of the rosettes (sparse dot packing) in a RWn tiling, the unit cell of the lattice is large, and the tiling may at first seem indistinguishable from a pseudo-Penrose tiling derived from a de Bruijn multigrid (cf. de Bruijn 2/14(1), for example).
A lattice star for n=9 is shown below. Because this lattice star produces a RW9 tiling with higher density than any alternative set of lattice points of the same symmetry, it is called dense. For illustration purposes, circles have been centered at each lattice point to indicate the positions (although not the sizes) of the rosettes. In the associated star grid, nine lines intersect at the center of every circle. These lines are truncated here by the circle boundaries.
Fig. C4.12.2
Fig. C4.12.3
This extremal star grid for n = 9 is a periodic array of replicas of the lattice star in Fig. C4.12.2.
(Note that a chain of three-bead Conway worm segments connects each pair of nearest-neighbor rosettes.)
It is instructive to compare an edge between a particular pair of lattice points in the star grid with the corresponding chain of rhombs in the tiling. In the star grid above, there are only two kinds of edges—those between nearest-neighbor dots and those between fourth-nearest-neighbor dots. Choose a particular edge and then follow the chain of rhombs ('ladder' with parallel rungs) between the two rosettes in RW9(1) that correspond to the pair of lattice points joined by the edge.
Fig. C4.12.4
An extremal star grid for n = 15, which is the dual of the tiling RW15(1).
Fig. C4.12.5
A zoom shot of the region just below the center of the image in Fig. C4.12.4
Fig. C4.12.6
A single lattice unit cell of the extremal star grid for n = 16, which is the dual of RW16(1)
.
Fig. C4.12.7
The lattice star for the extremal tiling RW21(1).
.
Fig. C4.12.8
Star grid for the extremal tiling RW21(1)
The star grid is a superposition of the lattice stars in Fig. C4.12.7.
(Lattice points here are approximately six times farther apart than in Fig. C4.12.7.)
In 1991, in the first edition of the instruction booklet for the tiling puzzle ROMBIX, I posed a special puzzle challenge called 'POLKA DOTS'. It is reformulated here as a puzzle for RWn.
Suppose you are required to tile a vast flat area—like the state of Rhode Island, for example—with Rhombic Wallpaper. Suppose further that there are rosettes ('dots') embedded in this tiling. All of the spaces (the 'mortar') between the dots must be tiled by the rhombs of RWn.
I'll call this problem connecting the dots. It's an example of a so-called extremal problem. In 1991 I didn't have a general solution and I still don't have one. For n = 2, 3, and 4, the problem is trivial, because solutions follow immediately from properties of regular and semi-regular tilings of the plane by squares, hexagons, and octagons.
When I revisited this problem in 2002, I replaced the rombiks of order eight
by the rhombs of SRIn.
Perhaps I was guided by what Halmos called "Polya's dictum":
I don't know whether the solution for every n is a periodic tiling, but if it is, then it can be found from an algorithm adapted from the Gessel-de Bruijn method of associating rhombs with their duals—the points of intersection in configurations of lines.
The algorithm makes it possible to investigate candidate solutions for any particular n by using a periodic variant of the multigrids invented by de Bruijn for his analysis of Penrose tilings. I call these modified grids star grids. I examined not only dense dot packings, but also sparse ones in which the rosettes are quite widely separated. The density of the rosettes in a tiling ('density') is determined by the design of its lattice star—the basic structural unit of the star grid dual of the tiling. Star grids and lattice stars are defined below.
For a summary of some of the results obtained so far, see dot spacing data. These results have been confirmed for small values of n, but they have not yet been proved.
(More to come—soon, I hope!)
FUTURE STUFF
- 2-dimensional puzzles
- QUARKS
- LOMINOES (for an illustrated ten-page description of LOMINOES, look here)
- 3-dimensional puzzles
- TETRONS, CUBONS, OCTONS, DODECONS, and ICONS
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- triply periodic minimal surfaces
- posters
- RHOMBBURST
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- H.A.Schwarz's minimal surface H