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## ROUNDEST POLYHEDRA

Alan H. Schoen

Comments are welcome!

Nested pair of truncated icosahedra, (sculptor unknown)

Photographed by Reiko Takasawa Schoen in Madison Square Park, New York City, October 21, 2012

THIS SITE IS CURRENTLY UNDER CONSTRUCTION.

A0. Random Polyhedral Honeycomb

The fifteen convex cells in this packing are the Voronoi polyhedra of

fifteen points located at random positions in the central region of a sea

of more than one hundred random points. The computer program used

to carry out the calculations was designed by the author and was coded

in FORTRAN by Randall Lundberg in 1969. In 1971, Robert Fuller

constructed this vinyl model.

A1. Roundest Polyhedra

The problem of roundest polyhedra in R ^{3}is:

Which convex polyhedron withnfaces

has the smallest value of the ratioA^{ 3}/V^{ 2}?

(= surface area andA= volume.)V

is called theA^{ 3}/V^{ 2}isoperimetric quotientand will be abbreviated

here as. Because it is dimensionless, it is independent of scale.IQIn 1897, Minkowski proved the

existenceof a roundest polyhedron

for everyn≥ 4. In 1899 it was proved by Lorenz Lindelöf that a

necessarycondition for a polyhedronto be roundest is thatP

circumscribes a sphereP

tangent to the faces ofatP

their respective centroids.For a polyhedron circumscribed about the unit sphere,

A/V= 27V= 9A.

Hence minimizingA^{ 3}/V^{ 2}is equivalent to minimizingA(orV).It has been conjectured — but never proved — that the roundest

polyhedron is always,simplei.e.,that its vertices are all of degree three.

My computational geometry 1986 IBM conference paper,

'A Defect-Correction Algorithm for Minimizing the Volume

of a Simple Polyhedron Which Circumscribes a Sphere',summarizes my 1986 computer investigation of roundest polyhedra.

Conjectured solutions for 4 ≤n≤ 35 and forn= 42 are described and

illustrated. Shortly before the conference, I issued a supplement that includes

conjectured solutions for 36 ≤n≤ 41 andn= 43. (This supplement also

contains William Tutte's analysis of the asymptotic number of

simple polyhedra as a function of the numbernof faces.)

I made final runs in late 1986 and obtained solutions for

n= 45, 46, and 47.Among the most interesting results of this study were

(a) the discovery of the first example (

n= 25) of an

polyhedron found to be roundest;asymmetric

(b) the discovery of the first example (

n= 33) of a

polyhedron— also conjectured to be roundest —

that;contains a heptagon(c) additional data confirming a discovery I had made in

October, 1975, disproving a conjecture of Victor Klee.

Klee conjectured that for everyn>= 4, the roundest

polyhedron and the polyhedron of maximum volume

among all polyhedra inscribed in the sphere are

what I will call,metrically duali.e.that the face centroids

of the roundest polyhedrawith the vertices ofcoincide

the inscribed polyhedra of maximum volume.Below is page 5 of Victor Klee's 1972 video booklet, "Shapes

of the Future — Some Unsolved Problems of Geometry, Part 2".

Klee's conjecture is stated near the bottom of this page.

The handwritten entry below it is a brief summary I scrawled

in my copy of the booklet on January 30, 1976, after I had demonstrated

by an elementary calculation that the vertices of the Berman-Hanes

inscribed 9-hedron Q_{9}(shown below) fail to coincide with the

centroids of the faces of the 8-hedron P_{8}I had found to have the

smallest volume among all polyhedra that circumscribe the sphere.

Page 5 of Victor Klee's video booklet "Shapes of the Future: Part 2"

The inscribed Berman-Hanes polyhedron Q_{9}

(brown) and the circumscribed polyhedron P_{8}(blue)The blue polyhedron

P_{8}, which has eight, is shown herefacesabout the unitcircumscribed

sphere. A second polyhedronQ_{8}, which has eight, is shownverticesin the sameinscribed

sphere. In 1967, Joel Berman and Kit Hanes proved thatQ_{8}has the largest volume among

polyhedra with eight vertices inscribed in the unit sphere, and they derived exact expressions

for its vertex coordinates. In 1934, Michael Goldberg conjectured thatP_{8}has thesmallest

volume among polyhedra with eight faces that circumscribe the unit sphere, but no analytic

expressions are known for its coordinates. When the two polyhedra are oriented so that the

axes of rotational symmetry and planes of reflection symmetry ofP_{8}coincide with those

ofQ_{8}, the face centroids ofP_{8}are almost coincident with the vertices ofQ_{8}. The gap

between the face centroids ofP_{8}and the vertices ofQ_{8}is equal to ~0.04700640080523555

on each quadrilateral face and ~0.003249596513702373 on each pentagonal face. Tiny yellow

dots mark the centroid locations on one pentagonal face and on one quadrilateral face.

Schlegel diagram Convex hullNeil Sloane's

Q9, the arrangement of nine points on the unit

sphereSfor which the convex hull has the largest volume.

Schlegel diagram Convex hull

, the polyhedron of largest volumeP9

that circumscribes the unit sphere S

Edge skeleton of the inscribed polyhedron Q9

Edge skeleton of Q122, a simplicial polyhedron with

122 vertices, 362 edges, and 240 faces. Q122 has full

icosahedral symmetry. (Discovered by Neil Sloanet al

Q122 is found to have thevolume amonglargest

polyhedra with 122that areverticesinscribed

in the unit sphere. It is the topological dual of a

Goldberg polyhedronP122 that in 2015 was

found by Wayne Deeter to have thesmallest

volume among polyhedra with 122thatfaces

areabout the unit sphere.circumscribed

The inscribed polyhedronQ122 (a Goldberg polyhedron)

Pied tiling ofQ122

It has 240 triangle faces: 60 yellow, 120 green, and 60 red.

The ratios of the triangle areas are [approximately]

yellow : green : red = 1 : 1.125 : 1.148.

Q122 — nested rings (axial view)

Q122 — nested rings (oblique view)

Schlegel diagram ofQ122

Colored Schlegel diagram of thecircumscribedpolyhedronP122

Uncolored Schlegel diagram of thecircumscribedpolyhedronP122

The blue polyhedron

P_{8}, which has eight, is shown herefacesabout the unitcircumscribed

sphere. A second polyhedronQ_{8}, which has eight, is shownverticesin the sameinscribed

sphere. In 1967, Joel Berman and Kit Hanes proved thatQ_{8}has the largest volume among

polyhedra with eight vertices inscribed in the unit sphere, and they derived exact expressions

for its vertex coordinates. In 1934, Michael Goldberg conjectured thatP_{8}has thesmallest

volume among polyhedra with eight faces that circumscribe the unit sphere, but no analytic

expressions are known for its coordinates. When the two polyhedra are oriented so that the

axes of rotational symmetry and planes of reflection symmetry ofP_{8}coincide with those

ofQ_{8}, the face centroids ofP_{8}are almost coincident with the vertices ofQ_{8}. The gap

between the face centroids ofP_{8}and the vertices ofQ_{8}is equal to ~0.04700640080523555

on each quadrilateral face and ~0.003249596513702373 on each pentagonal face. Tiny yellow

dots mark the centroid locations on one pentagonal face and on one quadrilateral face.

In the spring of 2015, I was happily surprised to receive an email from Wayne Deeter, who

informed me that he too had been investigating roundest polyhedra. He had accumulated an

enormous body of results, which he generously sent me, from runs of an optimizing computer

program of his own that he only later discovered is based on an algorithm very similar to

the one I designed for 'Lindelöf', the optimizing program I wrote in FORTRAN in 1986.Lindelöf begins by generating an

n-faced polyhedron circumscribed about the unit sphere. The

points on the sphere where the face planes are initially tangent are randomly distributed, but

in every iteration, the faces are appropriatelytilted, causing their tangent points to migrate on the

sphere in such directions that the gaps between the facecentroidsand their tangent points are reduced.

Eventually the face centroids and tangent points coincide nearly exactly. For each value ofn, I

anointed the polyhedron with the smallest area (or volume) as the roundest one, but since I

was using a statistical sampling method to generate these polyhedra, my conclusions were

only as good as the statistics would allow. The dramatic increase in computer speed that had

occurred during the 29 years after my 1986 study allowed Wayne to make hundreds or

thousands of times more runs than I made in the interval (4 ≤n≤ 47) covered by my study.

It is hardly surprising, therefore, that for one value ofn, 38, Wayne discovered a very slightly

rounder polyhedron than the one I had conjectured was roundest.Wayne has enormously expanded the scope of this field by examining polyhedra with far more

faces than any I considered in 1986. This great expansion has enabled him to draw conclusions

regarding many aspects of the problem that — for the first time — have good statistical support.In late 1986, my polyhedra research notebooks were accidentally lost while the contents of my

office were being moved to another building across campus, and they were never recovered.

I then decided to discontinue my computer studies of this problem. But Wayne has generously

allowed me to use his data to reconstruct both graphic images and physical models of the solution

polyhedra I found in 1986 for 4 ≤n≤ 47. Wayne will soon publish an article summarizing his results.

The 19th century German geometer Victor Schlegel introduced the use of flat drawings, which are now called

Schlegel diagrams, to represent the combinatorial structure of convex polyhdra. The geometry of such a

projection diagram is illustrated below for the cube. Let us assume that the faces of the cube are opaque

squares but that the faceFhas been removed so as to provide a peep hole for a [virtual] observer. Thestationis chosen to be so close to

pointFthat the interior surface of every cube face except forFitself is visible.

Projection scheme for Schlegel diagram of the cube

Schlegel diagram of the cube

The observer could of course choose instead to locate the station point (a) just outside a particularvertexV

or (b) just outside anedgeE. A Schlegel diagram for the cube under arrangement (a) is illlustrated below.

Projection scheme for Schlegel diagram of the truncated octahedron

Schlegel diagram of the truncated octahedron

SCHLEGEL DIAGRAMS of conjectured

roundest n-faced polyhedra(4 ≤n≤ 47)

4 5 6 7 8

9 10 11 12 13

14 15 16 17 18

19 20 21 22 23

24 25 26 27 28

29 30 31 32 33

34 35 36 37 38

39 40 41 42 43

44 45 46 47

Here are schlegel diagrams for the 25 faces of theasymmetricalpolyhedronP_{25}:

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

And here are Schlegel diagrams for the 43 faces ofP_{43}:

1-5

6-10

11-15

16-20

21-25

26-30

31-35

36-40

41-43

SCHLEGEL DIAGRAMS AND STEREOSCOPIC IMAGES

OF CONJECTURED ROUNDEST POLYHEDRA FOR 4 ≤n≤ 47

(Under construction!)

n=4

n=5

n=6

n=7

n=8

n=9

n=10

n=11

n=12

n=13

n=14

n=15

n=16

n=17

n=18

n=19

n=20

n=21

n=22

n=23

n=24

n=25

n=26

n=27

n=28

n=29

n=30

n=31

n=32

n=33

n=34

n=35

n=36

n=37

n=38

n=39

n=40

n=41

n=42

n=43

n=44

n=45

n=46

n=47

P44, probably theroundestpolyhedron with 44 faces

IBM Computational Geometry Conference, 1986If you can identify this conferee, please tell me who she is!

The problem of roundest polyhedra in R

^{3}asks:

Whichn-faced polyhedron has the smallest ratio ofS^{ 3}/V^{ 2}?

(= surface area andS= volume.)V

is called theS^{ 3}/V^{ 2}isoperimetric quotientand will be abbreviated

here as. Because it is dimensionless, it is independent of scale.I.Q.In 1897, Minkowski proved that there exists a roundest polyhedron

for everyn≥ 4. In 1899 it was proved by Lorenz Lindelöf that a

necessarycondition for a polyhedronto be roundest is thatP(i)

circumscribes a sphere,P

and

(ii) this sphere is tangent to

the faces ofat theirP

respective centroids.For a polyhedron circumscribed about the unit sphere,

S/V= 27V= 9S.

Hence minimizingS^{ 3}/V^{ 2}is equivalent to minimizingS(orV).It has long been conjectured — but never proved — that the roundest

polyhedron is alwayssimple,i.e.,that its vertices are all of degree three.

In 1986, I wrote a FORTRAN program called 'Lindelof' that repeatedly

tiltsthe faces of a polyhedron that circumscribes the sphere until every

face is (almost exactly) tangent to the sphere at its centroid. I'll call this

stateTAC(for "tangent at centroid"). The starting polyhedron in each

run was constructed by applyingnrandomly oriented tangent planes to

the sphere and determinng their intersections (although the program

also accepted pre-designed polyhedra as input). In each iteration, every

face plane was tilted in a direction that would reduce the gap between

its current centroid and its point of tangency. For eachn, the polyhedron

found to have the smallest area after a long sequence of tilts was anointed

as the [probable] roundest.

I summarized my results for 4 ≤

n≤ 35 and forn= 42 in an IBM conference

report:'A Defect-Correction Algorithm for Minimizing the Volume

of a Simple Polyhedron Which Circumscribes a Sphere'and in a May 1986 supplement that describes conjectured solutions for

36 ≤n≤ 41 andn= 43. (This supplement includes numerical data derived

from an analysis by William Tutte of the asymptotic number of simple

polyhedra as a function of the numbernof faces.)

In August, 1986, I ran Lindelof for the last time and derived solutions

forn= 45, 46, and 47.Below are links to sets of face templates for

n= 8, 25, 33, and 44.

You can use these templates to construct physical models of these

examples of conjectured 'roundest' polyhedra. For the casen= 8,

I have provided templates only for faces 1, 2, 3, and 4, since faces

5, 6, 7, and 8 are congruent to faces 1, 2, 3, and 4, respectively.For

n= 8, 25, and 44, eachvertexof the template is labeled by an

integer, but for the casen= 33, eachedgeis labeled by an integer.template for

n= 8: face 1

template forn= 8: face 2

template forn= 8: face 3

template forn= 8: face 4templates for

n= 25: ALL faces

templates for

n= 33: ALL faces

templates for

n= 44: faces 1, 2, 3

templates forn= 44: faces 6, 8, 14

templates forn= 44: faces 7, 9, 10

templates forn= 44: faces 5, 11, 12, 34

templates forn= 44: faces 13, 15, 16

templates forn= 44: faces 17, 18, 41

templates forn= 44: faces 19, 20, 44

templates forn= 44: faces 21, 22, 23

templates forn= 44: faces 4, 24, 26

templates forn= 44: faces 25, 31, 33

templates forn= 44: faces 27, 28, 29, 30

templates forn= 44: faces 35, 36, 37

templates forn= 44: faces 32, 38, 40

templates forn= 44: faces 39, 42, 43

The polyhedron P33, which I have conjectured is the

roundestpolyhedron with 33 faces, is composed of

one heptagon (green), thirteen pentagons (yellow),

and nineteen hexagons (red-orange).29 of the 33 faces of

P33define a connected assembly

with the same combinatorial structure as the faces of

the 'soccer ball' polyhedron (truncated icosahedron).Below is a Schlegel diagram of

P33:

To view a sequence of eight rotated images of

P33,

set the magnification at 80%.

The theory of(sometimes calledroundest polyhedra)best polyhedra

was first treated by Simon Antoine Jean L'Huilier (1750-1840). One

of several accomplishments for which L'Huilier is remembered today

is his generalization tonon-convexpolyhedra of the famous formula

of Euler forconvexpolyhedra, which Euler first described in 1750 in

a letter to Goldbach:v-e+f= 2;

v,e, andfare the numbers of vertices, edges, and faces, respectively.The history of this development is summarized in George G. Szpiro's

fascinating book, "Poincaré's Prize".The subject of roundest polyhedra is conventionally referred to as

'The Isoperimetric Problem for Polyhedra'.

In the 19

^{th}century the theory of roundest polyhedra was investigated

by Jakob Steiner, E. Kötter, Lorenz Lindelöf, and Hermann Minkowski,

and in the 20^{th}century by Ernst Steinitz, Michael Goldberg, and Victor Klee.Lorenz Lindelöf proved a fundamental theorem about roundest polyhedra

(cf.my 1986 conference paper) — first in 1869 and again in 1899. In 1986

Klee greatly deepened the analysis and extended it to dimensions beyond three.

Jakob Steiner (1796-1863) Lorenz Lindelöf (1827-1908) Hermann Minkowski (1864-1909)

Georgy Voronoy (1868-1908) Ernst Steinitz (1871-1928) Victor Klee (1925-2007)

Ms. Ivonne Vetter, a staff member at the Mathematisches Forschungsinstitut

Oberwohlfach, noticed yesterday (March 28, 2011) that Lorenz Lindelöf's

picture was missing from the pantheon above. She then kindly emailed me

the splendid Lindelöf photo, from the Oberwohlfach archives, that you see here.

Thanks, Ivonne!I now deeply regret that unlike Paul Halmos (1916-2006), I never made a

practice of taking a snapshot of every mathematician I met. As a consequence,

I have no photo of Michael Goldberg. If someone will send me one, I'll gladly post it.I am deeply grateful to Joseph Malkevitch, who is himself a prolific geometer,

Joseph Malkovitchfor his 2015 article about the work of Victor Klee.

My own interest in roundest polyhedra was rekindled in 1972 when I first saw Klee's

two films about geometry, 'Shapes of the Future — Some Unsolved Problems

in Geometry'. Shortly afterwards Prof. Klee (1925-2007) gave me copies of the filmscript

booklets for these two films, Part I: Two Dimensions and Part II: Three Dimensions.

With the permission of the Mathematical Association of America, I have made

digital copies of these films and uploaded them to YouTube:Part I: Two Dimensions (YouTube video)

and

Part II: Three Dimensions. (YouTube video)I met Michael Goldberg several times at AMS meetings in the late sixties. We

discussed minimal surfaces and Voronoi polyhedra, but unfortunately he never

brought up the subject of roundest polyhedra, in which I had a growing interest.

Like Victor Klee, Michael was an unusually modest man. I greatly regret that

he never mentioned to me either of his groundbreaking publications about polyhedra:'The Isoperimetric Problem for Polyhedra' (1934)

and

'A Class of Multisymmetric Polyhedra' (1937).In 1985 I discovered Goldberg's 1935 article, but only in 1991, when I bought

Ian Stewart's wonderfully witty book about mathematics, "Game, Set, and Math",

did I learn of Michael's 1937 article introducing what are now called Goldberg

polyhedra, a class of polyhedra based on the regular icosahedron.Donald Caspar and Aaron Klug independently rediscovered Goldberg polyhedra

in 1962. They are related to the structure of several common viruses and are described

in the Caspar-Klug publication, "Physical Principles in the Construction of Regular

Viruses", Cold Spring Harbor Symp. Quant. Biol. 27, pp. 1-24 (cf.VIRUSWORLD).

Front view ofP44

Rear view ofP44

Schlegel diagram ofP44, centered on the back of the polyhedron.

In the near future, I plan to compute — and display here — examples ofnets,

for the benefit of those who prefer to use them for the construction of models,

instead of using individual polygon templates.

Tetsuya Hatanaka and the

P44model

he assembled in Spring, 2011

In 2011, I constructed a new model of P44, using the face templates I

made in 1986. Below is a cross-eyed stereo pair of photos of this model.

Identifying itsD_{2d}symmetry required careful scrutiny of the model!

The model I constructed in 2011 of

P44, the putative 'roundest'

polyhedron with 44 faces (identfied in 1986). Its symmetry isD_{2d}.Below is a stereo image of a simpler convex polyhedron with

D_{2d}

symmetry. It has the simplest possible combinatorial structure of all

convex polyhedra withD_{2d}symmetry. It is shown inscribed in a

cube of edge length two. One of its two orthogonalc2axes is the

blueline, which coincides with thex-axis. The otherc2axis is the

redline, which coincides with the line between points at 0,1,1)

and (0,-1,-1).

P44also has two orthogonal planes of mirror symmetry, the horizontal

(equatorial)plane and the verticalx-yplane. It is invariant under twox-z

successive rotations—a half-turn about the [vertical]-axis, followed byz

a quarter-turn about the [horizontal]-axis.xLike the convex hexahedron shown above, if

P44is viewed from the

back, it appears the same as it does from the front, except for the fact

that its image is rotated by a quarter-turn.My 1986 conference paper on roundest polyhedra lists only two other

values ofn— 8 and 20 — for which the roundestn-faced polyhedron

hasD_{2d}symmetry (cf.pp. 162-164).Although a separate template is provided here for each of the forty-four

faces ofP44(cf.above), there are only eight different shapes of faces —

two pentagons (p_{1}andp_{2}) and six hexagons (h_{1},h_{2}, ...,h_{6}). The number

of specimens and the symmetry of each face are given below. Each of the

asymmetrical face shapes occurs in two four-specimen subsets. The faces in

one of these subsets are reflected images of the faces in the other subset.

pentagons

4p_{1}:d2symmetry

8p_{2}:c1symmetry (i.e.asymmetric)

hexagons

4h_{1}:d2symmetry

4h_{2}:d2symmetry

4h_{3}:d2symmetry

4h_{4}:d2symmetry

8h_{5}:c1symmetry

8h_{6}:c1symmetryThe results described in my 1986 conference paper and in the May 1986

supplement demonstrate that roundest polyhedra exhibit a great variety

of symmetries. Determining the symmetry of each of these polyhedra is

an interesting challenge. It is probably best accomplished by examining,

from different directions, either physical models or — more simply! —

virtual models, using a computer program likeMathematica, for example.In 1989, my student David M. Aubertin wrote a computer science masters thesis

in which he designed and implemented a program for obtaining and displaying

solutions for the4-dimensional versionof the isoperimetric problem for polyhedra.

Here is a hyperlink to his thesis, which is entitled'Optimization of Four-Dimensional Polytopes'.

It contains numerous stereoscopic perspective drawings of projections into R

^{3}of the

solution 4-polytopes. It also includes a listing of the elegantly structured FORTRAN

program Dave wrote for this study, together with a short BASIC program for

displaying the perspective images. He has informed me that you are quite welcome

to copy and run these programs.

Cloud City, a sculpture by Tomás Saraceno

Photographed by Reiko Takasawa Schoen at the Metropolitan Museum of Art, New York City, October 20, 2012

n=4

n=5

n=6

n=7

n=8

n=9

n=10

n=11

n=12

n=13

n=14

n=15

n=16

=17

n=18

n=19

n=20

n=21

n=22

Q_{22}is composed of 40 triangles of 10 distinct shapes.

There are 4 specimens of each shape.

front view rear view cage wireframe

Schlegel diagram of a 36-face subset ofQ_{22}

viewed along a diad axis

Flowered Schlegel diagram ofQ_{22}

viewed along a diad axis

Schlegel diagram ofQ_{22}

viewed along a quasi-diad axis

Schlegel diagram ofQ_{22}

viewed along a second quasi-diad axis

n=23

Q_{23}is composed of 42 triangles of 7 distinct shapes.

There are 6 specimens of each shape.

front view rear view cage wireframe

Symmetrical Schlegel diagram ofS_{1},

a 36-face subset of the 42 faces ofQ23

Symmetrical 'flowered' Schlegel diagram

of all 42 faces ofQ23

Symmetrical Schlegel diagram ofS_{23}(2),

a second 36-face subset of the 42 faces ofQ23

Symmetrical 'flowered' Schlegel diagram ofS_{23}(2)

Asymmetrical Schlegel diagram ofS_{23}(3),

a 37-face subset of the 42 faces ofQ23

AsymmetricalfloweredSchlegel diagram ofS_{23}

Schlegel diagram centered on an arbitrarily chosen face

ofQ23

n=122

Q_{122}is composed of 240 triangles of 3 distinct shapes.

There are 60 specimens of the largest and of the smallest triangles

and 120 specimens of the triangle of intermediate size.

Schlegel diagram of a 235-face subset of the 240 faces ofQ122,

dual of a Goldberg polyhedron, viewed along a 5-fold axis

Schlegel diagram ofQ122, dual of a Goldberg polyhedron,

viewed along a 5-fold axis

Click here for an image of far higher resolution .

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