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## 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

Cloud City, a sculpture by Tomás Saraceno

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

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

P44, probably theroundestpolyhedron with 44 faces

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

She's the person who volunteered to accept this paper model

I had assembled the night before. (I preferred not to carry it

already assembled on the flight from Illinois to New York.)

The problem of roundest polyhedra in R

^{3}is:

For which convex polyhedron withnfaces is the ratio ofS^{ 3}/V^{ 2}smallest?

(= 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 been conjectured — but never proved — that the roundest

polyhedron is alwayssimple,i.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. In May, 1986, I issued a supplement that includes conjectured

solutions for 36 ≤n≤ 41 andn= 43. This supplement also includes data

from the analysis by William Tutte of the asymptotic number of

simple polyhedra as a function of the numbernof faces.

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) 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 tobest polyhedranon-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. Klee greatly deepened the analysis and also 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 now see here.

Thanks, Ivonne!I regret that unlike Paul Halmos (1916-2006), I never developed the habit of taking a snapshot of every mathematician I met. As a consequence, I don't have a photo of Michael Goldberg. If someone will send me one, I'll gladly post it.

Mathematicians are indebted to the prolific geometer Joseph Malkevitch

Joseph Malkevitch

for his 2015 article about the work of Victor Klee.My interest in roundest polyhedra was kindled 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 mentioned the subject of roundest polyhedra, in which I had a growing interest. Like Victor Klee, Michael was an unusually modest man. That is possibly one of the reasons why he never told me about either of his groundbreaking publications about polyhedra:

'The Isoperimetric Problem for Polyhedra' (1934)

and

'A Class of Multisymmetric Polyhedra' (1937).In 1985 I discovered Michael's 1935 article, but it wasn't until 1991 — when I bought Ian Stewart's wonderfully witty book about mathematics, "Game, Set, and Math" — that I first learned of Michael's 1937 article introducing the concept now known as 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.