Alan H. Schoen

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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 the roundest polyhedron with 44 faces
IBM Computational Geometry Conference, 1986

If 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 R3 is:                                      

For which convex polyhedron with n faces is the ratio of S 3/V 2 smallest?
(S = surface area and V = volume.)

S 3/V 2 is called the isoperimetric quotient and will be abbreviated
here as I.Q.. Because it is dimensionless, it is independent of scale.

  In 1897, Minkowski proved that there exists a roundest polyhedron
for every n ≥ 4. In 1899 it was proved by Lorenz Lindelöf that a   
necessary condition for a polyhedron P to be roundest is that         

(i) P circumscribes a sphere,
(ii) this sphere is tangent to   
the faces of P at their
respective centroids.   

      For a 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 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 for n = 42 are described and
        illustrated. In May, 1986, I issued a supplement that includes conjectured
       solutions for 36 ≤ n ≤ 41 and n = 43. This supplement also includes data
from the analysis by William Tutte of the asymptotic number of      
simple polyhedra as a function of the number n of 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 case n = 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, each vertex of the template is labeled by an   
integer, but for the case n = 33, each edge is labeled by an integer.   

        template for n = 8: face 1
        template for n = 8: face 2
        template for n = 8: face 3
        template for n = 8: face 4

        templates for n = 25: ALL faces

        templates for n = 33: ALL faces

        templates for n = 44: faces 1, 2, 3
        templates for n = 44: faces 6, 8, 14
        templates for n = 44: faces 7, 9, 10
        templates for n = 44: faces 5, 11, 12, 34
        templates for n = 44: faces 13, 15, 16
        templates for n = 44: faces 17, 18, 41
        templates for n = 44: faces 19, 20, 44
        templates for n = 44: faces 21, 22, 23
        templates for n = 44: faces 4, 24, 26
        templates for n = 44: faces 25, 31, 33
        templates for n = 44: faces 27, 28, 29, 30
        templates for n = 44: faces 35, 36, 37
        templates for n = 44: faces 32, 38, 40
        templates for n = 44: faces 39, 42, 43

The polyhedron P33, which I have conjectured is the
roundest polyhedron with 33 faces, is composed of   
one heptagon (green), thirteen pentagons (yellow),   
and nineteen hexagons (red-orange).                          

    29 of the 33 faces of P33 define 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 roundest polyhedra (sometimes called 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 to non-convex polyhedra of the famous formula of Euler for convex polyhedra, which Euler first described in 1750 in a letter to Goldbach:

v - e + f = 2;

v, e, and f are 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 19th century the theory of roundest polyhedra was investigated by Jakob Steiner, E. Kötter, Lorenz Lindelöf, and Hermann Minkowski, and in the 20th 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)
         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)
         '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 of P44

Rear view of P44

Schlegel diagram of P44, centered on the back of the polyhedron.         

In the near future, I plan to compute — and display here — examples of nets,
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 P44 model
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 its D2d 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 is D2d.          

Below is a stereo image of a simpler convex polyhedron with D2d           
symmetry. It has the simplest possible combinatorial structure of all        
convex polyhedra with D2d symmetry. It is shown inscribed in a             
cube of edge length two. One of its two orthogonal c2 axes is the            
blue line, which coincides with the x-axis. The other c2 axis is the          
red line, which coincides with the line between points at 0,1,1)               
and (0,-1,-1).                                                                                                

P44 also has two orthogonal planes of mirror symmetry, the horizontal  
(equatorial) x-y plane and the vertical x-z plane. It is invariant under two
successive rotations—a half-turn about the [vertical] z-axis, followed by
a quarter-turn about the [horizontal] x-axis.                                               

Like the convex hexahedron shown above, if P44 is 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 of n — 8 and 20 — for which the roundest n-faced polyhedron       
has D2d symmetry (cf. pp. 162-164).                                                         

Although a separate template is provided here for each of the forty-four
  faces of P44 (cf. above), there are only eight different shapes of faces —
  two pentagons (p1 and p2) and six hexagons (h1, h2, ..., h6). 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.

4 p1: d2 symmetry
                           8 p2: c1 symmetry (i.e. asymmetric)

4 h1: d2 symmetry
4 h2: d2 symmetry
4 h3: d2 symmetry
4 h4: d2 symmetry
8 h5: c1 symmetry
8 h6: c1 symmetry

  The 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 like Mathematica, 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 the 4-dimensional version of 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 R3 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.