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

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).

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.

My interest in roundest polyhedra was kindled in 1972 when I first saw the late Victor Klee's two films, 'Shapes of the Future — Some Unsolved Problems in Geometry'. Afterwards Prof. Klee (1925-2007) gave me copies of his filmscript booklets, Part I: Two Dimensions and Part II: Three Dimensions. With the permission of the Mathematical Association of America, I have made digital copies of both 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 in the late sixties at AMS meetings. We talked mostly about minimal surfaces and Voronoi polyhedra. Like Victor Klee, Michael was a modest man. Unfortunately he never mentioned to me that he'd written a groundbreaking article in 1934, 'The Isoperimetric Problem for Polyhedra'.

The roundest polyhedron question for R3 is:

Among the convex polyhedra with n faces, which has the smallest ratio of S 3/V 2 ?
(S = surface area and V = volume.)

In 1897, Minkowski proved that a roundest polyhedron exists 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
                ( 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 (May, 1986) includes conjectured solutions for 36 ≤ n ≤ 41 and n = 43. The supplement includes data from William Tutte's analysis 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. These templates can be used to construct physical models 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 cases n = 8, 25, and 44, each vertex of the template is labeled with an integer, but for the case n = 33, each edge is labeled with 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

P44, [probably] the 'roundest' polyhedron with 44 faces
IBM Computational Geometry Conference, 1986
(If you can identify this participant, please tell me who she is!)

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 that its image  
is rotated by a quarter-turn.                                                                         

My 1986 conference paper on roundest polyhedra lists only two other   
     values of n for which the roundest n-faced polyhedron has D2d symmetry:  
8 and 20 (cf. pp. 162-164).                                                                         

Although a separate template is provided here for each of the forty-four faces of P44 (see above), there are only eight different face shapes — two pentagons (p1 and p2) and six hexagons (h1, h2, h3, h4, h5, h6). The number of specimens of each face and its symmetry are given below. Each of the asymmetrical faces occurs in two four-specimen subsets; the faces in one subset are reflected images of the faces in the other subset.

  • pentagons
    • 4 p1: d2 symmetry
    • 8 p2: c1 symmetry (i.e. no symmetry)
  • hexagons
    • 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 one of these polyhedra is an interesting challenge. It is probably best accomplished by examining, from different directions, either a physical model or a virtual model — in Mathematica, for example.

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.