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ROUNDEST POLYHEDRA

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


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 R3 is:                                      

Which convex polyhedron with n faces
has the smallest value of the ratio A 3/V 2?

(A = surface area and V = volume.)

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

  In 1897, Minkowski proved the existence of 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         

P circumscribes a sphere
tangent to the faces of P at
their respective centroids.

        For a polyhedron circumscribed about the unit sphere, A/V = 27V = 9A.
Hence minimizing A 3/V 2 is equivalent to minimizing A (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. Shortly before the conference, I issued a supplement that includes
      conjectured solutions for 36 ≤ n ≤ 41 and n = 43. (This supplement also
contains William Tutte's analysis of the asymptotic number of         
simple polyhedra as a function of the number n of 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
asymmetric polyhedron found to be roundest;       

(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 every n >= 4, the roundest
    polyhedron and the polyhedron of maximum volume
among all polyhedra inscribed in the sphere are     
            what I will call metrically dual, i.e. that the face centroids
      of the roundest polyhedra coincide with the vertices of
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 Q9 (shown below) fail to coincide with the
     centroids of the faces of the 8-hedron P8 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 Q9
(brown) and the circumscribed polyhedron P8 (blue)

The blue polyhedron P8, which has eight faces, is shown here circumscribed about the unit  
sphere. A second polyhedron Q8, which has eight vertices, is shown inscribed in the same     
sphere. In 1967, Joel Berman and Kit Hanes proved that Q8 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 that P8 has the smallest   
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 of P8 coincide with those      
of Q8, the face centroids of P8 are almost coincident with the vertices of Q8. The gap            
between the face centroids of P8 and the vertices of Q8 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 hull                              

Neil Sloane's Q9, the arrangement of nine points on the unit
sphere S for which the convex hull has the largest volume.



Schlegel diagram                                               Convex hull                              

P9, the polyhedron of largest volume
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 Sloan et al

     Q122 is found to have the largest volume among
polyhedra with 122 vertices that are inscribed
   in the unit sphere. It is the topological dual of a
    Goldberg polyhedron P122 that in 2015 was     
found by Wayne Deeter to have the smallest  
volume among polyhedra with 122 faces that
are circumscribed about the unit sphere.         



The inscribed polyhedron Q122 (a Goldberg polyhedron)



Pied tiling of Q122
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 of Q122



Colored Schlegel diagram of the circumscribed polyhedron P122



Uncolored Schlegel diagram of the circumscribed polyhedron P122



The blue polyhedron P8, which has eight faces, is shown here circumscribed about the unit  
sphere. A second polyhedron Q8, which has eight vertices, is shown inscribed in the same     
sphere. In 1967, Joel Berman and Kit Hanes proved that Q8 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 that P8 has the smallest    
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 of P8 coincide with those      
of Q8, the face centroids of P8 are almost coincident with the vertices of Q8. The gap            
between the face centroids of P8 and the vertices of Q8 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 appropriately tilted, causing their tangent points to migrate on the
                sphere in such directions that the gaps between the face centroids and their tangent points are reduced.
    Eventually the face centroids and tangent points coincide nearly exactly. For each value of n, 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 of n, 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 face F has been removed so as to provide a peep hole for a [virtual] observer. The station
   point
is chosen to be so close to F that the interior surface of every cube face except for F itself 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 particular vertex V
    or (b) just outside an edge E. 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 the asymmetrical polyhedron P25:


                    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 of P43:

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

If you can identify this conferee, please tell me who she is!


The problem of roundest polyhedra in R3 asks:                                

Which n-faced polyhedron has the smallest ratio of S 3/V 2 ?
(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,
and
(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 long been conjectured — but never proved — that the roundest
        polyhedron is always simple, i.e., that its vertices are all of degree three.


       In 1986, I wrote a FORTRAN program called 'Lindelof' that repeatedly
      tilts the 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
    state TAC (for "tangent at centroid"). The starting polyhedron in each
      run was constructed by applying n randomly 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 each n, 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 for n = 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 and n = 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 number n of faces.)                             

    In August, 1986, I ran Lindelof for the last time and derived solutions
for n = 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 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. 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 Malkovitch

for 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 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.

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

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 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.                                                                             




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

Q22 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 of Q22
viewed along a diad axis


Flowered Schlegel diagram of Q22
viewed along a diad axis


Schlegel diagram of Q22
viewed along a quasi-diad axis


Schlegel diagram of Q22
viewed along a second quasi-diad axis



n=23

Q23 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 of S1,
a 36-face subset of the 42 faces of Q23


Symmetrical 'flowered' Schlegel diagram
of all 42 faces of Q23


Symmetrical Schlegel diagram of S23(2),
a second 36-face subset of the 42 faces of Q23


Symmetrical 'flowered' Schlegel diagram of S23(2)


Asymmetrical Schlegel diagram of S23(3),
a 37-face subset of the 42 faces of Q23



Asymmetrical flowered Schlegel diagram of S23


Schlegel diagram centered on an arbitrarily chosen face
of Q23



n=122

Q122 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 of Q122,
dual of a Goldberg polyhedron, viewed along a 5-fold axis



Schlegel diagram of Q122, dual of a Goldberg polyhedron,
viewed along a 5-fold axis
Click here for an image of far higher resolution .



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