The Isoperimetric Problem for Polyhedra, first proposed in the 18^{th}
century by Simon Antoine Jean L'Huilier,
is the problem of finding the convex polyhedron with a given number of faces that
has the smallest surface area to volume ratio: *A ^{3}/V^{2}*.
In the 19

In his 1935 paper Michael Goldberg conjectured that for all *n* except
*n=11* and *13* the solutions are medial polyhedra: simple polyhedra
(three faces around each vertex), with faces being either b-gons or (b+1)-gons.
(There are no medial polyhedra with 11 or 13 faces.)
Prior to his paper solutions had been proven only for *n=4* (regular tetrahedron),
*5* (triangular prism) and *6* (cube), and a partial proof had been made
for *7* (pentagonal prism).
Goldberg provided a proof for *12* (regular dodecahedron).
He also provided putative solutions for *n* up to *16*, and for
*20, 32* and *42*.

In his 1986 paper and supplement Alan Schoen reported on his search for putative
solutions using a Monte Carlo method. The first case of note is for *n=25* which
has C_{1} (that is no) symmetry. (Goldberg speculated on this possibility in
his paper on the Isoperimetric Problem.) For *n=32* and *42* Goldberg had
proposed polyhedra having icosohedral symmetry: members of what are now known as
Goldberg polyhedra. Schoen's search results supported Goldberg's I_{h} solution for
*32* but came up with a better, D_{5h} solution for *42*.

I recreated Alan's Monte Carlo program in 2015. The increase in computing speeds
in the intervening three decades allowed for useful runs up to *n=200*:
Table 1-1: Summary of Monte Carlo runs for n=4 to 200.
This method works for such large *n* apparently due to several factors: the
number of polyhedra meeting the Lindelöf conditions grows much more slowly with increasing
*n* than does the number of distinct simple polyhedra, and the routine tends
to exclude polyhedra whose tangent points are not fairly uniformly distributed on the
surface of a sphere.

While Goldberg's conjecture regarding solutions being medial polyhedra appears to
be incorrect it may still be a useful approximation. To limit the number of cases
needing to be checked, only medial polyhedra having their 12 pentagons near the vertices
of the regular icosahedron were considered. These were then run (for *n=186* to
*504*) through Schoen's roll-toward centroid routine to attempt to "stabilize" them.
Polyhedra which did not converge were discarded.

In order to extend the search for cases having heptagonal faces, it is noted that
a patch of two pentagonal faces and a heptagonal face on an *n* faces polyhedron
may be substituted for a patch consisting of one pentagonal and three hexagonal faces
to obtain an *n-1* polyhedron. This procedure was performed on the 100 best results
of the previous stage until 12 heptagons had been added:
Table 1-2: Summary of constratined search runs.

Collection of best known roundest polyhedra.

While medial polyhedra having icosahedral symmetry, the Goldberg polyhedra, are not
neccessarily the best, they may still offer clues as to what makes a polyhedron good:
Table 1-3: Goldberg Polyhedra.
It is well known that there exist many Goldberg pairs.
As *G(a,b)* and *G(b,a)* are chiral forms of the same polyhedron when *a≠b*
and neither are zero we may consider only *a≠0, b≥0, a≥b*.
Note that the member of each pair which has a greater *b/a* also has a greater *IQ*.

By making use of rotational icosahedral symmetry computation of the Goldberg polyhedra
was reduced by a factor of 60 to produce:
Table 1-4: Goldberg Pairs.
This was to look for a possible ideal *b/a* less than one.
In all cases *a=b* is still best.
Several statistics were gathered by this program.
The relative size of the 12 pentagons decreases with increasing *n*.
There is more variability in hexagon sizes for smaller values of *b/a*.

The Goldberg polyhedra may be viewed as a mapping of a hexagonal lattice
onto a sphere. They have three zones. The first consists of the twelve pentagons
corresponding to the vertices of the master icosahedron, the defects neccessary
for the spherical shape. The next are 30 parallelograms corresponding to the edges
of the master polyhedron. For *G(a,b)*, the *a* and *b* define
the lengths of the parallelogram edges. The last are 20 triangles for the 20
icosahedron faces.

There are two "degenerate" cases, both having *I _{h}* symmetry.
When

The addition of heptagon-pentagon pairs for polyhedra having more than 200
faces often results in higher *IQ*. This has been noted for the minimum energy
point placement problems. For example, see Hardin & Saff.

Lengyel, Gáspár & Tarnai have looked at the roundeds polyhedra constrained to having the higher order symmetries: tetrahedral, octahedral, and icosahedral. This page updates their list.

The problem of finding the minimum volume polyhedra tangent to a sphere is equivalent to the
Isoperimetric Problem for Polyhedra. The dual of this problem is finding the maximum
volume polyhedra having all their vertices on the surface of a sphere.
In 1963 Grace used a computer search to find a solution for *n=8*. In 1970
Berman and Hanes proved solutions for *n=4* through *8*. In 1994 Hardin, Sloane
and Smith searched for putative solutions for *n≤130*.

My approach is based on the assumption that as both problems are "optimal point placement" problems, what is a good case for one would likely be good for the other. I therfore took the duals of the 100 best roundest polyhedra and optimized them for volume: Table 2-1: Summary of maximal volume polyhedra.

Collection of best known maximal volume polyhedra.

D. M. Aubertin, *Optimization of Four-Dimensional Polytopes*, Masters thesis,
Southern Illinois University, Carbondale, Illinois, (1989).

J. D. Berman & K. Hanes, *Volumes of polyhedra inscribed in the unit sphere in
E ^{3}*, Mathematische Annalen vol. 188 (1970), pp. 78-84.

P. W. Fowler, J. E. Cremona & J. I. Steer,
*Systematics of bonding in non-icosahedral carbon clusters*,
Theor Chim Acta, vol 73 1–26 (1988).

M. Goldberg, *The Isoperimetric Problem for Polyhedra*, Tohoku Mathematics
Journal vol. 40 (1935), pp. 226-236.

M. Goldberg, *A Class of Multi-Symmetric Polyhedra*, Tohoku Mathematics
Journal vol. 43 (1937), pp. 104-108.

D. W. Grace, *Search for largest polyhedra*, Math. Comp, 17, 197–199 (1963).

D. P. Hardin & E. B. Saff, *Discretizing Manifolds via Minimum Energy Points*,
Notices Amer. Math. Soc., vol 51 1186–1194 (2004).

D. P. Hardin, T. Michaels & E.B. Saff, *A Comparison of Popular Point Configurations
on S ^{2}*, Dolomites Research Notes on Approximation, vol 9 16–49 (2016).

R. H. Hardin, N. J. A. Sloane & W. D. Smith, *Spherical Codes*, book in preparation.
http://neilsloane.com/maxvolumes/index.html

V. Klee, *Viewer's Manual to accompany the film Shapes of the Future -
Some Unsolved Problems in Geometry, Part II: Three Dimensions*,
distributed by Modern Learning Aids (1972).

A. Lengyel, Z. Gáspár & T. Tarnai, *The Roundest Polyhedra with Symmetry Constraints*,
Symmetry 2017, 9(3), 41; doi:10.3390/sym9030041

S. Lhuilier, *De relatione mutua capacitatis et terminorum figurarum etc.*,
Varsaviae (1782).

L. Lindelöf, *Propriétés générales des polyhèdres etc.*, Bull. Acad. Sci. St.
Pétersbourg 14 (1869) p.257.

L. Lindelöf, *Recherches sur les polyèdres maxima*, Series: Acta Soc. Sci. Fenn.
vol 24, Officina Typographica Societatis Litterariae Fennicae, Helsingfors, 1899.

H. Minkowski, *Allgemeine Lehrsätze über die convexen Polyheder*, Göttingen, Nach.
Ges. Wiss. Math. phys. (1897), pp. 198-219.

A. Schoen, *A defect-correction algorithm for minimizing the volume of a simple
polyhedron which circumscribes a sphere*, in Proc. of 2nd Annual ACM Symposium on
Computational Geometry, Yorktown Heights, N.Y., 2–4 June 1986, ACM Press, 1986, p.159.

A. Schoen, *Supplement to a ‘defect-correction algorithm for minimizing the volume
of a simple polyhedron which circumscribes a sphere’*, Technical report No 86-01*,
Department of Computer Science, Southern Illinois University, Carbondale, Illinois, (1986),
p.1.

T. Tarnai, Z. Gáspár & A. Lengyel (2013) *From spherical circle
coverings to the roundest polyhedra*, Philosophical Magazine, 93:31-33, 3970-3982, DOI:
10.1080/14786435.2013.800652

Wayne Deeter - wrd@deetour.net

Last modified: March 21, 2018