Biggest Little Polyhedron

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A polyhedron has vertices. The greatest distance between vertices is 1. What is the maximum volume of the polyhedron? This is known as the biggest little polyhedron problem.

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For four vertices, the solution is trivially the regular tetrahedron.

Five vertices require an equilateral triangle and a perpendicular unit line; this was solved in 1976 [1].

Six vertices require a more complex solution, which was solved to four digits of accuracy in 2003 [2, 3].

The author found exact solutions for 6, 7, 8, 9, 10, 11, and 16 points [4, 5]. This Demonstration contains those solutions, as well as the best known solutions up to 128 points. Oleg Vlasii improved many of these values.

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Contributed by: Ed Pegg Jr (November 2015)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The unit-rod polyhedron shows a rod between all vertex pairs a unit distance apart.

The unit-star picture builds polygons from half-unit rods meeting at a vertex.

The unit-length graph shows how vertices a unit distance apart are connected.

The planar vertex map puts the vertices on a sphere, then unrolls the sphere into a planar form.

In the supported triangles image, all vertices of a blue triangle (with brown number) are at distance 1 from an opposing vertex with a matching green number.

The Initialization section contains various programs that may be able to improve some of the solutions.

References

[1] B. Kind and P. Kleinschmidt, "On the Maximal Volume of Convex Bodies with Few Vertices," Journal of Combinatorial Theory, Series A, 21(1) 1976 pp. 124–128. doi:10.1016/0097-3165(76)90056-X.

[2] A. Klein and M. Wessler, "The Largest Small -dimensional Polytope with Vertices," Journal of Combinatorial Theory, Series A, 102(2), 2003 pp. 401–409. doi:10.1016/S0097-3165(03)00054-2.

[3] A. Klein and M. Wessler, "A Correction to 'The Largest Small -dimensional Polytope with Vertices,'" Journal of Combinatorial Theory, Series A, 112(1), 2005 pp. 173–174. doi:10.1016/j.jcta.2005.06.001.

[4] E. Pegg Jr. "Biggest Little Polyhedra" from Wolfram Community—A Wolfram Web Resource. (Oct 28, 2015) community.wolfram.com/groups/-/m/t/463699.

[5] E. Pegg Jr. "Biggest Little Polyhedron—New Solutions in Combinatorial Geometry" from Wolfram Blog—A Wolfram Web Resource. (May 20, 2015) blog.wolfram.com/2015/05/20/biggest-little-polyhedronnew-solutions-in-combinatorial-geometry.



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