Powered Clique Polyhedra
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Can a tetrahedron be divided into similar tetrahedra of different sizes? The answer is not known, but a solution for triangles was found by Zak . This triangle, shown further in , has edges that are all powers of , a root of . In my Demonstration , I showed other polygons that could be wheel-divided into similar powered triangles. These wheels used many polynomials, but was special among them, occurring more than three times as many times as any other in the set of solution polygons.[more]
This Demonstration looks at in three dimensions. Up to 15 points are shown as spheres. All distances between two points are powers of . Since they are basically complete graphs, these can be called cliques. One of these polyhedra might lead to a tetrahedron with a similar self-dissection.[less]
Contributed by: Ed Pegg Jr (July 2017)
Open content licensed under CC BY-NC-SA
 A. Zak, "A Note on Perfect Dissections of an Equilateral Triangle," Australasian Journal of Combinatorics, 44, 2009 pp. 87–93. ajc.maths.uq.edu.au/pdf/44/ajc_v44_p087.pdf.
 E. Pegg Jr. "Zak's Triangle" from Wolfram Community—A Wolfram Web Resource. community.wolfram.com/groups/-/m/t/851275.
 E. Pegg Jr. "Wheels of Powered Triangles" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/WheelsOfPoweredTriangles.