# Powered Clique Polyhedra

Powered Clique Polyhedra

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 [1]. This triangle, shown further in [2], has edges that are all powers of , a root of --1=0. In my Demonstration [3], I showed other polygons that could be wheel-divided into similar powered triangles. These wheels used many polynomials, but --1=0 was special among them, occurring more than three times as many times as any other in the set of solution polygons.

ν=1.1509639252577…

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This Demonstration looks at --1=0 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.

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