 Powered Clique Polyhedra

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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

Snapshots   Details

References

 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.

Permanent Citation

Ed Pegg Jr

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send