11453

# 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 . In my Demonstration [3], 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.
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.

### DETAILS

References
[1] 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.
[2] E. Pegg Jr. "Zak's Triangle" from Wolfram Community—A Wolfram Web Resource. community.wolfram.com/groups/-/m/t/851275.
[3] E. Pegg Jr. "Wheels of Powered Triangles" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/WheelsOfPoweredTriangles.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.