[an error occurred while processing this directive]

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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


[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.
[an error occurred while processing this directive]

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+