WOLFRAM NOTEBOOK

WOLFRAM|DEMONSTRATIONS PROJECT

Powered Clique Polyhedra

​
-
+
pick a polyhedron
1
power of ν
usagecount
colorused
-5
3
-3
3
-2
9
-1
6
0
9
1
12
2
15
3
21
4
15
5
15
6
18
7
15
8
12
9
12
10
3
12
3
with
6
ν
-
2
ν
-1 = 0
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
ν=1.1509639252577…
, a root of
6
ν
-
2
ν
-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
6
ν
-
2
ν
-1=0
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
6
ν
-
2
ν
-1=0
in three dimensions. Up to 15 points are shown as spheres. All distances between two points are powers of
ν
. 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.
Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.