### Basic Equations

• Euler's Equation: V + F = E + 2
 Name Vertices Faces Edges V + F E + 2 Tetrahedron 4 4 6 8 8 Cube 8 6 12 14 14 Octahedron 6 8 12 14 14 Rhombic Dodecahedron 14 12 24 26 26 Icosahedron 12 20 30 32 32 Dodecahedron 20 12 30 32 32 Rhombic Triacontahedron 32 30 60 62 62
• Sum of Surface Angles = V * 360° - 720°
 Name Sum of Angles1 Face Faces Total Sum ofSurface Angles Vertices V*360° V*360°-720° Tetrahedron 60°*3=180° 4 720° 4 1440° 720° Cube 90°*4=360° 6 2160° 8 2880° 2160° Octahedron 60°*3=180° 8 1440° 6 2160° 1440° Rhombic Dodecahedron 360° 12 4320° 14 5040° 4320° Icosahedron 60°*3=180° 20 3600° 12 4320° 3600° Dodecahedron 108°*5=540° 12 6480° 20 7200° 6480° Rhombic Triacontahedron 360° 30 10800° 32 11520° 10800°
• Volume equations
 Name Volume Equation Tetrahedron 1 (Edge Length)^3 Cube 3 (Face Diagonal)^3 Octahedron 4 (Edge Length)^3 Rhombic Dodecahedron 6 (Long Face Diagonal)^3 Icosahedron Dodecahedron Rhombic Triacontahedron

## How The Polyhedra Are Related One To Another

### Intersecting Tetrahedra In Cube

We have just seen how two intersection Tetrahedra define a cube. Two Tetrahedra Define a Cube

### Octahedron In Intersecting Tetrahedra

The intersection of 2 Tetrahedra defines an Octahedron.
 Intersecting Tetrahedra Define Octahedron

### Intersecting Cube And Octahedron Define VE

Intersecting Cube and Octahedron define a Cuboctahedron. Fuller calls the Cuboctahedron the "Vector Equilibrium" because all radial vectors from the center of volume out to a vertex is the same length as the edge vectors. This polyhedron is also called the "VE" for short.
 Intersecting Cube and Octahedron Define VE

### Tetrahedra In Dodecahedron

It is possible to define 10 Tetrahedra within the Dodecahedron utilizing only the vertices of the Dodecahedron.
 10 Tetrahedra In The Dodecahedron

Each of the Dodecahedron's vertices is shared with 2 Tetrahedra. We can eliminate this redundancy by removing 5 Tetrahedra. 5 Tetrahedra In The Dodecahedron

This suggests a spiral vortex motion in each of the Dodecahedron's faces. 5 Tetrahedra In The Dodecahedron Suggests Spiral Motion

### Cubes In Dodecahedron

It is possible to position 5 Cubes within the Dodecahedron.

 5 Cubes In The Dodecahedron

## Rotating Cubes

The model of 5 Cubes in the Dodecahedron suggested to me that the cubes might be positioned within the Dodecahedron by rotations from a single cube position. This is indeed the case.

Consider a single cube. It has 4 Vertex to Vertex rotation axes. 4 Vertex to Vertex Rotation Axes

If we assign a Cube to each of these axes, we have a total of 5 Cubes (original 1 plus 4). We then rotate the 4 Cubes about these 4 axes.  Four Rotating Cubes, One Stationary 5 Cubes in Dodecahedron

## The 120 Polyhedron

We will now put all of the polyhedra together into a single polyhedron. The resulting polyhedron will have 120 triangular faces. It is called the 120 Polyhedron.

This polyhedron was originally described to me by Lynnclaire Dennis. For more information on Ms Dennis and her work, see the Pattern web site at http://www.pattern.org/.  Dodecahedron

Add in all 10 Tetrahedra.  Dodecahedron, Tetrahedra

Add the 5 Cubes.  Dodecahedron, Tetrahedra, Cubes

Add the duals to each of the 5 Cubes; 5 Octahedra.  Dodecahedron, Tetrahedra, Cubes, Octahedra

Recall that each Cube/Octahedron pair defines a rhombic Dodecahedron.  Dodecahedron, Tetrahedra, Cubes, Octahedra, rhombic Dodecahedra

Add the dual to the (regular) Dodecahedron; the Icosahedron.  Dodecahedron, Tetrahedra, Cubes, Octahedra, rhombic Dodecahedra, Icosahedron

Recall that the Icosahedron/Dodecahedron pair defines the rhombic Triacontahedron.  Dodecahedron, Tetrahedra, Cubes, Octahedra, rhombic Dodecahedra, Icosahedron, rhombic Triacontahedron

Connect all the outer vertices together. This defines the 120 Polyhedron.  Dodecahedron, Tetrahedra, Cubes, Octahedra, rhombic Dodecahedra, Icosahedron, rhombic Triacontahedron, 120 Polyhedron  120 Polyhedron

(Note that there are other polyhedra with 120 triangular faces. They will not be discussed here.)