Mike Summers titleDesign
The ICOSA and DODECAHEDRON'S nested solids
On the previous page we looked at how the ICOSAHEDRON and it's dual the DODECAHEDRON have 31 axis of rotation and the circles that these define. You will notice that they intersect at regular locations around a sphere. These correspond to the apices and mid edge points of the ICOSA and DODECAHEDRON'S if they are projected onto the surface of a sphere. By connecting these locations with straight lines we find that TETRAHEDRONS, CUBES and OCTAGONS are defined in multiple orientations.
5 TETRAHEDRA fit inside the 31 great circles of the ICOSAHEDRON and DODECAHEDRON. These align to the apices of the DODECAHEDRON. [move over the image to see an animation] 5 interlocking tetrahedrons
There are in fact 10 TETRAHEDRA that fit inside the 31 great circles of the ICOSAHEDRON and DODECAHEDRON. These align to the apices (vertices) of the DODECAHEDRON. [move over the image to see an animation] 10 interlocking tetrahedrons
There are 5 CUBES that fit inside the 31 great circles of the ICOSAHEDRON and DODECAHEDRON. These like the tetrahedra align to the apices of the DODECAHEDRON. [move over the image to see an animation] 5 interlocking cubes
There are 5 OCTAGONS that fit inside the 31 great circles of the ICOSAHEDRON and DODECAHEDRON. These align to the middle of the edges of the ICOSAHEDRON and DODECAHEDRON. [move over the image to see an animation] 5 Interlocking octagons

Also...

A pentagonal plane can be defined across the base of 5 triangular faces of an ICOSAHEDRON. There are 12 of these pentagonal planes that make a 60 faced convex polyhedra. [move over the image to see an animation]

12 interlocking pentagons
This animation is of a CAD model of an ICOSAHEDRON where the 12 pentagonal planes have been substituted by model starfish with their arms intertwined. [move over the image to see an animation] 12 intwined starfish
A further pentagon can be defined between the pentagonal faces of the DODECAHEDRON. There are 12 of these which meet at the apices of the pentagon faces and form pentagonal stars across them. [move over the image to see an animation] 12 more interlocking pentagons
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