# Spectral Icosahedra

Stella 4d: Polyhedron Navigator has a “put models on vertices” function which I used to build this cluster of 61 icosahedra. If you’d like to try this software for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.

# One Dozen Icosahedra

I made this using Stella 4d, software you can try for yourself right here.

# Augmenting the Icosahedron with Great Icosahedra

These two polyhedra are the icosahedron (left), and the great icosahedron (right).

Since the faces of both of these polyhedra are equilateral triangles, it is possible to augment each of the icosahedron’s twenty faces with a great icosahedron. Here is the result.

I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

# An Icosahedron, Constructed from Smaller Polyhedra

The polyhedra at the vertices are rhombic triacontahedra, and the yellow edges are elongated rhombic prisms. This was made using Stella 4d, software you may try for free at this website.

# A Rhombic Ring of Icosahedra, Leading to a Rhombic Dodecahedron Made of Icosahedra

As it turns out, eight icosahedra form this rhombic ring, by augmentation:

Measured from the centers of these icosahedra, the long and short diagonal of this rhombus are in a (√2):1 ratio. How do I know this? Because that’s the only rhombus which can made this polyhedron, a rhombic dodecahedron, dual to the cuboctahedron.

This rhombic dodecahedral cluster of icosahedra could be extended to fill space, since the rhombic dodecahedron itself has this property, an unusual property for polyhedra. Whether space-filling or not, the number of icosahedron per rhombic-dodecahedron edge could be increased to 5, 7, 9, or any greater odd number. Why would even numbers not work? This is a consequence of the fact that opposite faces of an icosahedron are inverted, relative to each other; a pair of icosahedra (or more than one pair, producing odd numbers > 1 when added to the vertex-icosahedron) must be attached to the one at a rhombic-dodecahedron-vertex to make these two inversions bring the triangular face back around to its original orientation, via an even number of half-rotations, without which this consruction of these icosahedral rhombi cannot happen.

Here’s another view of this rhombic dodecahedron, in “rainbow color” mode:

All images above were produced using Stella 4d, software which may be tried for free right here.

# Three Polyhedral Clusters of Icosahedra

In the last post on this blog, there were three images, and the first of these was a rotating icosahedron, rendered in three face-colors. After making it, I decided to see what I could build, using these tri-colored icosahedra as building blocks. Augmenting the central icosahedron’s red and blue faces with identical icosahedra creates this cubic cluster of nine icosahedra:

If, on the other hand, this augmentation is performed only on the blue faces of the central icosahedron, the result is a tetrahedral cluster of five icosahedra:

The next augmentation I performed started with this tetrahedral cluster of five icosahedra, and added twelve more of these icosahedra, one on each of the blue faces of the four outer icosahedra. The result is a cluster of 17 icosahedra, with an overall icosahedral shape.

All of these images were made using Stella 4d, which is available at http://www.software3d.com/Stella.php.

# On Icosahedra, and Pyritohedral Symmetry

In this icosahedron, the four blue faces are positioned in such a way as to demonstrate tetrahedral symmetry. The same is true of the four red faces. The remaining twelve faces demonstrate pyritohedral symmetry, which is much less well-known. It was these twelve faces that I once distorted to form what I named the “golden icosahedron” (right here: https://robertlovespi.wordpress.com/2013/02/08/the-golden-icosahedron/), but, at that point, I had not yet learned the term for this unusual symmetry-type.

To most people, the most familiar object with pyritohedral symmetry is a volleyball. Here is a diagram of a volleyball’s seams, found on Wikipedia.

Besides the golden icosahedron I found, back in 2013, there is another, better-known, alteration of the icosahedron which has pyritohedral symmetry, and it is called Jessen’s icosahedron. Here’s what it looks like, in this image, which I found at http://en.wikipedia.org/wiki/Jessen%27s_icosahedron.

The rotating icosahedron at the top of this post was made using Stella 4d, a program which may be purchased, or tried for free (as a trial version) at http://www.software3d.com/Stella.php.