dodecahedron | RobertLovesPi.net

Dodecahedral Cluster of Truncated Dodecahedra

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Dodecahedral Cluster of Truncated Dodecahedra

I made this using Stella 4d, which you can find right here: http://www.software3d.com/Stella.php.

Seven Variants of the Truncated Dodecahedron

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Several Variants of the Truncated Dodecahedron

The smaller pics may be enlarged with a click.

I used Stella 4d to assemble these polyhedra. You may try this program for free at http://www.software3d.com/Stella.php .

A Polyhedral Cage Which Includes the Dodecahedron, the Icosahedron, and the Rhombic Triacontahedron

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A Polyhedral Cage Which Includes the Dodecahedron, the Icosahedron, and the Rhombic Triacontahedron

The dodecahedron’s edges pass through the purple squares (edge midpoints) and red hexagons (vertices), and have blue decagons above their pentagonal face-centers. The blue decagons’ centers also mark the vertices of the triangular faces of the icosahedron, each of which has a purple square as a side-midpoint, and a red hexagon over its face-center. The rhombic triacontahedron’s faces have blue decagons at the vertex of each acute angle, and red hexagons at the obtuse angle vertices, with purple squares above the rhombic faces’ centers.

I used Stella 4d to make this image, and you can find that program at http://www.software3d.com/Stella.php.

A Truncated Icosahedron / Dodecahedron Blend

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A Truncated Icosahedron / Dodecahedron Blend

I used Stella 4d, software you can find at http://www.software3d.com/Stella.php, to make this image.

The Twelfth Stellation of the Icosahedron/Dodecahedron Compound

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The Twelfth Stellation of the Icosahedron/Dodecahedron Compound

Software credit: I made this using Stella 4d, which you can find at http://www.software3d.com/Stella.php.

The Hyperspace Analog of the Dodecahedron/Icosahedron Compound

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The Hyperspace Analog of the Dodecahedron/Icosahedron Compound

The dodecahedron and the icosahedron are dual to each other, and can be combined to make this well-known compound.

In hyperspace, the analog to the dodecahedron is the hyperdodecahedron, also known as the 120-cell, as well as the hecatonicosachoron. Its dual is the 600-cell, or hexacosichoron, made of 600 tetrahedral cells. The image at the top is the compound of these two polychora, rotating in hyperspace.

These images were made using Stella 4d, available at http://www.software3d.com/Stella.php.

Two Polyhedra Based on the Icosahedron, the Dodecahedron, and the Rhombic Triacontahedron

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Two Polyhedra Based on the Icosahedron, the Dodecahedron, and the Rhombic Triacontahedron

Both of these were made using Stella 4d: Polyhedron Navigator. It’s available at http://www.software3d.com/Stella.php.

If you want to enlarge this second image, simply click on it.

A Dodecahedron, Augmented with Twelve Pentagonal Cupolae

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A Rhombicosidodecahedron, with Twelve Pentagonal Cupolae Excavated

Software credit: I made this using Stella 4d, available at http://www.software3d.com/Stella.php (free trial download available).

Three-Color Arrangement for the Dodecahedron

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Three-Color Arrangement for the Dodecahedron

There are well-known symmetrical 4- and 6-color arrangements for the dodecahedron, and the rhombic dodecahedron has such arrangements in 3, 4, and 6 colors. What’s different about the (Platonic) dodecahedron that the 3-color arrangement you see here doesn’t make the cut, yet there is one for the rhombic dodecahedron?

The answer: in the other arrangements mentioned, faces of the same color do not share edges. Here, they do, so this one is usually not listed with the others.

Software credit: please visit http://www.software3d.com/Stella.php and try the free download of Stella 4d available there. It’s the program I used to make this image. And, as for the color arrangements mentioned above, they’re pre-loaded into the Stella interface as easy-to-find options.