icosahedron | RobertLovesPi.net

A Faceted Version of a Truncation of the Icosahedron

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A Faceted Version of a Truncation of the Icosahedron

I made this with Stella 4d, a program you can find at http://www.software3d.com/Stella.php.

The Truncated Truncated Icosahedron

The icosahedron has twenty triangular faces. Truncate it once, and the triangles become hexagons, with pentagons appearing under the pyramids removed in the truncation. This is the “soccer ball” shape familiar to millions.

If you take this figure and truncate it again, the twenty hexagons become twenty dodecagons, the twelve pentagons each become decagons, and sixty isosceles triangles appear under the pyramids removed by this second truncation.

I made this image using Stella 4d, a program you can find at www.software3d.com/Stella.php. Also, just for fun, here’s a version of it with the colors switched around, and with a slight bounce as it rotates in the other direction.

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.

An Icosahedron-Variant Featuring Regular Decagons

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An Icosahedron-Variant Featuring Regular Decagons

I used Stella 4d, software you can find at 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.

Faced-Based Zonish Versions of the Icosahedron and the Icosidodecahedron

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Faced-Based Zonish Versions of the Icosahedron and the Icosidodecahedron

I’ve had some success lately finding near-misses to the Johnson solids by making face-based zonish versions of various polyhedra. These were found during that search, and are certainly not near-misses, but I still find them interesting, primarily due to their symmetry. Like the others, they were found using Stella 4d, which you can try or buy at http://www.software3d.com/Stella.php.

The top image was formed by making this modification to the icosahedron, and the one below (which you can enlarge with a click) was created by doing the same thing to an icosidodecahedron.

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.

Four Hollow Polyhedron-Variants Related to the Rhombic Triacontahedron and the Icosahedron

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Four Hollow Polyhedron-Variants Related to the Rhombic Triacontahedron and the Icosahedron

To make any of these smaller images as large as the first one, simply click on the smaller image of your choice.

These are all members of the same stellation-sequence, just like the two octahedron-variants in the last post.

I made these in just a few minutes with software, written by a friend of mine, called Stella 4d: Polyhedron Navigator. You can check it out for yourself at http://www.software3d.com/Stella.php. A free trial download is available.