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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.

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.

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 .

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 Truncated Form of the Rhombic Dodecahedron

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A Truncated Form of the Rhombic Dodecahedron

This is not the only truncated form of the rhombic dodecahedron. In this polyhedron, square-based pyramids have been “truncated away” from the rhombic dodecahedron’s four-valent vertices, but the three-valent vertices remain untouched.

If this truncation is done in such a way as to leave the hexagonal faces equilateral (which is not done here), so that all edges of the polyhedron have the same length, the result is called a “chamfered cube.” This closely-related figure may be seen at https://en.wikipedia.org/wiki/Chamfered_cube.

[Image credit: this rotating model was created using Stella 4d, software you can find right here, with a free trial download available.]

20-Thex: A Four-Dimensional Polytope

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The 20-Thex: A Four-Dimensional Polytope

In hyperspace, or four-space, there are six regular polychora, analogous to the Platonic Solids in three-space. Beyond the Platonics in the study of polyhedra comes, of course, the Archimedean Solids, which include many truncated forms of Platonic polyhedra.

In hyperspace, there are varieties of progressively-less regular polychora, also, and one of these, in a group called the truncates, is called 20-thex, or simply the “thex.” (Those are short names for this polychoron; it’s also called the truncated hexadecachoron, or truncated 16-cell.) What you see above is a (seemingly) three-dimensional projection of a thex, as it rotates in hyperspace.

Just as polyhedra have polygons as faces, polychora have polyhedra as unit cells. This is the net for the thex. As you can see, the thex is composed of both truncated tetrahedra and octahedra.

Both of these images were created using Stella 4d, which you can try for yourself at http://www.software3d.com/Stella.php.

A Variant of Kepler’s Stella Octangula

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A Variant of Kepler's Stella Octangula

Johannes Kepler named the compound of two tetrahedra the “stella octangula,” thus helping make it one of the best-known polyhedral compounds today. This variant uses triakis tetrahedra in place of the Platonic tetrahedra in that compound. The triakis tetrahedron is a Catalan solid, and is dual to the truncated tetrahedron.

Software credit: see http://www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.