The Truncated Truncated Icosahedron

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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.

truncated trunctaed icosahedron

A Pentacontahedron Featuring Six Regular Dodecagons, Eight Equilateral Triangles, Twenty-Four Trapezoids, and Twelve Rectangles

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A Pentacontahedron Featuring Six Regular Dodecagons, Eight Equilateral Triangles, Twenty-Four Trapezoids, and Twelve Rectangles

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

Tessellation Using Regular Hexacontakaihexagons, Regular Dodecagons, and Two Different (and Unusual) Concave, Equilateral Polygons

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Tessellation Using Regular Hexacontakaihexagons, Regular Dodecagons, and Two Different (and Unusual) Concave, Equilateral Polygons

Hexacontakaihexagons have 36 sides, and dodecagons, of course, have twelve. When a regular hexacontakaihexagon is surrounded by twelve regular dodecagons, in the manner shown here, adjacent dodecagons almost, but not quite, meet at vertices. The gaps between these near-concurrent vertices are so small that they cannot be seen in this diagram — a zoom-in would be required, with thinner line segments used for the sides of the regular polygons.

As a result, the yellow and purple concave polygons aren’t what they appear, at first, to be. They look like triconcave hexagons, but this is an illusion. The yellow ones, in sets of two regions that aren’t quite separate, are actually tetraconcave, equilateral dodecagons with a very narrow “waist” separating the two large halves of each of them. As for the purple ones, they appear to occur in groups of four — but each set of four is actually one polygon, with three such narrow “waists” separating four regions of near-equal area. These purple polygons are, therefore, equilateral and hexaconcave icosikaitetragons — that is, what most people would call 24-gons.