The Truncated Truncated Icosahedron

Image

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

Six Pairs of Parallel Decagons

Image

Six Pairs of Parallel Decagons

Each pair is a different color. Because these decagons intersect in space, but do not meet at edges, they do not form a true polyhedron. They are merely a symmetrical configuration of twelve decagons in space, surrounding a central point.

I made this out a “true polyhedron” by hiding all the other faces from view. Before the hiding and recoloring of faces, this looked this way (you can click on it to enlarge it):

Augmented Convex hull

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

A “Bowtie” Polyhedron Featuring Regular Decagons and Nonagons

Image

This is the sequel to the previous post. Making the decagons and nonagons regular, using Stella 4d, proved to be quite easy!

Software credit: see http://www.software3d.com/stella.php (free trial download available).