12 | June | 2014 | RobertLovesPi.net

Stellated Polyhedron Featuring Self-Intersecting Regular Decagons

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Stellated Polyhedron Featuring Self-Intersecting Regular Decagons

I created this using Stella 4d: Polyhedron Navigator, a program you can find athttp://www.software3d.com/Stella.php.

To make this, I started with the dual of the great rhombicosidodecahedron, a polyhedron known as the dysdyakis triacontahedron. I then augmented half of its faces with tall prisms (thereby creating the chirality in this polyhedron), and took the convex hull of the result. The sixty red triangles are the tops of the augmentation-prisms.

A stellation of the above polyhedron, and a color-change, produced this result, also chiral. It may be enlarged with a click.

These polyhedra were created using Stella 4d, a program which you may buy — or try for free, as a trial download — at http://www.software3d.com/Stella.php.

A Polyhedron Featuring Sixty Irregular, Convex Hexagons and Thirty Rhombi

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A Polyhedron Featuring Sixty Irregular, Convex Hexagons and Thirty Rhombi

I created this using Stella 4d: Polyhedron Navigator, a program you can find at http://www.software3d.com/Stella.php.

A Polyhedron Featuring Sixty Octagons and Sixty Triangles

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A Polyhedron Featuring Sixty Octagons and Sixty Triangles

If someone had asked me if it were possible to form a symmetric polyhedra out of irregular triangles and octagons, using exactly sixty of one type each, I would have guessed that it were not possible. Why does it work here? Part of the reason is that each triangle borders three octagons, and each octagon borders three triangles — a necessary, but not sufficient, condition. This is a partial truncation of an isomorph of the pentagonal hexacontahedron, the dual of the snub dodecahedron. As such, no surprise — it’s chiral.

This was made while stumbling about in the wilderness of the infinite number of possible polyhedra using Stella 4d: Polyhedron Navigator. You can get it here: http://www.software3d.com/Stella.php.