Tag Archives: geometry

A Twice-Zonohedrified Dodecahedron

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If one starts with a dodecahedron, and then creates a zonohedron based on that solid’s vertices, the result is a rhombic enneacontahedron.

If, in turn, one then creates a new zonohedron based on the vertices of this rhombic enneacontahedron, the result is this 1230-faced polyhedron — a twice-zonohedrified dodecahedron. Included in its faces are thirty dodecagons, sixty hexagons, and sixty octagons, all of them equilateral.

Stella 4d: Polyhedron Navigator was used to perform these transformations, and to create the rotating images above. You can try this program for yourself, free, at http://www.software3d.com/Stella.php.

Two Symmetrohedra Featuring Decagons

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Symmetrohedra are polyhedra which have some form of polyhedral symmetry, as well as having regular polygons for many (not necessarily all) of their faces. I found these two symmetrohedra while playing around with Stella 4d: Polyhedron Navigator — a program you can try for free at this website. Each of these symmetrohedra have 74 faces, with twelve of them being regular decagons.

From the Great Rhombicosidodecahedron to Something Much Stranger

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This is the great rhombicosidodecahedron, one of the thirteen Archimedean solids.

Here’s its dual, the disdyakis triacontahedron.

I use a program called Stella 4d to make these .gifs and manipulate polyhedra, and one of Stella‘s functions is “try to make faces regular.” I performed this function on the disdyakis triacontahedron, which has ten triangles meeting at some vertices — so 600 degrees’ worth of triangle-angles tried to squeeze in around those points when the faces were made to be regular. This forces the polyhedron to become non-convex — to the point of looking wrinkled.

“That’s weird looking,” I thought. “I wonder what its dual looks like?” With Stella, I could find out with one mouse-click, and I was most surprised by the result.

In this polyhedron, there are thirty orange rectangles, twelve light blue 10/4-gons, and twenty violet 6/2-gons. None of them are regular. Here are what the faces look like in isolation, starting with an orange rectangle, then a light blue 10/4-gon, and lastly a violet 6/2-gon.

If you’d like to try Stella for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.