About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things, a large portion of which are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.

Snowfall in Blue

Like much of America, we’ve had a lot of snow lately, with more on the way. Making the piece above helped me unwind after a very stressful drive to the store and back on snow- and ice-covered roads. The car went skidding all over the place a few times, but I did make the trip without actually running into anything. I’m not going out again until the melt arrives.

A Tessellation Featuring Regular Heptagons and Triangles, as well as Six- and Four-Pointed Stars

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It’s a complete accident that this tessellation popped up a second time here — I had forgotten about the first one. At least it has different colors!

Creating a Polyhedron Featuring 72 Regular Pentagons

This polyhedron contains 72 regular polygons, 20 equilateral triangles, and 302 faces in all. But where did it come from?

I made it, starting with the snub dodecahedron, using Stella 4d (available here).

This is the dual of the snub dodecahedron. It is called the pentagonal hexecontahedron.

There are various ways of combining polyhedra with their duals, and the one I used here is called morphing duals by expansion. Here’s the 50%-morphed version.

The next step was to use Stella’s “try to make faces regular” function, which worked best for the 72 pentagonal faces, as well as the twenty of the equilateral triangles.

From there, simple changes of face-color produced the polyhedron shown at the top of this post.

An Intermediate Form Between the Icosidodecahedron and the Rhombic Triacontahedron

This polyhedron combines the faces of an icosidodecahedron (red and blue) with the those of a rhombic triacontahedron (green). The gaps between those two sets of polygons are the yellow rectangles. I made this using the “morph duals by expansion” function of Stella 4d: Polyhedron Navigator. You can try this program for yourself, free of charge, at http://www.software3d.com/Stella.php.