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.

# Tag Archives: Mathematics

## 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.

## A Blue Tessellation of Rhombi and Regular Hexagons

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# Three Pentagrams (from 2011)

I ran into this on Facebook, where I posted it in 2011. That was the year before I started this blog.

# 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.

# At 53, My Age Is a Prime Number Again

It has long been a tradition of mine to make star polygons to mark my birthdays. Since I’m 53 today, here’s a 53-pointed star.

# A Tessellation for the New Year MMXXI

Here’s another version, with different colors:

# A Tessellation of Regular Hexagons and Squares, as Well as Equilateral and Elongated Hexagons

Here’s the same tessellation, using different colors:

# A Tessellation Featuring Regular Enneagons and Triangles, as Well as Equilateral Three-Pointed Stars

Here’s a second version, with different colors: