This is a zonohedrified pentagonal prism. It has ninety rhombic faces, and they are based on the vertices of that prism. I made it using Stella 4d, which you can buy, or try for free, at

Me: Hello, Gem. Are you doing well today? Google Gemini: Hello there! I'm doing great, thank you for asking. I'm ready and available to help with any questions or tasks you might have. What can I do for you today? Me: I'm a math and science teacher at the high school level. This next year will be my 31st year teaching.

I made this using Stella 4d, which you can try for free at

Enneahedra are nine-faced polyhedra. This particular one has three square faces, and six kite faces. I made it from a triangular prism, using Stella 4d, which you can try for free at

Enneahedra are nine-faced polyhedra. This particular one has three square faces, and six kite faces. I made it using Stella 4d, which you can try for free at

I use an iPad, but I don't like Siri. When she's turned on, it's by mistake, and without the need for her, um, "help." The other day, I accidentally activated Siri. After she said her standard opening bit, I barked at her, "&%$# it, Siri! Go away!" She replied, in the tone of one with hurt feelings, "I'm just trying to help." I hadn't even thought about the fact that she was listening. Someone actually went to the trouble to give proto-A.I., which is what Siri is, the ability to sulk.

All these images were produced using Google Gemini, an A.I., at Helen is a superhero character from a role-playing game called Champions.

I set out to have an AI create some action scenes for Helen Anderson, a superhero character I created years ago for Champions, a role-playing game. Helen is a human/demon hybrid who is super-strong, flies, and can breathe fire. I started with the AI Microsoft Copilot, since it came built-in on this laptop. The big drawback to Microsoft Copilot is its content filters.

I made this using Stella 4d, which you can try for free at

My friend Tony once ran a Champions game (think Dungeons & Dragons with superheroes), and I had a superhero character in his game named Helen Anderson, when she wasn’t simply going by the…

My friend Tony once ran a Champions game (think Dungeons & Dragons with superheroes), and I had a superhero character in his game named Helen Anderson. As an experiment, I tried to use ChatGPT to make a convincing image of this character. The first thing I requested at was "Create an image of a female demon/human hybrid. She has white horns, white fangs, white hair in a 'Mohawk' hairstyle, and white, feathered wings.

When using AI, it is vital to be careful and explicit with your instructions. Mine were simple: "A stack of multicolored dodecahedra." This wasn't a hard question for a thing with access to Google . . . and there's not a pentagon in sight. ChatGPT, you fail this test.

Here's an octagonal antiprism. Next, the dual of this antiprism: a trapezohedron. Its faces are sixteen kites. If you stellate this trapezohedron twice, you get a two-part compound. Now, the compound of the octagonal antiprism and its dual. The next thing I did was to form the convex hull of this compound. The next thing I did was to examine this convex hull's dual.

This polyhedron has sixty faces which are kites, and twelve which are regular pentagons. It was created, by stellating a snub dodecahedron, using software you can find at

Much work has been done on the medians of triangles: segments that connect vertices with the midpoints of opposite sides. This morning, I decided to explore what happens if the trideans are examined, rather than the medians. The reason you don't know the word "tridean" is simple: I just made it up. It's related to a median, though. To find trideans, you must first trisect, rather than bisect, each side of a triangle.

First, the heptagonal antiprism. The rest of the polyhedra in this post were made from this antiprism, using the functions available in Stella 4d: Polyhedron Navigator, such as stellation. If you'd like to try this program yourself, you can do so, free, at

I couldn't figure out how to silence the Microsoft Copilot I don't want installed on my computer, so I asked it for assistance . . . and it worked!

This mandala features ten squares, ten regular hexagons, ten golden triangles, and one ten-pointed star.

This mandala features ten squares, ten regular hexagons, ten golden triangles, and one ten-pointed star.

My wife, Dee Austin, drew this on the whiteboard at the school where we both teach. I like it, so I decided to post it here.

I’m sure I’ll get back to virtual polyhedra soon enough, but, in the meantime, enjoy this icosidodecahedron made using the traditional Euclidean tools, plus scissors, card stock, and ta…

I'm sure I'll get back to virtual polyhedra soon enough, but, in the meantime, enjoy this icosidodecahedron made using the traditional Euclidean tools, plus scissors, card stock, and tape. No computers were used to make this polyhedron.

The obtuse triangles are golden gnomons, isosceles triangles with vertex angles of 108 degrees. They are facelets; the actual faces are the larger regular pentagons of which the golden gnomons are parts. Much of each face is hidden from view, hidden from view in this polyhedron's interior. No computer programs were involved in the construction of this model.

This is one example of an unusual feature sometimes seen in Bill Watterson’s Calvin and Hobbes — a single panel from this cartoon can be presented alone, and actually be funnier, by its…

This is one example of an unusual feature sometimes seen in Bill Watterson's Calvin and Hobbes -- a single panel from this cartoon can be presented alone, and actually be funnier, by itself, than the original full comic strip it came from. Here's the full original comic from which the above image was taken. Here's another example of a C&H…

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Clockwise from bottom right, these polyhedra are the cuboctahedron, the rhombic triacontahedron, the rhombic dodecahedron, the compound of the cube and the octahedron, and the rhombicosidodecahedro…

I made this using Stella 4d, which you can try for free here.

I made this using Stella 4d, which you can try for free here.

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Media used: acrylic on canvas. Although I have not yet seen it in person, this painting is based on images I have seen of Delicate Arch, in Utah’s Arches National Park.

1991-2003, acrylic on canvas, 16″ x 24″

I've loved this particular panel from Bill Watterson's Calvin and Hobbes since I first saw it, back in the 1990s. It appealed initially to my inner smart-ass (which is quite strong), but, over the years, my thinking has changed -- to a simple appreciation of Calvin's question. Slightly altered versions of the question can also be asked about anything we study in mathematics, not just integers.

This is the familiar "magic snake" toy which has been around for many years. The most common version of it is made of 24 right triangular prisms. It can be twisted into many different shapes. Of course, me being me, I wanted to make a polyhedron with it. Here are three of these standard-sized snake-toys, twisted into rhombicuboctahedra. While it isn't easy, it is possible to find longer variants of this toy.

In the image above, the faces of this faceted truncated octahedron are colored by face type. In the one below, the faces are colored by number of sides: blue for triangles, red for quadrilaterals, and yellow for hexagons. I made these using Stella 4d, which you can try for free at this website.

I made this polyhedron (using Stella 4d, which you can try for free here) by modifying the tetrated dodecahedron. Its 32 faces include four regular hexagons, twelve squares, four equilateral triang…

I made this polyhedron (using Stella 4d, which you can try for free here) by modifying the tetrated dodecahedron. Its 32 faces include four regular hexagons, twelve squares, four equilateral triangles, and twelve isosceles trapezoids.

The faceted snub cube shown above is colored by face type. The one below has faces colored by number of sides, with red triangles and yellow quadrilaterals. Here’s the dual of this particular…

The faceted snub cube shown above is colored by face type. The one below has faces colored by number of sides, with red triangles and yellow quadrilaterals. Here's the dual of this particular faceting, shown in "rainbow color mode." I made these virtual polyhedron models using Stella 4d, which you can try for free right here.

Always remember: President Donald Trump is a convicted felon.

Always remember: President Donald Trump is a convicted felon.

I made this using multiple stellations, some of them with tetrahedral symmetry, to modify the cubohemioctahedron, one of the uniform polyhedra. I did this using Stella 4d, which you can try for you…

I made this using multiple stellations, some of them with tetrahedral symmetry, to modify the cubohemioctahedron, one of the uniform polyhedra. I did this using Stella 4d, which you can try for yourself, free, at Also, here's what the cubohemioctahedron looks like, without modification.

This is the same tessellation as the one in the last post, except that this one has had its colors inverted, using MS-Paint.