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.