A Rotating Zonohedron 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

The State of Things with Artificial Intelligence (Not Fiction) 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…

Modified Square Antiprism with Pentagons as Lateral Faces I made this using Stella 4d, which you can try for free at

#topten #music #Beatles #TheBeatles

An Enneahedron 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

An Involuntary Conversation with #Siri

Fight Scenes with Helen Anderson, a/k/a Defiance All these images were produced using Google Gemini, an A.I., at Helen is a superhero character from a role-playing game called Champions.

A Comparison of the Image-Creation Abilities of Microsoft Copilot and Google Gemini 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…

A Compound of Two Square-Based Pyramids I made this using Stella 4d, which you can try for free at

Using ChatGPT To Make Images of Role-Playing-Game Characters: Helen Anderson 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…

I asked ChatGPT to Make an Image of a Stack of Dodecahedra. Here is the result. 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…

Seven Polyhedra with Eightfold Dihedral Symmetry, Starting with the Octagonal Antiprism 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…

A Polyhedron Featuring Kites 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

Triangles and Their Trideans 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"…

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Four Polyhedra With 7-Fold Dihedral Symmetry 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,…

Uninstall Yourself, You A.I. You! 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!

robertlovespi.net/2025/05/31/m... #mandala #mathematics #geometry #mathsky

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

Psycho Bunny 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.

robertlovespi.net/2025/05/22/t... #mathsky #geometry #polyhedra #polyhedron #icosidodecahedron #icosidodecahedra #mathematics

Two Views of a Paper Model of an Icosidodecahedron 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.

Two Views of a Paper Model of the Great Dodecahedron 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,…

robertlovespi.net/2025/05/10/o... #CalvinandHobbes #comicstrip #screwdriver #oneordinaryscrewdriver #funny #humor #comic #Calvin

robertlovespi.net/2025/05/10/o... #Calvin #CalvinandHobbes #screwdriver #oneordinaryscrewdriver #cartoon #funny #comicstrip

“One Ordinary Screwdriver” and Single-Panel-Only Cartooning in “Calvin and Hobbes,” By Bill Watterson 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…

robertlovespi.net/2015/09/22/r... #art #rainbow #spectrum #reflect #reflection #reflected

robertlovespi.net/2025/05/07/t... #polyhedra #polyhedron #geometry #mathematics #mathsky

robertlovespi.net/2025/05/04/a... #mathsky #mathematics #polyhedra #polyhedron #geometry

A Great Stellated Dodecahedron, Augmented With More Great Stellated Dodecahedra I made this using Stella 4d, which you can try for free here.

robertlovespi.net/2025/04/26/t... #hexagon #tessellation #geometry #mathematics #mathsky

Tessellation of Regular Hexagons, Regular Star Hexagons, and Rhombi

robertlovespi.net/2016/01/16/a... #Arch #ArchesNationalPark #art #acrylic #paint #painting

robertlovespi.net/2013/01/31/p... #art #acrylic #portrait #paint #painting

Khalil was NOT GRANTED DUE PROCESS If they don't like your thoughts...⬇️⬇️

So What Is the Point of Learning, Anyway? 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…

Foldable Snake Toys, of Two Sizes 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…

Two Views of a Faceted Truncated Octahedron 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,…

robertlovespi.net/2025/03/16/a... #mathsky #mathematics #geometry #polyhedra #polyhedron #symmetrohedra #symmetrohedron

A 32-Faced Symmetrohedron With Tetrahedral Symmetry 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.

robertlovespi.net/2025/03/15/a... #mathsky #mathematics #faceted #faceting #facet #geometry #polyhedra #polyhedron

A Faceted Snub Cube, and Its Dual 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…

robertlovespi.net/2025/02/22/o... #potus #president #donald #trump #criminal #felon #convicted

On the Felon-in-Chief Always remember: President Donald Trump is a convicted felon.

robertlovespi.net/2025/02/21/f... #mathsky #mathematics #geometry #polyhedron #polyhedra

Four Triangular Dipyramids, Surrounding a Common Point 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…

robertlovespi.net/2025/02/20/a... #mathsky #geometry #mathematics #tessellation