iiinova — where your digital self keeps evolving

Peter Kagey @peterkagey.com · 30/06/2026

Have you seen the Swirled Series? (isohedral.ca/swirled-seri...) "Early in November, Daniel Piker (aka @KangarooPhysics) suggested that a group of people could get together online, and each create a short segment of animation, arranged so that all the start and end frames are identical..."

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Peter Kagey @peterkagey.com · 18/06/2026

Check out Alexander Graham Bell’s Tetrahedral Kites from Public Domain Review! (More photos if you follow the link!) publicdomainreview.org/collection/a...

Four people standing in a field with a kite that looks like a torus

An airplane-looking kite made out of a tetrahedral-octahedral configuration

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Peter Kagey @peterkagey.com · 06/06/2026

See even more renderings on Jiabao Li's website! www.jiabaoli.org/math-playgro...

A rendering of a mathematical climbing structure with children playing on it.

A rendering of a mathematical merry-go-round.

"Inside the merry-go-round is a zoetrope. The inside wall contains a sequence of models and the outside has narrow slits. As the carousel spins, people sitting on it see the still models animate."

Rendering of a Lissajous Swing Set.

"Swings hang from two pivot points so each rider creates a Lissajous curve in the air. A small sandbag under the seat releases sand as the swing moves. The motion leaves patterns in the sand that look like flowers or knots. Each person creates a unique pattern because the curve depends on the person’s weight, the swinging rhythm and speed."

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Peter Kagey @peterkagey.com · 06/06/2026

Check out Jiabao Li's project of "transforming abstract math into climbable sculptures," complete with a rendering of a mathematical playground! "What if others could literally play with math—climb on it, move around it, think about it differently?" infinitesums.simonsfoundation.org/math-ambassa...

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Peter Kagey @peterkagey.com · 17/05/2026

Check this amazing photo which was taken by Mona Hassan Abo-Abda of the Giza Pyramids during Forever is Now exhibition in Cairo, Egypt which won the Wiki Loves Monuments competition in 2023. www.wikilovesmonuments.org/galleries/20...

Giza Pyramids behind a wireframe sculpture of hands during Forever is Now exhibition by Mona Hassan Abo-Abda.

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Peter Kagey @peterkagey.com · 10/05/2026

I made an animation of a camera traveling through a Boerdijk–Coxeter helix (en.wikipedia.org/wiki/Boerdij...)

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Peter Kagey @peterkagey.com · 08/04/2026

I'm not sure I can explain it, but this is @motivickyle.bsky.social.

Top few rows of parity triangle which vaguely look like a face

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Peter Kagey @peterkagey.com · 27/02/2026

Inspired by @robinhouston.mathstodon.xyz.ap.brid.gy's G4G16 gift of tetragonal disphenoidal blocks, I've made some magnetized versions, which are satisfyingly click-y!

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Peter Kagey @peterkagey.com · 27/12/2025

A video of the pen plotter in action!

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Peter Kagey @peterkagey.com · 11/12/2025

I asked a Code Golf Stack Exchange question about this too, if you want to read more. codegolf.stackexchange.com/q/181203/53884

An illustration of a non-self-intersecting maximal loop on a torus.

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Peter Kagey @peterkagey.com · 28/10/2025

I just heard from David that he found (and fixed!) a subtle bug in the collision checking. It turns out that you can have two tetrahedra, T₁ and T₂ whose interiors intersect, but where T₁ (in red) does not have any edges that intersect T₂'s faces, as shown in this illustration that David sent me.

A yellow and red tetrahedron colliding, but the red edges do not intersect any yellow faces.

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Peter Kagey @peterkagey.com · 21/08/2025

A Mathematica demo I made of wandering around a parameter space of the convex hull of various scalings the vertices of a icosahedron, dodecahedron, and icosidodecahedron. Given the right parameters, we can recover all Platonic and Catalan solids with full icosahedral symmetry.

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Peter Kagey @peterkagey.com · 20/08/2025

Here's a 4-second video showing that little robot at work! ✍️🤖

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Peter Kagey @peterkagey.com · 20/08/2025

Made some Escher-inspired magnetic refrigerator magnets by using my AxiDraw V3 pen plotter to draw some tiles on a magnetic vent cover.

AxiDraw V3 in progress drawing the tiles

Square magnets with identical patterns on them attached to a refrigerator door

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Peter Kagey @peterkagey.com · 18/08/2025

Painted my flower pot with Truchet tiles. (1 of 7429) (oeis.org/A368306)

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Peter Kagey @peterkagey.com · 31/07/2025

I learned how to use a CNC machine and wrote G-code today for the first time.

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Peter Kagey @peterkagey.com · 28/07/2025

Funhouse mirror icosahedron selfie made possible by @joshmillard.bsky.social

A face distorted through a picture taken through a stained-glass icosahedron.

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Peter Kagey @peterkagey.com · 10/07/2025

Am I tempted to travel to Austria just for the purpose of climbing through a 97 m³ spatial net? 😏 berliner-playequipment.com/us/referenze...

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Peter Kagey @peterkagey.com · 07/07/2025

Just this past month, I've been working with Pontus von Brömssen and Bert Dobbelaere on analogous problems for other space-filling polyhedra. Here are all of the ways of placing 4 cells (each 1/12 of a rhombic dodecahedron) of the rhombic pyramidal honeycomb face-to-face. oeis.org/A385274

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Peter Kagey @peterkagey.com · 03/07/2025

Hay 187 formas posibles que se pueden formar a partir de 5 de estas fichas.

The 187 possible shapes that can be formed from 5 of the Valencia tiles.

Las 187 posibles formas que se pueden formar a partir de 5 fichas de Valencia.

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Peter Kagey @peterkagey.com · 03/07/2025

Up to rotation and translation, there are 3 distinct tiles, 4 connected components of two tiles, 15 of three tiles, 47 of four tiles, and 187 components of five tiles, shown here. Want to guess how many there are for six, seven, and eight tiles?

The 187 ways of choosing 5 Valencia tiles so that each tile is connected to another tile along an edge.

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Peter Kagey @peterkagey.com · 26/06/2025

A few others (e.g. Problems 31, 79, 132) have answers or bounds now because I've worked on the problem. For example, Problem 31 is discussed in my JIS paper with Bill Keehn, "Counting Tilings of the n × m Grid, Cylinder, and Torus."

Problem 31, which asks about ways of tiling grids with various tile designs.

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Peter Kagey @peterkagey.com · 26/06/2025

Some of the problems have answers or bounds now because I've asked about them (e.g. Problem 1, Problem 17, Problem 110)

Problem 17, which asks about ways of placing queens on a n x n chessboard to maximize number of available moves.

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Peter Kagey @peterkagey.com · 26/06/2025

Also, some of the problems, such as Problems 30, 53, 107, and 108, have solutions that were either already known or found after I wrote the problem down.

Problem 53, which asks about convexity classes of n-gons.

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Peter Kagey @peterkagey.com · 26/06/2025

For example, the main problem in Problem 68 is definitely *not* open, although some subquestions might be. If we close off the start/end of the maze, then we get a bijection with spanning trees on a grid graph, so counting the mazes is at most a step away from a standard result.

Problem 68, which asks about mazes on the square grid.

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Peter Kagey @peterkagey.com · 26/06/2025

This collection has 130+ questions that mathematicians might not know the answers to, but that a 10–14 year old can think deeply about (and potentially even solve!) (Here's the specific problem that the student was working on.)

Problem 48 from my Open Problems Collection which asks about the number of folds it takes to fold a strip of equilateral triangles down into a single triangle.

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Peter Kagey @peterkagey.com · 26/06/2025

Last week, I was a faculty member at Camp Conway, a math camp for ~10–14 year olds outside of Los Angeles, CA. The campers have time during their day to think about whatever they want, and I just got an email that a camper has been thinking about a problem from my "Open Problems Collection."

A pencil, paper, and a partially folded piece of paper folded into a strip of equilateral triangles.

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Peter Kagey @peterkagey.com · 18/06/2025

We played GoL with wraparound—so I showed them why we call this a torus too!

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Peter Kagey @peterkagey.com · 18/06/2025

I'm a faculty member at "Camp Conway" this week and I was asked to give a talk during the morning session. Of course, every student who goes to Camp Conway should come home knowing a little bit about Conway's Game of Life, so I made Conway's GoL starting with initial conditions of Conway.

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Peter Kagey @peterkagey.com · 04/06/2025

Don’t sleep on the Ellsworth Kelly stamps! about.usps.com/newsroom/nat...

Ten USPS stamps featuring Ellsworth Kelly art

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Peter Kagey @peterkagey.com · 29/05/2025

And precisely 500 ways of choosing six of them!

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Peter Kagey @peterkagey.com · 29/05/2025

And 123 ways of choosing five of them.

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Peter Kagey @peterkagey.com · 29/05/2025

There are 30 ways of choosing connected components of four of these Highland Park, CA driveway tiles up to rotation, reflection, and translation.

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Peter Kagey @peterkagey.com · 25/05/2025

Next step: buy a 3D printer and print a puzzle!

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Peter Kagey @peterkagey.com · 25/05/2025

First I designed a paper model, and played around with that for about ten minutes without luck. After concluding that I'm better at (and better enjoy) writing code than solving puzzles, I wrote a backtracking algorithm which solved the problem in a few minutes.

Paper model of a puzzle on the faces of a pentagonal icositetrahedron.

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Peter Kagey @peterkagey.com · 21/05/2025

Check out this video from Lars Blomberg illustrating Langton's ant on a snub trihexagonal tiling. Watch all 15 minutes and 43 seconds of this at oeis.org/A309293!

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Peter Kagey @peterkagey.com · 17/05/2025

Here's my (very small) version: POK I printed this on a SLS (powder) printer when I was at Harvey Mudd.

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Peter Kagey @peterkagey.com · 15/05/2025

The triakis icosahedron has 60 faces and A383492(6) = 10 6-forms. And the rhombic triacontahedron has 30 faces, A383490(5) = 12, and so you could potentially use these to tile *two* copies of the rhombic triacontahedron. If you spot any solutions to these or want to work together, let me know!

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Peter Kagey @peterkagey.com · 15/05/2025

There are a few other contenders for puzzles that I spotted, but these require allowing not just rotation of the pieces, but reflection too—which is less good for a puzzle in our 3D world. The tetrakis hexahedron has 24 faces and A383802(4) = 6 4-forms.

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Peter Kagey @peterkagey.com · 15/05/2025

But the sequences for the other twelve Catalan solids were missing from the OEIS, so I filled them in. Here are the A383802(7)=36 distinct ways of choosing 7 connected faces of the tetrakis hexahedron, up to its rotational/reflectional symmetries. oeis.org/A383802 en.wikipedia.org/wiki/Tetraki...

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Peter Kagey @peterkagey.com · 15/05/2025

Other folks have written code to count these objects on the faces of the Platonic solids (see oeis.org/A030135 and oeis.org/A030136 for examples from David W. Wilson) and in 2021 Hilarie Orman counted this in the specific case of the deltoidal hexecontahedron (oeis.org/A340635).

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Peter Kagey @peterkagey.com · 15/05/2025

During this, I've thought of a puzzle in the style of @two-star.bsky.social's puzzles. There are 8 ways of choosing a connected subset of 3 faces of the pentagonal icositetrahedron, showed above. This is perfect, because it has 24 faces. Can we use them to tile? I've fit seven out of eight!

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Peter Kagey @peterkagey.com · 15/05/2025

For the past couple of weeks, I've been playing around with generalizations of polyominoes on the faces of polyhedra—mostly Platonic and Catalan solids. We look at connected subsets of faces, where we consider two subsets the same if you can rotate/reflect from one to the other.

Four examples of 3-subsets of the faces of a deltoidal icositetrahedron.

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Peter Kagey @peterkagey.com · 07/05/2025

My first time doing watercolors since I was a kid. I've colored Lambert azimuthal equal-area projections of the spherical versions of Johnson solids J₂₅ and J₆₆

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Peter Kagey @peterkagey.com · 04/05/2025

I've been playing around with the Lambert azimuthal equal-area projection of a sphere onto a disk. Here are some examples with various spherical polyhedra. Here are examples of an icosahedron, a rhombic dodecahedron, and Johnson solid J₂₅.

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Peter Kagey @peterkagey.com · 03/05/2025

Not to mention the gyroelongated square pyramid I saw yesterday!

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Peter Kagey @peterkagey.com · 03/05/2025

Have you ever seen a snub dodecahedron in the wild?

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Peter Kagey @peterkagey.com · 03/05/2025

Have you ever seen a deltoidal icositetrahedron in the wild?

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