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Cubes_Art @cubesart.bsky.social · 18h

I’ve probably doodled this dozens of times, but I just really like this knot.

A 9-layer Rubik's cube solved to depict an 18-crossing mathematical knot.

A trace over a faded copy of the photograph used to create a planar embedding of the 18-crossing knot.

A planar embedding of the 18-crossing knot from the Rubik's cube.

A stylized version of the pattern on the 9-layer Rubik's cube. This shows the embedding of the knot on the cube as a single color.

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Cubes_Art @cubesart.bsky.social · 27/09/2026

Here’s a knot on the 9-cube that my colleague @cubicskies.bsky.social and I call the great-great-granny knot. It’s the connected sum on four trefoil knots.

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Cubes_Art @cubesart.bsky.social · 15/09/2026

Here's a knot on a 9-layer Rubik's cube to cheer you up.

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Cubes_Art @cubesart.bsky.social · 23/08/2026

This is the FTO (face-turning octahedron) twisty puzzle. It’s now an official WCA event And I think it’s lovely. Here, I’ve solved it to a pattern that is striped when viewed from 4 vertices, and concentric squares from 2 vertices.

I imagine other folks have done this pattern, but I just had to try it. I don't know what to call it... Stripes? I don't think it's octahedron-in-an-octahedron. One thing I really like about this pattern is that (I think) there's a sort of parity to the colors. l've learned that this puzzle has two classes of colors - blue/red/ purple/yellow and white/gray/orange/green. If you notice here, the blue & red pieces are swapped and so are the yellow and purple pieces (as a permutation, this is two disjoint 2-cycles).
These colors form a tetrahedron on the octahedron (no two are adjacent). There is a dual tetrahedron formed by the other four colors! But those colors are not swapped in pairs; the three white triangles moved to the gray face, the gray triangles to the orange face, the orange triangles to the green face, and the green triangles to the white face (as a permutation, this a 4-cycle!).

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Cubes_Art @cubesart.bsky.social · 19/06/2026

Well that was awesome! Got to meet members of Wilco last night and gave Glenn a 9-layer cube with the band’s name a sun on top. ☀️ I didn’t take a photo of the back, but it also says “Wilco”.

Dave hands Glenn Kotche a 9-later Rubik’s cube solved to say “Wilco”

A 9-layer Rubik’s cube solved to say Wilco with a band on stage in the background.

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Cubes_Art @cubesart.bsky.social · 25/05/2026

I've been exploring this family of patterns. These are torus knots/links with 3n crossings (for n = 0,..., 5) on odd-sized Rubik's cubes from 3 layers through 13 layers. I tried to make the colors exactly the same, but the even links forced me to use edges from the r/w/b half of the cube.

Six images of nxnxn rubik's cubes solved to states that depict mathematical knots/links. The sizes of cubes increase by 2 layers, beginning with a classic 3-layer Rubik's cube (solved to show an "unknot") and ending with a 13-layer cube (solved to show the 15-crossing torus knot). Each knot/link is a torus knot/link with 3n crossings from n = 0 to n = 5. The 6- and 12- crossing links each have two components.

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Cubes_Art @cubesart.bsky.social · 24/05/2026

Super proud of a couple of 18-year olds who graduated from Clarke Central HS (that’s what the “C” is for) yesterday. Also super proud of Amy (their mom; my partner) for raising two wonderful people.

A 9-layer Rubik’s cube solved to depict a mortarboard hat and stylized “C” for “Clark Central”

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Cubes_Art @cubesart.bsky.social · 09/05/2026

Here’s a link I doodled on the 7cube yesterday.

Eight photographs of a 7-layer Rubik’s cube solved to a state that is an 18-crossing link of three components. Each photograph centers a different corner of the Rubik’s cube and they are arranged to give a sense of how the pattern continues when you rotate the cube in your hand.

A (kind of) stereographic projection of an 18-crossing link of three components. This projection allows the observer to experience all six sides of the patterned Rubik’s cube simultaneously.

A more abstract planar embedding of the same link from the Rubik’s cube in the first image. All threads are colored black.

A more abstract planar embedding of the same link from the Rubik’s cube in the first image. All threads are colored differently to emphasize the 3-fold symmetry of the link. They are colored black, light blue, and pink. Trans rights are human rights.

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Cubes_Art @cubesart.bsky.social · 05/05/2026

Here’s a photo I took a couple years ago, the first time I was exploring these patterns.

Five photos of odd-sized Rubik’s cubes with similar braid patterns. The number of threads in each braid increases by 1 as the cubes get larger. The sixth photo is Artemis the cat.

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Cubes_Art @cubesart.bsky.social · 04/05/2026

Yesterday‘s pattern has me revisiting the same theme on the 9cube. Unlike yesterday's pattern, this link is not Brunnian.

8 perspectives of a 9-layer Rubik's cube solved to a state in which four threads weave among each other to form a braided link. The link is alternating and has 36 crossings.

A planar embedding of the link in the other photo. The link is alternating, has 36 crossings, has 4 components, and each component crosses with the other three six times.

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Cubes_Art @cubesart.bsky.social · 03/05/2026

Here's a planar embedding of the link. It's an 18-crossing Brunnian link (see alt text for more). katlas.org/wiki/Brunnia...

A digital hand drawing of an 18-crossing link with alternating crossings. There are three components (black, pink, purple). Each component is an equilateral triangle with rounded corners and rotated 40º from the others. The black always crosses in front of purple, purple always crosses in front of pink, and pink always crosses in front of black. This makes this an 18-crossing Brunnian link.

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Cubes_Art @cubesart.bsky.social · 03/05/2026

Here’s a clip of me finishing up an 18-crossing braid link pattern on a 7-layer Rubik’s cube.

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Cubes_Art @cubesart.bsky.social · 17/02/2026

He said “Yes I am” and proceeded to ask me about the mathematics of solving a Rubik’s cube. He was kind and attentive and actually engaged intellectually with what I was saying. And he flew Delta Comfort +

David Plaxco standing next to Jon Ossoff in the Reagan Airport terminal holding a 9-layer Rubik’s cube that says “vote”. Jon had great hair and was very charismatic and smart. I think this guy is going places.

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Cubes_Art @cubesart.bsky.social · 30/01/2026

Project update: Tanya and I have been writing up the results of this project and we've realized that all of the knots in this class of patterns are at most 6 "tangle moves" from each other and at most three tangle moves from what we're calling the "great-great-granny knot" (see alt text).

This is a representation of only some of the ways our class of knots are connected via a specific tangle move in which places on the knot/link are cut and then those four ends are connected (I know this isn't a great description of this, sorry).

We've realized that all of the knots in this class of patterns are at most 6 "tangle moves" from each other and at most three tangle moves from what we're calling the "great-great-granny knot" (center of photo).

Also, I swear, we're *not* starting a cult.

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Cubes_Art @cubesart.bsky.social · 18/01/2026

Fun stuff! I’ve done color modding with 3x3s through 11x11s. Mostly, I make “solvable” cubes with nonstandard color schemes, but if you use two cubes and don’t mind ambiguity, there are a lot of fun possibilities. Here are a couple 3-color 9x9s. I call them Ice-Nine and Hades cube.

A 9-layer Rubik’s cube with only “cool” colors - white, blue, & green. This cube has three pairs of adjacent faces that are the same color. In this photo, we can see the two adjacent blue faces and pieces from the white faces form a pattern that is a completely white stripe traversing all three visible faces. This is unusual because it requires edge pieces that are white on both sides.

This photo shows two 9-layer Rubik’s cubes - a “warm” color cube called Hades and “cool” color cube called Ice-Nine. Each cube has unique properties that are impossible on a typical 9-layer Rubik’s cube.

This photo shows two 9-layer Rubik’s cubes - a “warm” color cube called Hades and “cool” color cube called Ice-Nine. The Hades cube has the property that opposite faces are the same color. So there are four yellow/red edges, four yellow/orange edges, and four red/orange edges.

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Cubes_Art @cubesart.bsky.social · 05/01/2026

If you’re at #jmm check out these infinite polyhedra sculptures near the conference center, just south of the Carnegie Apple Store. Sorry I didn’t take photos of them from the outside. maps.app.goo.gl/EWqHqNQbwkkd...

This image shows the view of the inside of a sculpture that is a truncated icosahedron (soccer ball) made out of one-way mirrors with lights running along the edges of the polyhedron.

This image shows the view of the inside of a sculpture that is a dodecahedron made out of one-way mirrors with lights running along the edges of the polyhedron.

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Cubes_Art @cubesart.bsky.social · 13/12/2025

Mwahahaha... Join me in about 30 minutes to see what the hell might do with 6 9-layer Rubik's cubes. www.gathering4gardner.org/news-2025-12...

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Cubes_Art @cubesart.bsky.social · 09/12/2025

Great puzzle. Kinda proud. I try to solve all categories in my head before guessing, but had to just take a leap today and guess for purple. Process of elimination for the blue, then home free. I was *really* tempted by some red herring categories, one of which had 6 or 7 possible tiles.

Screenshots of the NYT game “Connections”

It shows a “reverse rainbow”🌈 result in which the categories were found in order from most difficult to least difficult. The second screenshot shows that the puzzle was “extremely tricky” and shows which percentages of players had 0, 1, 2, 3, & 4 mistakes.

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Cubes_Art @cubesart.bsky.social · 23/11/2025

Made this 2-sided this morning and realized that playing with the crossings generates two different mathematical links: One is L6n1 (which is apparently three fibers in the Hopf Fibration?) and the other is Borromean rings (L6a4). Urls for their katlas pages in the alt text.

A 7-layer Rubik's cube solved to depict the mathematical link L6n1.

https://katlas.org/wiki/L6n1

A 7-layer Rubik's cube solved to depict Borromean rings.

https://katlas.org/wiki/L6a4

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Cubes_Art @cubesart.bsky.social · 14/11/2025

I found this knot pattern this week and generated a planar embedding by first using stereographic projection onto the disk, tracing, and tweaking—all in PowerPoint and Photoshop. I really enjoy this process as it allows me to experience the structure of the knot throughout the very slow process. 😊

A 9-layer rubik's cube that has been solved to a pattern that is an embedding of an 18-crossing knot.

An 18-crossing mathematical knot. This is the knot that is on the 9-layer Rubik's cube.

A stereographic projection of a 9-layer rubik's cube onto a unit disk. The cube has been solved to a pattern that is an embedding of an 18-crossing knot.

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Cubes_Art @cubesart.bsky.social · 31/10/2025

I think you’re right. The trick might be in having an even number of each type of tile In the given knot. Here’s the most basic. :)

A Rubik’s cube solved to a state called “cube-in-a-cube-in-a-cube” which can also be thought of as an embedding of the unknot.

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Cubes_Art @cubesart.bsky.social · 08/06/2025

BTW: This is the Bible verse referenced in the graffiti.

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Cubes_Art @cubesart.bsky.social · 07/06/2025

Here are all 54 distinct knots/links that Tanya and I have identified in our exploration of a specific class of knots on 9-layer Rubik's cubes. We have also determined that this list contains no redundancies & that any of the 4096 patterns that we began with is equivalent to one of these 54 knots.

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Cubes_Art @cubesart.bsky.social · 04/06/2025

I've been investigating a class of knots with Tatyana Khodorovskiy. We began with 4,096 initial patterns on a 9-layer cube and have reduced this set to identify 54 distinct knots in this class. You can read more about it in my Bridges paper from last year, but we're currently writing up the results!

A mathematical knot superimposed on a disk that represents a projection of a Rubik's cube. This knot has 12 crossings and is one of 54 such knots that I have found with Tatyana.

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Cubes_Art @cubesart.bsky.social · 20/05/2025

Apparently, “notationally” is not one of the six pangrams today.

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Cubes_Art @cubesart.bsky.social · 28/04/2025

Here’s a candid shot from the time I met Ernö Rubik in 2018. He very generously signed several of my cubes and chatted about my patterns. Thanks to @cardcolm.bsky.social for setting up the opportunity.

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Cubes_Art @cubesart.bsky.social · 27/04/2025

Brutal start for me today. Ended up getting it in 4.

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Cubes_Art @cubesart.bsky.social · 26/04/2025

I use onenote, but this exercise stands out in my memory.

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Cubes_Art @cubesart.bsky.social · 25/04/2025

Teaching people to be critical of lies they hear is NOT an infringement of the liars’ free speech.

This screen capture from a cnn article about the resignation of the NSF head included these paragraphs describing the recent cancellation of NSF grants.

https://www.cnn.com/2025/04/24/science/national-science-foundation-director-resign/index.html

The text in the photo reads:

But the list also included other topics, such as the “Spread of Unsubstantiated Information” and “Addressing Vaccine Information Integrity.”

In a post about the canceled grants, the NSF said that the misinformation research funding was halted to comply with Trump’s January 20 executive order on “restoring freedom of speech.”

The agency said it does “not support research with the goal of combating ‘misinformation,’ ‘disinformation,’ and ‘malinformation’ that could be used to infringe on the constitutionally protected speech rights of American citizens across the United States in a manner that advances a preferred narrative about significant matters of public debate.”

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Cubes_Art @cubesart.bsky.social · 22/04/2025

Viva Papa. I had the chance to see Pope Francis when we traveled to Italy a few years ago. I didn’t wholly agree with him but I admired his humanity. He was imperfect, caring, and humble. That’s refreshing to see in someone at that level of power in the world. I wish peace to all.

A cube solved to read “Viva Papa” held in Saint Peter’s Square (which is oval-shaped… 🤔).
On the steps of the basilica is a small stage on which sits Pope Francis in the summer of 2022.

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Cubes_Art @cubesart.bsky.social · 15/04/2025

NYT Games added a “Purple First” counter to their Connections stats! Ever since I learned that the colors implied difficulty, I’ve made it a side quest to solve them in “perfect reverse” order: purple, blue, green, yellow.

A screen shot of the statistics page for the game Connections in the NYT Games app. I have circled the statistic “purple first” with a purple highlighter.

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Cubes_Art @cubesart.bsky.social · 03/04/2025

I made this pattern to demonstrate the Fire and Ice color scheme on my 9-layer 🔥 & 🧊 cube. I dis- and re-assembled the cube from its standard color scheme to create this color scheme. I like how the warm and cool colors contrast. Which side do you prefer: cool colors on warm or warm on cool?

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Cubes_Art @cubesart.bsky.social · 03/04/2025

Wow! What an incredible, beautiful book. This makes my heart so happy.

Photo of the book “The Five Sides of Marjorie Rice - How to Discover a Shape” written by Amy Alznauer and illustrated Anna Bron. The title is framed by a pentagon, which is flanked by several organic objects with intricate shapes. A cat sits in a laundry basket in the lower-left and a woman stands looking at the pentagon in the lower-right.

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Cubes_Art @cubesart.bsky.social · 01/04/2025

This video might seem to contradict my earlier PSA, but there’s and *important* difference! Can you tell it?

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Cubes_Art @cubesart.bsky.social · 01/04/2025

Reminder: today is April Fool’s Day, so if you see a 9-layer cube that purports to have the word “Fool” written on it in pixels of all one color, it’s NOT real. There are only four of each color pixel in each permutation class, so you would run out of pixels if you tried to write “Fool”

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Cubes_Art @cubesart.bsky.social · 31/03/2025

Following yesterday’s theme, two variations on the same knot. 🪢 Here are two knot patterns that share the same edge crossings.

A 9-layer Rubik’s cube solved to depict a 15-crossing knot on three faces.

A 9-layer Rubik’s cube solved to depict a 9-crossing knot on three faces.

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Cubes_Art @cubesart.bsky.social · 30/03/2025

A small tweak in one knot pattern creates a second, different knot. I believe these are the 9-crossing torus knot and a direct sum of three trefoils. (Artemis is adorably sleepy)

A 9-layer Rubik’s cube solved to depict a mathematical knot with 9 crossings (I think it’s a torus knot). 🪢

A 9-layer Rubik’s cube solved to depict a different mathematical knot with 9 crossings (I think it’s the direct sum of three trefoils). 🪢

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Cubes_Art @cubesart.bsky.social · 24/03/2025

Lovely hike today in El Yunque National Forest, Puerto Rico. I saw a 360° rainbow while standing under a waterfall and ate Puerto Rican-Asian fusion plantains at a beach. Here’s a photo of a pattern I made in between activities.

A 9-layer Rubik’s cube solved to read “PUERTO” along two adjacent faces and “rico” on the third visible face. Tropical plants are in the background.

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Cubes_Art @cubesart.bsky.social · 23/03/2025

Beautiful day walking around Old San Juan, Puerto Rico.

A 9-layer Rubik’s cube solved to read “SAn• JUAn” with a Sun-filled piazza in the background.

A 9-layer Rubik’s cube solved to depict a spirally knot pattern. Topologically, this is the un-knot.

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Cubes_Art @cubesart.bsky.social · 20/03/2025

Here’s a knot on an 11-layer Rubik’s cube. I call this type of knot a permutation knot (see alt text for an explanation of what I mean by that). This cube has a nonstandard color scheme. I dis- and re-assembled it to have a “hot” corner and “cold” corner. I call this the fire and ice color scheme.

An 11-layer Rubik’s cube solved to a state that depicts a 12-crossing knot. I call this type of knot a permutation knot because the pattern on each face can be thought of as a permutation of the positions that threads enter and exit the face. For instance, on the white face, a blue thread enters the face on the right in the last (5th) position and exits the left edge of the face in the middle (3rd) position. Similarly, the orange thread enters in the 4th position and leaves in the 5th position, and so on. So, this knot can be thought of as encoding the permutation (12453) on each of the three faces.

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Cubes_Art @cubesart.bsky.social · 16/03/2025

One reason I like this puzzle is because it’s great for beginners. Here, you can see it in action. www.thecubicle.com/products/moy...

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Cubes_Art @cubesart.bsky.social · 16/03/2025

One of my favorite non-cubic twisty puzzles is the bead pyraminx. Every scramble is ≤ 8 moves from solved (I’ve heard the radius is actually 4, but I haven’t proven it myself). Once the corners are solved, the “beads” can be permuted to a solved state with one of two commutators.

A cat sits on a turquoise sofa next to a tetrahedral twisty puzzle that has a dot in the center of each face. Each of the four dots (or “beads”) of the puzzle is moved whenever an adjacent corner is turned.

Artie (the cat) fell asleep next to the puzzle while I was typing this post.

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