Gerard Westendorp

Trying to understand nature, in the meanwhile making visualizations of math and physics concepts.
https://westy31.nl/

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npub1a3xqgaqgtgmdkechl6aq9at0nvmqmwjhy2glaxanlgawsx0e3vzqpngyzl

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GerardWestendorp@mathstodon-xyz.mostr.pub

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nprofile1qqswcnqywsy95dkmvutlawsz74hekdsdhftj9y07nwel5whgr8uckpqpzemhxue69uhhyetvv9ujumn0wd68ytnzv9hxgqgkwaehxw309aex2mrp0yh8qunfd4skctnwv46q2jy8x5

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relay.nostr.band

relay.primal.net

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npub1a3xqgaqgtgmdkechl6aq9at0nvmqmwjhy2glaxanlgawsx0e3vzqpngyzl

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Last Notes

2025-05-24 22:39:09

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@nprofile…q8c7
Aha, think I got something. You can tile the plane with both the original quad, and the sub quads.
https://media.mathstodon.xyz/media_attachments/files/114/565/203/631/864/562/original/34035f286e019aa1.jpg

2025-05-24 09:48:13

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@nprofile…umph @nprofile…cfz8 @nprofile…5rpe
I tried warping higher genus uniform polyhedra into mult-holded doughnuts. I only succeeded for the Octahemioctahedron, which has genus 1.
But an easier project would be to warp the hyperbolic fundamental polygon into the polyhedron. "Easy" in principle, but quite a bit of work.

2025-05-17 10:08:41

“Lord of the Rings”, “Star Wars”, and “Harry Potter” are 3 fantasy worlds that seem to me to have striking similarities. I put an “equivalence table” in the picture.
But I have notice that the fan-bases of these 3 words are often quite disjoint. I wonder how widespread this phenomenon is. So I thought I’d try a poll. (It is my first poll.)
Unfortunately, I can’t seem to put a picture and a poll in one post. *So I put it in a reply* Also, the poll has a max number of 2 options, I would have like to have used 8.
https://media.mathstodon.xyz/media_attachments/files/114/522/616/163/243/998/original/e13e4c8d66fb6c0b.jpg

2025-05-12 00:18:02

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@nprofile…fa4k Some of the articles talk about "increased electrical conductivity". If there is conductivity, then there is energy consumption. In the video you see rotation, but also translation.

2025-05-11 13:58:52

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@nprofile…fa4k I once forced a mid day nap on a child that I was baby sitting 2X per week. The first time, he cried for at least an hour (made me doubt my idea...) The second time, he cried for 5 minutes. The third time, he asked if he could go for a nap.

2025-05-08 09:02:46

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@nprofile…fa4k I can imagine that if you take the Fourier transform of the pattern, (for a dot you have to use something like a gaussian), then you can get some insight.
The case that is "uniform" in space, is less unform in Fourier space. It will have a wavelength peak near the average distance between dots.

2025-05-04 21:36:53

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@nprofile…5rpe Reminds me a bit of @nprofile…fa4k recent post on atomic nuclei that spin themselves to death.

2025-05-04 21:33:14

My new 3D printer is good. I wanted to print a kind of Seifert surface, (a surface bounded by a knot) but in the first try, the surface was so thin it printed just the knot.
https://media.mathstodon.xyz/media_attachments/files/114/451/686/120/830/293/original/a6d02bb17935dc4c.jpg

2025-04-24 05:36:01

One small leap for Woman,
one giant free fall for Womankind
https://media.mathstodon.xyz/media_attachments/files/114/391/314/740/218/308/original/c51ad41fbd22a400.jpg

2025-04-18 17:09:07

I got myself a new maxwell equation circuit equivalent shirt, embroidered version.
https://westy31.nl/Electric.html#Maxwell
I am planning to do some new work on this subject.
https://media.mathstodon.xyz/media_attachments/files/114/360/066/885/690/581/original/4d82cd94991fd2f8.jpg

2025-04-08 07:59:11

The set of 30 possible dice. Together with the 24 ways of orienting each die, they form the 30X24=720 elements of S6, the group of permutations of 6 objects. I posted a couple of other pictures relating to this lately.
There is a nice weblog by Peter Cameron on this:https://cameroncounts.wordpress.com/2010/05/11/the-symmetric-group-3/
He writes that he used to have a set of 30 dice like this, but lost them.
The arrangement of the dice has some special properties:
In each row and column, each opposite pair occurs once and only once. This means that the rows and columns are “synthematic totals”, as Peter Cameron explains.
The dice on opposite sides of the diagonal are mirror images.
https://media.mathstodon.xyz/media_attachments/files/114/301/196/073/626/284/original/d71cb623c111d790.gif

2025-04-07 10:10:43

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@nprofile…fa4k @nprofile…yu7x
While trying to better understand the relationship between the Cremona Richmond configuration and the Tutte Coxeter graph, I noticed that in the Tutte Coxeter graph, there is an obvious trajectory that passes through all duads twice: going round the large circle. But in the Cremona Richmond configuration, this doesn’t work. So I made a 3D version of it, in which you can follow the same trajectory in Cremona Richmond as in Tutte Coxeter.
As you can check, it works.
I was hoping to get something with a higher “Aha factor”. But I’ll post it anyway. The Tutte Coxeter graph image is by @nprofile…yu7x

2025-04-03 15:35:38

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@nprofile…paff
Sounds cool!
https://media.mathstodon.xyz/media_attachments/files/114/274/763/546/807/562/original/ee6513439957966a.png

2025-04-03 10:58:15

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@nprofile…fa4k @nprofile…yu7x
Now for the relationship with the Tutte Coxeter graph. I've printed the "synthemes" (triplets of duads) on triangles, with the mirror image on the back. So I can "Triomino" a labeling of the Tutte Coxeter graph. I've convinced myself that laying down 2 adjacent synthemes conpletely fixes the labeling. Makes sense, there are 720 ways of doing that, and 720 inner automorphisms of the graph.
https://media.mathstodon.xyz/media_attachments/files/114/273/632/953/793/240/original/e898c8dadb604a19.jpg

2025-04-02 09:11:18

I want to understand group theory a bit better, so I’ll be making a couple of pictures. One statement that sparks curiosity is “The only symmetric group (all permutations of N elements) that has an outer automorphism is S6.” @nprofile…fa4k and @nprofile…yu7x have already made some cool picture about this at:
https://blogs.ams.org/visualinsight/2015/08/15/tutte-coxeter-graph/
Here is an animated version of the Cremona Richmon configuration. Each point is a permutation of a pair of numbers between 1 and 6. Each of the 15 points is in a unique combination of 3 of the 15 lines. Each of the 15 lines has a unique combination of 3 points. Also, dual to the 6 numbers, there are 6 “synthematic totals”, sets of 5 disjoint lines. The 5 thick lines represent one such total, it can be rotated to produce 4 more. The 6th one is the set of all green lines.
More to come...
https://media.mathstodon.xyz/media_attachments/files/114/267/569/199/275/167/original/369136dd301226c6.mp4

2025-04-01 13:27:23

This sowing thread winding carton is interesting. We CAN divide the circle in 48 positions, 24 "teeth" with 2 sides. the "winding rule" is (-1,17,-1,19,-1,17). These 6 moves advance the thread by 2 positions, or 1 tooth. The position is periodic after 48 lines, but the entire line pattern is only periodic after 3*48 = 144 lines.
https://media.mathstodon.xyz/media_attachments/files/114/262/887/144/082/832/original/639906397e6a895c.png

2025-03-26 08:44:45

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@nprofile…puk6 Looks a bit like he surface of the sun. Maybe the rules of your model have some relation to magneto hydrodynamics.

2025-03-09 23:52:18

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@nprofile…qe76 @nprofile…8fmq
Got one too!
https://media.mathstodon.xyz/media_attachments/files/114/135/158/803/839/524/original/f377197fa4593bfd.jpg

2025-02-24 20:09:34

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@nprofile…4aqs I more or less agree, except for the word "Trivially". Understanding history, the present, and predicting the future is difficult.

2025-02-24 19:00:33

Interesting correlation between region and voting result in the German election.
https://media.mathstodon.xyz/media_attachments/files/114/060/391/693/235/486/original/e05152fc307bcfec.png

2025-02-23 10:30:11

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@nprofile…khyw @nprofile…32va
Als je het helemaal doortrekt, 3000 puzzels van 1 stukje zijn direct af.

2025-02-22 18:23:45

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@nprofile…fqqu
I actually played rugby for about 1 or 2 years long ago, but I was not very good...

2025-02-22 17:43:31

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@nprofile…yajh @nprofile…fqqu @nprofile…q8c7
>>Not quite! Gaussian integers _do_ have unique factorization.
Hmm, 13 = (2+3i)(2-3i), so 13 is not a Gaussian prime?

2025-02-22 17:27:45

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@nprofile…yajh @nprofile…fqqu @nprofile…q8c7
I tried to make pictures of the generalization of the modular group resulting from replacing integers by Gaussian integers.
Made a website on it:
https://westy31.nl/ComplexFarey/ComplexFarey.html
https://media.mathstodon.xyz/media_attachments/files/114/048/704/463/172/297/original/36a236946f6c99a0.png

2025-02-22 17:10:39

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@nprofile…yajh @nprofile…fqqu @nprofile…q8c7
I remember reading that the failure of unique factorization into primes for Gaussian integers led to some failed proofs of Fermat's Last Theorem by reasonably famous mathematicians.

2025-02-22 13:54:52

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@nprofile…yajh @nprofile…fqqu @nprofile…q8c7
By the way, I left out the restriction that the sum of squares is itself a square. That makes the number of cases a lot smaller.

2025-02-22 13:41:43

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@nprofile…yajh @nprofile…fqqu @nprofile…q8c7
The smallest case with 8 pairs is
\(1105 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2 = 23^2 +24^2
\)

2025-02-22 13:07:55

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@nprofile…yajh @nprofile…fqqu @nprofile…q8c7
I didn't believe you, until I checked it numerically. by brute force So far, up to (a,b) <= 20, 6 pairs is the max.
https://media.mathstodon.xyz/media_attachments/files/114/047/654/802/462/545/original/ca0b62eb98bae25d.png

2025-02-21 22:53:33

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@nprofile…fqqu @nprofile…yajh @nprofile…q8c7
If I understand it correctly, you could use such numbers to draw a nice circle on a square grid, such that the circle passes through the grid points (a,b), (c,d), e,f) and (g,h).
But that would be a bit weird, I would think you could construct sines and cosines with rational numbers from that. There must be a (2X2) rotation matrix for example that turns (a,b) into (c,d).

2025-02-15 10:45:01

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@nprofile…yu7x
Nice!
I made a hinged tessellation of that one.
https://westy31.nl/Hingedpolyhedra/Hingedpolyhedra.html
https://media.mathstodon.xyz/media_attachments/files/114/007/490/738/603/286/original/bd98e0fff6385c73.mp4

2025-02-15 10:38:44

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@nprofile…yu7x
I made a hinged tessellation out of that one.
https://westy31.nl/Hingedpolyhedra/Hingedpolyhedra.html
https://media.mathstodon.xyz/media_attachments/files/114/007/466/549/180/986/original/17c6a343bedda0a5.png

2025-02-09 22:47:48

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@nprofile…fa4k @nprofile…p3dv
I like also the Mathpages article on Noether's theorem:
https://www.mathpages.com/home/kmath523/kmath523.htm
After reading it, I thought *now* I get it!

2025-01-19 12:28:10

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@nprofile…fa4k Another thought: Viruses are like computer viruses: The do not copy themselves to other computers, they *get* copied. Computers do things, they need free energy. But viruses are just passive code.

2025-01-19 12:08:39

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@nprofile…fa4k Well, I think you can have a jar with fluid, containing viruses, that is in thermal equilibrium. I remember a talk I listened to once, in which they described how they changed the pH of the fluid, the viruses discarded their protein capsule (probably this has some technical name). They changed the pH back, and the protein capsules reformed. So they switch between 2 equilibria.
But when viruses interact with other life, there is no longer (global) equilibrium, although if you imagine a box around the virus, you could argue that this box is in equilibrium during the time that no matter or energy is passing through the box boundary.
You could say "a virus injects itself in a cell", but actually it is the cell that is doing the work.

2025-01-19 00:43:30

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@nprofile…fa4k Viruses are actually in thermal equilibrium, the don't "do" anything.
Some people go to Tbilisi, Georgia to get bacterial infections treated by special viruses:
https://en.wikipedia.org/wiki/Phage_therapy

2025-01-11 13:32:34

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@nprofile…p7w0 I'm tempted to make one with Rubic cubes, that are transforming while moving.
But that would be quite a bit of work...

2025-01-11 13:30:55

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@nprofile…p7w0 Nice!
I was confused at how the number of cubes per edge seems to change from 8 to 7. But of course, it is always 7, except I double count the corners when they are all aligned.

2024-12-31 12:07:03

I made this animation with the online tool by @nprofile…m6g0, (https://hcschuetz.github.io/polyhedron-star/dist/) following up on the thread by @nprofile…fa4k
This one is a near-miss Johnson solid, the hexagons are folded through the middle by about 10 degrees.
It turns out this figure is actually a compound of 2 triangular prisms, 6 square pyramids, and 3 tetrahedra. So I was able to look up the exact bend angles. ("autobend" feature was still having some trouble, but that could be because i did something wrong)
https://media.mathstodon.xyz/media_attachments/files/113/747/321/305/807/766/original/763fc07111e0740b.mp4

2024-12-29 23:06:12

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@nprofile…m6g0
It is quite possible I made a mistake!
I sent the input file to:
https://github.com/hcschuetz/polyhedron-star/discussions/1

2024-12-29 16:15:23

@nprofile…m6g0 :
I had a go at trying you program. Managed to do the hexagonal prism. The "autobend" feature had some difficulty here, but all angles are easy to calculate manually in this case.
https://media.mathstodon.xyz/media_attachments/files/113/736/975/242/467/227/original/90651f563a6809bf.png
https://media.mathstodon.xyz/media_attachments/files/113/737/001/619/223/024/original/382c656f262fb663.png

2024-12-28 12:37:18

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@nprofile…yg3m @nprofile…qmju
An easy example: Suppose in the year 3000, nobody has friends, except Santa Claus. So everybody except Santa has 1 friend, and he has millions. So the average number of friends of your friends is hugely larger than than your own number of friends.

2024-12-23 14:50:03

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@nprofile…m6g0 @nprofile…fa4k
Ah, nice!
I was thinking about making something like that, but no time to do it...
I'll take a good look at it!
https://media.mathstodon.xyz/media_attachments/files/113/702/679/005/664/485/original/76266effbec26a49.jpg

2024-12-03 07:52:40

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@nprofile…zccc @nprofile…fa4k
I suspect you can prove generalizations of Pick's theorem by induction: You "glue" together disjoint shapes that already satisfy Pick. You could create an abstract concept of "move", as any operation that preserves all the numbers in the Pick formula. The "minus 1" term should be generalized to "minus number of disjoint objects".
Maybe even generalize to higher dimensions, defining volumes etc.

2024-12-03 00:19:24

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@nprofile…fa4k Right! I guess you know Pick's theorem? There is probably a version of that on a triangular grid too.

2024-12-03 00:19:24

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@nprofile…fa4k Right! I guess you know Pick's theorem? There is probably a version of that on a triangular grid too.

2024-12-02 23:58:04

I am still playing around a bit with the "Thurston Polyhedra" following the recent discussion started by @nprofile…fa4k, I made a tetrahedron on a square grid. I was surprised that the tetrahedron was tiled exactly by 14 squares. Thinking about it further, I realized that a tetrahedron made from congruent triangles admits sets of parallel geodesic loops of the same length. On the sphere, this is impossible: Latitude lines are not geodesics, and longitude lines intersect at the poles.
https://media.mathstodon.xyz/media_attachments/files/113/585/902/212/831/458/original/23fadca83fb826d6.jpg

2024-12-02 23:58:04

2024-11-25 08:35:59

I realised cycling to my work, after I wrote the post, that the electron would appear to freeze on the event horizon because of time dilation.
So the change from black hole to white hole would appear to take infinitely long to a distant observer.

2024-11-25 08:31:19

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@nprofile…fa4k @nprofile…e5g0 @nprofile…yu7x @nprofile…cnse @nprofile…j7pr
But wait: The electron would would appear to "freeze" on the event horizon, because of time dilation.

2024-11-25 06:55:58

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@nprofile…fa4k @nprofile…e5g0 @nprofile…yu7x @nprofile…cnse @nprofile…j7pr
Hmm, I don't understand the diagram yet, but I am puzzled by the fact that time reversal symmetry appears to be broken: You can fall into a black hole, but the time reverse of that would be forbidden.
In practice, time reversing a rocket would violate thermodynamics, but an individual particle like a single electron does not have entropy. So if it can go into a black hole, why can't it go out in the time reverse of how it went in?

2024-11-25 06:55:58

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2024-11-20 01:06:54

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Impressive fold work by Henk van der Vorst
https://media.mathstodon.xyz/media_attachments/files/113/512/597/921/190/869/original/8013aa20a765b78b.jpg

2024-11-20 01:01:19

Last Saturday I was at the Mathematics & Art day "Ars et mathesis" in Eindhoven. I’ll post about a couple of nice things I saw/learned.
The first is about the so-called “Plastic ratio”. It is a bit like the Golden ratio, instead of the solution to \(x^2 = x+1 \) it satisfies \(x^3 = x+1 \); it is about 1.43. Instead of the Fibonacci numbers you have the Padovan numbers, whose recurrence relation is \( n_i = n_{i-2} + n_{i-3} \) .
I figured out how to do a spiral of squares whose sized as the Padovan numbers, similar to the Fibonacci-Golden Ratio spiral.
The Plastic ratio is used in Architecture.
Wikipedia: https://en.wikipedia.org/wiki/Plastic_ratio
https://media.mathstodon.xyz/media_attachments/files/113/512/493/249/109/581/original/ea3fa893c9e8b01e.jpg

2024-11-07 23:22:37

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@npub1pyk…cthf In Dutch 'snurk' means 'snore'.

2024-11-07 23:02:12

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@npub1tmp…ydrh She was often on Dutch TV during covid, staying calm in spite of all kinds of death threats. Seems

2024-11-07 22:37:12

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@npub1tmp…ydrh
You could try Marion Koopmans:
https://www.erasmusmc.nl/nl-nl/research/researchers/koopmans-marion

2024-11-07 18:53:52

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@npub1knz…c73p I read recently that optimists know that life sucks, while pessimists painfully rediscover it every day.
That's a joke of course...
We should do our best to do as little damage to our planet as possible.

2024-11-07 18:42:54

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@npub1knz…c73p By, then we will have created some smart AI robots that don't mind these temperatures.

2024-11-03 22:33:17

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@npub1knz…c73p
So, here is the 10 hexa-node case. Last time I had it creased across non-grid lines, but not this time. These things can "crumple" in multiple ways, it is hard to figure out what the "correct" way is. maybe it is even possible to make it convex.
We could conjecture that "All generalized buckyballs can be realized by folding only along grid lines".
It might be difficult to prove, it is a bit like protein folding into enzymes, a trendy topic.
https://media.mathstodon.xyz/media_attachments/files/113/421/370/482/622/687/original/ef0ca0eaf0dcd6c1.jpg

2024-11-03 21:31:00

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@npub1knz…c73p
This one seems to fold only on the grid lines. It has 5 hex-nodes.
I seem to remember that the very first thing i glued, did have no-grid folds. I'll redo it. (Starting to get handy at it...)
https://media.mathstodon.xyz/media_attachments/files/113/421/145/199/381/062/original/e5b91966fb2a5498.jpg

2024-11-03 13:59:22

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@npub125j…4as0 Ah, thanks! That would have taken me quite a bit of time to figure out. I thinks you can restore the 60 degree angle by folding the hexagons along the vertical lines? Each folded hexagon would break down into 2 sets of co-planar equilateral triangles. (But coplanar polygons are not allowed for Johnson solids)

2024-11-03 12:54:34

Here is another surprising result I got looking for interesting generalized buckyballs. This thing has 26 triangles, 12 5-nodes and 3 6-nodes.
the 3 6-nodes seem to be the center of regular hexagons, but then this figure would be a Johnson solid, but it is not on the list of Johnson solids!
The "Hexagons" might be slightly non-planar. Maybe I'll check numerically.
https://media.mathstodon.xyz/media_attachments/files/113/419/122/797/495/463/original/d29cc4d4d1a693fd.jpg

2024-11-03 12:53:44

2024-11-03 12:43:13

@npub1knz…c73p challenged me to find a generalized buckyball in which you are forced to fold on a line other than the triangular grid line.
I tried to make the simplest case that is not the icosahedron. I intended it to have just 1 hexagon. But it turned out I made the hexagonal antiprism, which has 2 hexagons!
The thing is, I can't seem to make a case with only 1 hexagon. It would not violate the Euler characteristic being 2, that always works as long as there are 12 5-nodes. So there must be other restrictions.
https://media.mathstodon.xyz/media_attachments/files/113/419/051/898/203/864/original/95d0dc3875737e40.jpg

2024-10-28 09:32:19

Years ago, I spent a lot of time trying to design a soccer ball shaped carton that is easy to fold. It sort of worked, but not good quite enough. I glued some of the proto's together as a kind of art work.
https://media.mathstodon.xyz/media_attachments/files/113/384/328/215/635/926/original/7ae5d20f23fbd194.jpg

2024-10-28 08:21:25

@npub1knz…c73p
Here is the Buckyball, in soccer ball style.
(I have the coordinates automated now)
https://media.mathstodon.xyz/media_attachments/files/113/384/069/276/131/743/original/07e41dc9c4694ac1.jpg

2024-10-27 18:07:06

Generalized bricks exist! They have zero Gaussian curvature everywhere except in 8 points with angular deficit 90 degrees, like a cube. You can design them using the method suggested by Thurston and recently discussed by @npub1knz…c73p.
https://media.mathstodon.xyz/media_attachments/files/113/380/706/086/362/208/original/00803b6900029b26.mp4

2024-10-27 11:24:35

The octahedron can also be "Thurstonized"! So can the tetrahedron, but that cut-out is just a triangle with 4 sub triangles. So now only the dodecahedron remains to be Thurstonized from the 5 Platonics. Maybe impossible...
https://media.mathstodon.xyz/media_attachments/files/113/379/107/392/998/958/original/00d58b7e1752d65c.jpg

2024-10-27 10:13:13

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@npub1knz…c73p
It also works for Generalized cubes!
I can’t visualize yet how a generalized cube would look like other than a brick, but I’ll try and make one.
https://media.mathstodon.xyz/media_attachments/files/113/378/851/581/849/902/original/989fce3ee1b466e7.jpg

2024-10-26 14:29:23

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@npub1knz…c73p
The "ordinary" icosahedron is not so obvious. I figured it out eventually, by making paper icosahedra and cutting them up.
https://media.mathstodon.xyz/media_attachments/files/113/374/188/795/004/439/original/d35726062b10bc22.jpg

2024-10-15 18:54:32

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@npub1hlk…jq8c
My first failure is to notice the international day of failure 4 days too late.
Another interesting failure I had some time ago is that I searched for days for a bug in my computer code: It froze the computer the moment it started. Eventually, I found out that the code was working perfectly fine, it just needed a lot more computation time than I anticipated: A couple of minutes instead of a fraction of a second!
#InternationalDayOfFailure

2024-06-27 06:18:43

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@npub150e…rrqv Well, what I did was not so fancy. First I did all {a,b,c,d} from 1 to N. But then as N gets large, this gets out out of hand. So I used a slightly better method by assuming a>=b>=c>=d, and also a*b*c*d <Max, where Max is about 15000. Then it takes a could of minutes in VBA.

2024-06-26 22:50:58

Found a reference and a mistake in my list.
The reference:
https://oeis.org/A027563
The largest number is now 52061=79*659
I post the new list, and a graph.
https://media.mathstodon.xyz/media_attachments/files/112/685/329/420/451/064/original/114ef2db5c308a14.jpg
https://media.mathstodon.xyz/media_attachments/files/112/685/363/226/598/113/original/8504a6d684c8fd87.jpg

2024-06-23 20:41:23

A uniform tiling that fold-transforms into another uniform tiling,
https://media.mathstodon.xyz/media_attachments/files/112/667/866/826/318/452/original/cdf8a9676cb5a5e4.mp4

2024-04-26 13:16:51

A rigidly foldable tessellation, based on the "square twist". Square twists are alternately connected to their mirror image.
https://media.mathstodon.xyz/media_attachments/files/112/337/684/704/027/718/original/dd49d029d53b5801.mp4

2023-09-24 09:37:19

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@npub1knz…c73p
Interesting!
I have some unfinished work, which I left lying for a couple of years, that is related.
The idea was to relate equations like the Klein Quartic (xy^3+yz^3 + zx^3) to tilings of the complex plane. The equation (u+v+w=0) can be interpreted as a triangle, and the tiling would correspond to different ways of mapping (xy^3 -> u, yz^3 -> v, zx^3 -> w).
As I mentioned, I didn’t finish this project. But I found one tantalising link to Ramanujan’s taxi cab number 1729!
The triangle pattern in the picture does not have reflection symmetry in the circles, I was still figuring out how to do that.
I have a lot of other projects, but maybe I could continue the work….
https://media.mathstodon.xyz/media_attachments/files/111/119/440/984/107/139/original/1c7a05f4a16ed0cd.jpg