Gerard Westendorp

https://mathstodon.xyz/@GerardWestendorp

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

Website: https://westy31.nl/

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Last Notes

2026-01-07 16:15:24 GMT

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It gets a bit trickier with friction though. A particle falling through a viscous fluid reaches a constant momentum when the force of gravity equals the friction.

2026-01-06 16:52:39 GMT

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Pretty good approximation.
https://media.mathstodon.xyz/media_attachments/files/115/849/186/708/830/928/original/111dab41e08e7340.jpg

2025-12-27 22:53:04 GMT

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I have to admit I don't hear the difference between the Turkish, Persian and Arabic quarter tones, which in certain tuning traditions are subtly different. But that may be a matter of practice.
2 years ago I started singing, it is significantly increasing my awareness for these things, so maybe I will learn it still.
You might like the channel of Peter Pringle, he brings to life ancient instruments and songs, like this song in ancient Sumerian:
https://www.youtube.com/watch?v=QUcTsFe1PVs

2025-12-27 12:20:12 GMT

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In "Eastern" music (Arabic, Persian, Turkish) they also have quarter tones:
https://en.wikipedia.org/wiki/Quarter_tone
It sounds a bit like these notes are deliberately out of tune. A musician on Youtube explains this is done to create a certain mood called a "Maqam"
https://www.youtube.com/watch?v=ZLbQcs3W0Bs
I am certainly not an expert on this, but it is cool when a real expert is passionate about it.

2025-12-02 21:46:42 GMT

One of the things I saw on " Ars et Mathesis" day, Masterdam nov 25, were these 3D printed "coral" lamps by Maarten Lemmens. You can print them in "vase mode", a single spiraled line.
https://media.mathstodon.xyz/media_attachments/files/115/652/150/622/145/032/original/284b19d06b329001.jpg

2025-11-07 00:24:02 GMT

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I also like the play "Copenhagen", about Heisenberg an Bohr meeting during the war. I saw it once in a theatre. There is a Youtube clip featuring the author of the play:
https://youtu.be/PPI6Hl6V8Sk

2025-11-06 00:19:31 GMT

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The book "The making of the atomic bomb" has quite an extensive account on the correspondence between Meitner, Hahn, Frisch. Meitner had already fed to Sweden, but continued to correspond with Hahn about his experiments with bombarding Uranium with neutrons, asking for her opinion. Frisch and Meitner were hesitant, but then Frisch talked to Bohr, who immediately bought the idea that the bombarding neutron triggered the fission.

2025-11-01 14:36:38 GMT

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Hmm, I wonder how this looks like for the limiting case of the sphere.

2025-09-30 06:16:44 GMT

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Ah, I looked a bitter better at the equations. So an (t+s) dimensional space time, with a cosmological constant Lambda would have expansion coefficient \( Lambda^(s-t-1) \) .

2025-09-29 22:15:42 GMT

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Hmm, but most systems involving pressure have the pressure decreasing as volume increases, \( pV^k = constant \) . So the expansion of the Universe would slow down over time. I thought that recent evidence suggests that the rate actually increases. So this pressure is a bit weird,

2025-09-29 11:05:39 GMT

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Intuitively, I would guess these things would be highly unstable, they would explode, or evaporate in some kind of Gamma Burst, due to the large blue-shift.
Dark Energy appears to have negative mass, at least it powers the accelerating expansion of the universe. If you could concentrate a lot of dark energy into a small space, you might actually get a negative mass black hole.

2025-09-27 13:29:33 GMT

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I thought Antichrist was just Christ made of antimatter...

2025-09-27 11:32:56 GMT

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The Edinburgh Tapes?

2025-07-18 20:58:26 GMT

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A candidate for a most stretched transformed Buckyball. You can also produce squares from adjacent pentagons, octagons from adjacent heptagons, etc. So the parameter space is pretty complicated...
https://media.mathstodon.xyz/media_attachments/files/114/876/215/780/035/052/original/fd8421931474743f.jpg

2025-07-16 14:57:45 GMT

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You could start with a simpler case; The idea works for any polyhedron.

2025-07-16 09:14:44 GMT

After reading a post on the Azimuth weblog by @npub1nf4…nqe4 about "Stone-Wales transformations" , (https://johncarlosbaez.wordpress.com/2025/07/12/stone-wales-transformation/)
I spent some time exploring ways to mutate the Buckyball using Stone Wales transformations. Here is an interesting one:
Instead of 20 hexagons it has 10 heptagons and 10 extra pentagons, so it has 22 pentagons and 10 heptagons.
22+10 = 32 = 12+20: With these transformations, the number of vertices, edges and faces remains invariant. Also the valence of each vertex stays invariant. The sides per polygon changes, incrementing 2 while decrementing 2 others.
https://media.mathstodon.xyz/media_attachments/files/114/862/137/115/459/700/original/13484ad361809541.jpg

2025-07-11 09:01:32 GMT

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Ik took me a while to figure out what you were talking about, but I think I get it now. If you split a "discretized" cube into 6 tetrahedra, the cubes on the inner boundaries of tetrahedra have to decide in which tetrahedron they should be if we do not want to double-count cubes.
Pretty amazing that the number of double counts for an N-cube is related to the Eulerian numbers!
https://media.mathstodon.xyz/media_attachments/files/114/833/724/237/315/815/original/c34ec51366ad7965.jpg

2025-06-26 20:39:36 GMT

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We are in a time of great changes, which come with negative consequences for many people. The populists ( the gods of stupidity) claim to have easy answers. They blame the immigrants for stealing our jobs, blame women for outperforming men, blame the Chinese for out-manufacturing the West, and blame the established politicians for letting all this happen. And many established politicians have been sleeping. Hopefully they, or some new ones, are waking up. Mayve i should wake up too...

2025-06-26 16:38:42 GMT

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Amazing...
It is science that took us out of the Middle Ages, doubling life expectancy.
All kinds of statistical indicators, like the amount of poverty, death by violence, disease, illiteracy are all going the right way because of science, even though of course they are still too high.
Even problems like overpopulation and climate change will be solved, perhaps a bit late, unless we keep worshipping the gods of stupidity.

2025-06-22 08:43:47 GMT

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Hmm, 3D printing could make make such objects I guess.

2025-06-20 12:05:48 GMT

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I am not necessarily trying to prove anyone wrong, just thinking if a cosmology without energy conservation could correspond to physical reality. In the example you gave, you show is possible mathematically, but could our universe be like that, ie not have conserved energy?

2025-06-20 10:30:36 GMT

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Hmm, I guess if you describe them with an Action that is explicitly time dependent, then you can't use Noether's theorem involving time-shift symmetry. But I feel that nature is not like that. If space-time expands, it should be because of some local field, so that the action is not explicitly time dependent.

2025-06-20 10:13:51 GMT

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Cool!
Is there some way of including an energy term of space-time itself, such that the total energy of the universe remains constant?

2025-06-16 18:28:14 GMT

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Hmm, if a super-AI would need a lot of power, that could motivate it to get rid of humans, who use up a lot of power. On the other hand, it might not be in a hurry, assuming it could keep itself "alive" for a couple of million years.

2025-05-12 00:18:02 GMT

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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 GMT

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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 GMT

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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.

2025-02-15 10:45:01 GMT

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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

2025-02-09 22:47:48 GMT

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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!

2025-02-03 06:54:23 GMT

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Cool!
If you change the ellipse to a circle, by changing 2 sin v to sin v, does it still work?
The factor seems as if you could choose any number there.

2025-01-19 12:08:39 GMT

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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 GMT

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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

2024-12-31 12:07:03 GMT

I made this animation with the online tool by @npub13nc…2fmp, (https://hcschuetz.github.io/polyhedron-star/dist/) following up on the thread by @npub1nf4…nqe4
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

2024-12-03 07:52:40 GMT

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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 GMT

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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 GMT

I am still playing around a bit with the "Thurston Polyhedra" following the recent discussion started by @npub1nf4…nqe4, 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

2024-11-25 06:55:58 GMT

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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?

2024-11-20 01:06:54 GMT

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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 GMT

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

2024-11-15 17:27:15 GMT

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I just registered with BlueSky, as an experiment. The Layout is quite a bit like Twitter. Mastodon is getting a lot better I think at supplying interesting ideas and discussions, at least for me. The problem of how to organize such a platform is in itself interesting, although, like may interesting things, a lot of work before you can say something meaningful about.
https://media.mathstodon.xyz/media_attachments/files/113/488/113/534/340/772/original/85f8fe143017c15d.jpg

2024-11-07 18:53:52 GMT

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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 GMT

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

2024-11-04 18:35:53 GMT

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I think I'll make a version with increased scale and tidiness.

2024-11-04 13:26:05 GMT

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“So, when I ask whether there are creases across non-grid lines, I am only asking about the unique convex version.”
Aha, I think I get it now. There are lots of confusing ways of buckling the shape, but if the convex case is *unique*, then that is a huge tidying up criterion. I tried to make it convex, and in that case, I think you need to add fold over no-grid lines. Some of these correspond to the lines drawn on the print-out, which I think was by Thurston?
There are 3 lines on the print-out. 2 of them seem to clearly require bending, but the third is a bit subtle, could be sensitive to inaccuracies of the glued construction.
https://media.mathstodon.xyz/media_attachments/files/113/424/908/552/320/825/original/8f2026a7ef5e112e.jpg

2024-11-03 22:33:17 GMT

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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 GMT

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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 12:43:13 GMT

@npub1nf4…nqe4 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

2024-11-01 23:57:06 GMT

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I was thinking, Thurston’s method of defining a positive curvature polyhedron by leaving out wedges that extend outward could also be generalised to hyperbolic space. Instead of cutting out wedges at the points of positive Gaussian curvature, you have to fold some excess surface in a lettuce-like fold. I did this (see picture) in an attempt to construct a C1 isometric embedding of the hyperbolic plane, which should exist according to Nash an Kuiper.
https://media.mathstodon.xyz/media_attachments/files/113/410/376/941/074/573/original/076fc828e96785ea.png

2024-10-28 19:23:36 GMT

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Here is the latest version. It is now easy to change the colors, line thickness, etc, if needed.
https://media.mathstodon.xyz/media_attachments/files/113/386/675/241/878/195/original/438f65205767adcd.jpg

2024-10-28 10:01:36 GMT

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Ah, I can see how to easily generate a "Thurston cut-out" of these. It is just like the icosahedral one, and like the buckyball I just posted: You just subdivide all triangles. You could then color the hexagons.
In fact, I will probably make one this evening!
Hmm, but these Goldberg polyhedra also have a slight angular deficit for each hexagon vertex, so that it becomes more spherical. It would get very messy if you also did that in the cut-out. If you leave out those angular deficits, you get large flat hexagons "painted" by sub-hexagons.

2024-10-27 23:16:49 GMT

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Changing the icosahedron to the buckyball would be just doing 2x (per direction) as many triangles, leaving the wedges the same. Should be doable...

2024-10-27 23:12:57 GMT

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We could do the buckyball, AKA truncated icosahedron, that has plenty of 6-nodes. So far I generated the triangular grid by a Macro and then added the triangular wedges by hand. But I think I'll program those, so that the coordinates are much more precise. Also making changes will be less work after that.

2024-10-27 18:08:46 GMT

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Wow!

2024-10-27 10:15:07 GMT

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I edited the green triangles by hand, but I could write a macro so that the coordinates are exact.

2024-10-27 10:13:13 GMT

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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

2024-10-27 10:05:09 GMT

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It would be cool to be on the AMS weblog!
Here is the icosahedron cut out without the text. All the black gridlines are fold lines.
I think I am starting to understand this a bit better.
Suppose you have a “Generalised Buckyball”. It always has exactly 12 vertices where 5 triangles meet. Choose 1, call it the “North Pole”. Then draw geodesics from the North Pole to the other 11 5-nodes. Then cut out along these geodesic segments. It should now fold flat on the table, because all the non-zero Gaussian curvature points have been cut open. The geodesics will be straight lines when folded flat, and they will meet at 60 degree angles. So you get Thurston’s 11-gon.
There may be some trouble if several 5-nodes lie exactly on the same geodesics. We may have to choose a different North pole. You would have to prove this always works, but it works in practice in all cases I can think of.
https://media.mathstodon.xyz/media_attachments/files/113/378/820/247/143/506/original/f8fd1a20f384df8c.jpg

2024-10-26 14:29:23 GMT

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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

2024-10-25 23:41:00 GMT

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Well, cutting it out and glueing it together was successful. It seems to be convex, but I am not sure.
https://media.mathstodon.xyz/media_attachments/files/113/370/688/640/036/876/original/0c1ce65c248269e1.jpg