𝓙. 𝓜.

"Not everything written is important; not everything important is written."
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I strive to write about the mundane, the sublime, and all else in between. Sometimes unjustly accused of actually knowing what I'm talking about.
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Interested in consulting opportunities; DM me for further inquiries.

WordPress: https://tpfto.wordpress.com/
GitHub: https://github.com/tpfto
Shadertoy: https://www.shadertoy.com/user/tpfto
Ko-fi: https://ko-fi.com/pleasureoffiguring

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tpfto_at_mathstodon.xyz@momostr.pink

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Last Notes

2026-06-02 07:49:44 UTC

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a parallel line of research would be figuring out if the nitroplast organelle (cf. https://doi.org/10.1126/science.ado8571 & https://asm.org/articles/2024/june/beyond-endosymbiosis-discovering-first-nitroplast) can somehow be transplanted (perhaps transgenically) into easier to cultivate algae or bacteria.

2026-06-02 06:16:47 UTC

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RE: https://mathstodon.xyz/@mjd/116499301673586782
@npub12cu…d38p this calls to mind a toot from a few weeks ago (change "computer" to "robot"):
#note1fc3…nlee

2026-05-31 12:12:30 UTC

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I've been looking at (generalized) Kempner series (https://en.wikipedia.org/wiki/Kempner_series) recently, where if you remove terms from the harmonic series containing a particular string of digits, the result is convergent. I guess you can do something similar to the sum of prime reciprocals...

2026-05-29 14:40:22 UTC

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I slightly adjusted the color wheel in the following cartoon so that the rhombuses look a bit more distinctive. Let me know if it helps:
https://media.mathstodon.xyz/media_attachments/files/116/656/503/928/497/853/original/59000a80e8eaddfc.mp4

2026-05-29 13:20:06 UTC

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As @npub1nf4…nqe4, considering the faces of the rhombicosidodecahedron all at once can get messy quickly. The following cartoon attempts to sidestep this by treating the triangular, square, and pentagonal faces separately under the transformation \(\phi\mapsto-1/\phi\):
https://media.mathstodon.xyz/media_attachments/files/116/658/055/702/838/737/original/839eea87ee6da02a.mp4

2026-05-29 06:20:08 UTC

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well, the equianharmonic case is based on a triangular lattice, but if you visualize the actual ℘ function, the "period parallelogram" comes out looking like rhombuses instead. Here's a plot of the actual elliptic function being approximated, for comparison (slightly zoomed out):
https://media.mathstodon.xyz/media_attachments/files/116/656/408/504/172/291/original/c7c5ab77200ad8d6.png

2026-05-29 05:41:22 UTC

Quite a while back, @npub1nf4…nqe4 blogged about an animation by David Chudzicki depicting the (lemniscatic case of the) Weierstrass elliptic function ℘ being gradually approximated by its defining lattice sum, visualized through domain coloring (https://blogs.ams.org/visualinsight/2014/01/15/weierstrass-elliptic-function). I couldn't find an equivalent animation for the other important special case of Weierstrass ℘, the equianharmonic case (the case corresponding to a triangular lattice), so I made one:
#math
https://media.mathstodon.xyz/media_attachments/files/116/656/203/734/032/088/original/899567b332067108.mp4

2026-05-26 08:45:27 UTC

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in the same vein as "if it's not from the Champagne region of France, it's just sparkling wine" ;) But yes, replicas have their use for people who want the experience, but can't travel very far. (cc. [@j_bertolotti](https://mathstodon.xyz/@j_bertolotti ))

2026-05-25 21:55:09 UTC

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yeah, there are apparently multiple reports of Oppenheimer being a fidgety sort. But he had excellent administrative skills, so I guess it balanced out. (cc. @npub1t0h…j96l)

2026-05-25 12:48:45 UTC

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> And in Britain, [Oppenheimer] had a rather difficult time: He tried joining an experimental group, and there was some sort of serious trouble. I can’t remember what the trouble was, but it was really quite serious. He left Britain and went to Germany. And there, he began working under Max Born and decided that he was a theorist, not an experimenter. He would never have been a decent experimenter, he was altogether too nervous. He never stopped smoking, he always had a cigarette in his mouth. He was a very nervous, tense man.
From https://thebulletin.org/premium/2023-07/he-did-not-speak-the-ordinary-language-memories-of-oppie-from-a-manhattan-project-physicist/
(cc. @npub1nf4…nqe4)

2026-05-25 12:39:38 UTC

This article on subjecting silkworm silk to heat and pressures ~2000× atmospheric pressure (https://doi.org/10.1038/s41893-026-01821-y) is terribly interesting. Apparently, no solvent was needed for the teatment, but the treated silk exhibits enhanced tensile strength and remains biocompatible. If this pans out, people will be beating a path to their labs.
#chemistry

2026-05-25 09:07:16 UTC

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I guess this is related to the "Pauli effect" (https://en.wikipedia.org/wiki/Pauli_effect).

2026-05-24 13:54:04 UTC

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did you have your own darkroom and developed these yourself?

2026-05-24 13:04:36 UTC

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I was also thinking about how the mapping from the real line to a (punctured?) circle \(u\mapsto\left(\tfrac{1-u^2}{1+u^2},\tfrac{2u}{1+u^2}\right)\) (basically the 1D version of the Riemann sphere) is effectively a mapping between two seemingly unrelated groups.

2026-05-23 12:39:34 UTC

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what's an easy way to connect the circle group with stereographic projection?

2026-05-21 09:05:19 UTC

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without actually trying to do the math, it looks like the radius of the circle enclosing the points grows exponentially at each step...
...which is related to what I'm about to say next. Remembering our exchange on icosahedra earlier, I tried the same operation on the vertices of the icosahedron and dodecahedron, and quickly ran into the limits of a free-tier Wolfram Cloud account. So, the following pictures were the best I could do, and someone else will have to go further.
https://media.mathstodon.xyz/media_attachments/files/116/611/755/247/891/354/original/da46b3ee06e11488.png
https://media.mathstodon.xyz/media_attachments/files/116/611/755/770/588/119/original/109ab1558c7353d0.png

2026-05-21 08:06:30 UTC

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had a quick look at the paper. Their addition operation is \(x\vdash y=\phi^2 x-\phi y\), where \(phi\) is the golden ratio. (The paper actually uses \(\tau\) instead o \(\phi\), which is a minor annoyance.). In the figure you attached, it looks like they're considering a set of points a pair at a time and applying the addition operation to it. I've attached my own re-implementation of the idea, corresponding to Figure 1 in the paper (as attached in your previous toot).
https://media.mathstodon.xyz/media_attachments/files/116/611/528/831/660/737/original/b692d19a0c4cff5a.png

2026-05-21 07:11:36 UTC

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"I don't know what the hell I'm doing..." - but in at least your case, this is followed by "...but I'm going to try my best to find out and maybe have fun with it."... right? (cc. @npub197g…q2n4)

2026-05-21 01:44:35 UTC

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in this regard, even if I may be an impostor, I think I still come out ahead as long as I don't tick too many items of @npub1nf4…nqe4's Crackpot Index.

2026-05-19 14:09:49 UTC

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I got the joke ;), but possibly contaminating the water table with chloroform can be problematic, and people actually are concerned about chloroform contamination in drinking water. This is because chlorofom is often a result of the chlorine used to disinfect water reacting with organic material that has not been carefully removed by filtration or other means (cf. https://doi.org/10.1080/15287390252807957 and https://www.who.int/docs/default-source/wash-documents/wash-chemicals/trihalomethanes.pdf)

2026-05-19 13:56:04 UTC

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very interesting, thanks for pulling that one up! It's wild that lidocaine actually works on plants that move. Indeed, the inhalational anesthetics are poorly understood with respect to their mechanism of action, but we at least know that their being lipophilic (i.e., how well they dissolve in fats and other non-aqueous media) correlates well with their potency.

2026-05-19 13:34:49 UTC

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in fact, there has been some research on "plant nervous systems", in that even though plants do not have full blown neurons like animals do, they display traits like depolarization waves, neurotransmitter analogues, etc.; see for instance the work of Brenner et al. (https://doi.org/10.1016/j.tplants.2006.06.009) and Sibaoka (https://doi.org/10.1007/BF02493405). On that note, I wonder if these anesthetics would also influence the drooping of Mimosa pudica:
https://media.mathstodon.xyz/media_attachments/files/116/601/494/636/335/362/original/0a683dd46cd8a809.mp4

2026-05-19 07:03:43 UTC

RE: https://mathstodon.xyz/@johncarlosbaez/116589957362204579
@npub1nf4…nqe4's thread on transforming between the icosahedron and great icosahedron (https://en.wikipedia.org/wiki/Great_icosahedron) got a little tangled as it proceeded, so I thought I'd take this chance to make a fresh thread.
Skipping a bunch of details, John shows that you can obtain a great icosahedron from an icosahedron by making the change \(\sqrt 5\mapsto-\sqrt 5\); in terms of the golden ratio, this corresponds to the mapping \(\phi\mapsto-\tfrac1{\phi}\). Prompted by that, I decided to look into what happens when you try to interpolate between the two (or "tween", in graphics parlance).
(1/n)
#mathematics #math
#note157j…2vtr

2026-05-18 08:17:21 UTC

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my personal favorites for "unusual" museums are the Mütter Museum (https://en.wikipedia.org/wiki/M%C3%BCtter_Museum) and the Meguro Parasitological Museum (https://en.wikipedia.org/wiki/Meguro_Parasitological_Museum).

2026-05-18 01:25:13 UTC

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if Pierre's life hadn't been in cut short by that road accident, he might have ended up the same way as Marie.

2026-05-17 14:47:11 UTC

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no, I did no shading tricks whatsoever; this is the "mess" I was expecting to happen when I added faces, since Mathematica (the software I used for these cartoons) starts acting funny when faces in 3D graphics start intersecting in weird angles.

2026-05-17 14:36:43 UTC

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thank you for doing it on my behalf; my Mastodon client is not being very cooperative today. Here's the cartoon for the great dodecahedron: https://i.postimg.cc/QdpX7BTs/gdof.gif ; your predictions seem spot on, as expected.

2026-05-17 14:17:15 UTC

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The icosahedron worked out, surprisingly, but the dodecahedron got messy near the end: https://i.postimg.cc/7hTG7H43/icocf.gif https://i.postimg.cc/x8bJHfQX/doocf.gif

2026-05-17 14:04:51 UTC

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I could try adding faces (both for the icosahedron and dodecahedron), but I have a good feeling it'll get messy during the tweening. Let me get back to you... (cc. @npub1scq…0tlv) I'd be delighted if you think these cartoons I made are blog-worthy. :)

2026-05-17 13:47:08 UTC

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yes, I seem to get exactly the great stellated dodecahedron if I do that mapping (as expected, due to the duality between the icosahedron and the dodecahedron): https://i.postimg.cc/6QTrHNL1/dooc.gif

2026-05-17 13:23:24 UTC

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thanks, I decided to look at what happens when you "tween" (https://en.wikipedia.org/wiki/Inbetweening) the conversion between the icosahedron and great icosahedron (through the \(\sqrt 5\to-\sqrt 5\) substitution). It looks pretty neat: https://i.postimg.cc/kMSCt68g/icoc.gif

2026-05-16 13:06:02 UTC

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when I wrote my previous toot, I completely forgot about "Pele's tears" (https://en.wikipedia.org/wiki/Pele%27s_tears). On that note, all the glass we ever use in applications are amorphous and not quasicrystalline; one wonders how "violent" we need to get to get past boring, amorphous glass and obtain something weird yet exploitable. :)

2026-05-15 10:36:20 UTC

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"These waves were locked in when the alloy quickly cooled after impact, and they've been sitting there frozen in the structure ever since." - quenching (as opposed to annealing) is a truly magical process, a little like catching lightning in a bottle. (See also: Prince Rupert's drops.)

2026-05-11 11:27:03 UTC

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I wonder if one might be able to use a percolation model to analyze this behavior...

2026-05-10 07:58:34 UTC

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in theory, the nearest university library uses LCC (https://www.loc.gov/catdir/cpso/lcco/); e.g. PL for Asian fiction, PS for American fiction, etc. In practice, as the semesters draw along, an author's body of work eventually get lumped together, even if they were originally in disparate classes.

2026-05-10 04:48:29 UTC

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In cases like these, a retraction feels not unlike closing the barn door after the entire herd has already fled.

2026-05-08 16:09:29 UTC

The thread on surprising facts you found out recently by @npub1nf4…nqe4 was getting a bit long, so I'm posting this entry separately: if I'm understanding https://en.wikipedia.org/wiki/Mohorovi%C4%8Di%C4%87_discontinuity#Exploration correctly, no manmade device has yet reached the Moho, the layer demarcating the boundary between the Earth's crust and mantle.

2026-05-08 00:34:50 UTC

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the carbon dioxide is why some wags point to Venus as what the Earth might eventually become if we don't curtail our release of greenhouse gases in the atmosphere.

2026-05-07 12:27:43 UTC

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Roland Bulirsch (https://en.wikipedia.org/wiki/Roland_Bulirsch), one of the two people who wrote a numerical analysis textbook I frequently refer to, as well as someone who wrote quite a bit on the subject of elliptic integrals, was apparently a gym buddy of Arnold Schwarzenegger. Bulirsch, along with their other gym friends, took up a collection to help Schwarzenegger emigrate to America.

2026-05-02 07:16:34 UTC

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all the focus on "growth" in various companies' quarterly reports sometimes reminds me of the "cancer cell" analogy; after all, if there's anything cancer cells are widely known for, it's unchecked growth. Unchecked, because the limits on cell division (e.g. telomeres) that are usually in place have been completely bypassed. You sometimes wonder if "we're doing okay, because this quarter, we earned just as much as the previous one" couldn't be ever considered enough.

2026-04-28 20:22:34 UTC

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one of my favorite biomimicry examples is the redesign of Japan's bullet trains based on the kingfisher's head and beak, which made them more efficient and streamlined. See e.g. https://biomimicry.org.nz/the-shinkansen-and-the-kingfisher-a-tale-of-biomimicry-in-high-speed-rail-design/

2026-04-23 23:44:52 UTC

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The last one not being threaded was actually deliberate, since it was a parenthetical remark at best. :) Thanks for checking!

2026-04-23 21:58:40 UTC

(This is all pretty well-known, but I kind of wanted to practice how to do threaded toots, the same way @npub1nf4…nqe4 sometimes does with various interesting topics.)

2026-04-21 11:32:44 UTC

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"trawling" and "trolling" are two completely different methods of fishing: https://alaskagoldbrand.com/blogs/news/what-is-the-difference-between-trolling-and-trawling-alaska-gold-seafood ; the latter is what lent its name to the annoying internet activity.

2026-04-20 23:26:47 UTC

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most people, when they think of green-colored gemstones, tend to prefer jade. I, for one, like peridot's colors more. :D

2026-04-19 23:55:08 UTC

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it's interesting that Seidel's wiki entry makes no mention whatsoever of the fact that she was about to be evicted from her residence a little while before she passed. I suppose it's the nature of the thing, but just how many wiki articles on people gloss over documented facts about their lives?

2026-04-18 21:26:41 UTC

Since @npub1nf4…nqe4 posted about tetrahelices (https://en.wikipedia.org/wiki/Boerdijk%E2%80%93Coxeter_helix) the other day, I thought I'd share something I recently made on Shadertoy: https://www.shadertoy.com/view/sfBXDR . If you look at the implementation closely, I generate the points of the tetrahelix through a recursion relation, instead of using the trig formulae in the wiki page. #math

2026-04-16 17:15:51 UTC

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at an extreme, consider the following (non-elementary!) hypergeometric function:\( a\, {}_4 F_3\left({{\tfrac15,\tfrac25,\tfrac35,\tfrac45}\atop{\tfrac12,\tfrac34,\tfrac54}}\middle|\frac{3125 a^4}{256}\right) \) which can be shown to be a root of the quintic \( x^5-x+a \) (see e.g. Glasser's paper, https://doi.org/10.1016/S0377-0427(00)00287-9 for details).

2026-04-16 07:46:24 UTC

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as someone who works with "special functions": I have a feeling someone (or multiple someones) is conflating "algebraic" with "elementary". Let me explain, starting with an analogy: in the so-called "casus irreducibilis" of cubic equations, we readily accept the use of trigonometric or hyperbolic functions (certainly elementary!) to represent the roots of a cubic polynomial. For quintics, we need theta functions or hypergeometric functions, which are most assuredly not elementary, but evaluate to algebraic numbers. (Then we need Riemann theta for roots of polynomials of even higher order!)

2026-04-13 09:23:31 UTC

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if the tetrahelix has unit edge length, the inner cylinder has radius \( \frac{\sqrt{18}}{20} \), and the outer cylinder has radius \( \frac{\sqrt{27}}{10} \) . So the circumradius-inradius ratio is \( \sqrt{6}\approx 2.4495 \), which is less than the corresponding ratio of 3 for the regular tetrahedron.

2026-04-13 02:13:06 UTC

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personally, I find the fact that you can inscribe and circumscribe cylinders on the tetrahelix very cute.
https://media.mathstodon.xyz/media_attachments/files/116/394/971/934/415/956/original/46e1754adb21cc02.png

2025-10-04 10:57:18 UTC

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if it isn't already apparent, the supramolecular guys have a most whimsical naming sense.

2025-10-04 10:45:37 UTC

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one very interesting analogue is the drug sodium nitroprusside (https://en.wikipedia.org/wiki/Sodium_nitroprusside), which is used in emergency situations that require lowering a person's blood pressure very quickly. A neat resource for looking up crystal structures of these beasties is the CSD, e.g. https://www.ccdc.cam.ac.uk/structures/Search?Ccdcid=DOJJAP05

2025-10-04 10:22:39 UTC

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...and well, if you or anyone else reading this thread has found host-guest chemistry (supramolecular chemistry) to be really interesting, I would then invite you to look up cryptands, calixarenes, scorpionates, pillararenes... it's a really deep and fascinating rabbit hole! ;D

2025-10-04 10:09:42 UTC

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in the bonding model I am familiar with, each oxygen is basically "sharing" an electron pair with the coordinated cation (i.e., a dative bond); I am not up to date with the supramolecular literature to check if they have refined this model since I last looked at it decades ago... :D

2025-10-04 10:06:13 UTC

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it took me quite a while to find a "nice" picture modeling the bonding in the cesium electrode (and also something you can cite, in case you want to go down this rabbit hole). Page 232 of Steed and Atwood (https://www.wiley.com/en-us/Supramolecular+Chemistry%2C+3rd+Edition-p-00029542) has the following lovely figure:
https://media.mathstodon.xyz/media_attachments/files/115/315/292/897/916/411/original/9446b78d26cfef37.png

2025-10-04 09:12:02 UTC

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and at one interesting extreme, you have electrides (https://en.wikipedia.org/wiki/Electride), where you have a large enough cation like cesium trapped in a crown ether, and a single electron serves as the corresponding anion. Pretty neat stuff!

2025-10-04 09:07:43 UTC

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most commonplace crown ethers are ultimately built out of ethylene oxide, which has two carbons and one oxygen; this is why they have crown sizes a multiple of 3 (12-crown-4, 15-crown-5, 18-crown-6, etc.)