I've admittedly gotten a little lost wading through the source for #Numpy (https://github.com/numpy/numpy), so I thought I'd ask anyone with better familiarity with the internals.

In Numpy, you can do something like

import numpy as np
x = np.linalg.solve(np.array([[1, 1], [3, 5]]), np.array([-1, 2]))

or even

np.linalg.solve(np.array([[1, 2], [3, 5]]), np.array([1, 2+3j]))

and Numpy is smart enough to figure out what the array type of the result ought to be, despite the mixed nature of the input. Which (internal?) function(s) are doing the appropriate type conversions behind the scenes, so that one gets the appropriate result?

I need operations like this for an algorithm I am trying to implement, and I thought it best to not reinvent the wheel.

(Otherwise, how do other Python libraries handle mixed types sanely?)

#python

It looks like the #DLMF website does not play nice with browsers set to dark mode. That the LaTeX formulae remain black even in dark mode is only a little more frustrating. Shouldn't #MathJax be able to detect whether dark mode is enabled, at the very least?

#math

#math #creativecoding #generativeart

A little while back, I did a short write-up regarding the result of Denton, Tao, et al. (https://doi.org/10.1090/bull/1722) regarding the computation of the (squares of) eigenvector components from eigenvalues (https://tpfto.wordpress.com/2019/11/16/on-constructing-eigenvectors-from-eigenvalues/). Back then, I had already noted that matrices like the Wilkinson matrix (https://blogs.mathworks.com/cleve/2013/04/15/wilkinsons-matrices-2/) would spell trouble for the so-called "eigenvector-eigenvalue identity".

For this thread, I thought I'd write up an explicit demonstration in #Python, using Numpy.

(1/2)

#math

My rough estimate, based on papers I've already read, is that in papers that have statements along the lines of "A program implementing XXX of this paper is available from the author", of those papers that are > 5 years old, ~90% of those programs/source code/etc. are effectively lost media.

This is pretty conservative, I think. I have had the disappointment of writing to authors about papers written only a bit more than two years ago, and all I was able to obtain was an apology. Which, last I checked, cannot be executed on a computer.

@skua's response made me realize that I forgot to talk about how I lined stuff up.

So, as I explained earlier, I filled in the p-block first (transactinides at the bottom not included). I more or less still remembered that the s and d orbitals are completely filled by respectively 2 and 10 electrons, so I used that to count backwards, and fill in the s-block next. From there, I had a good chance of majorly screwing up where scandium was supposed to be (again, I struggled with the transition metals!), so it's basically sheer luck that I ended up lining things up correctly. (And it was only hours after doing the exercise that I remembered the sequence "4s2, 3d10, 4p6".)

Is it just me, or did the AMS just break access to their archives of the journal Mathematics of Computation? I can't seem to access papers like https://doi.org/10.1090/S0025-5718-00-01174-1 or https://doi.org/10.1090/S0025-5718-1989-0972369-4 .

#math

I might have talked about Andrew Glassner's deceptively simple idea of geometric substitution (cf. https://doi.org/10.1109/38.135881) before in making #mathart, but I like coming back to it every now and then. The examples in Glassner's article used polyhedra as examples, but I mixed it up and sometimes included tori and torus knots as well in my various attempts. The one attached to this toot, which is based on the trefoil, was from ~15 years ago.

#mathematics

I've attached my own attempt to start things off. As for the questions:

1. I'm most accustomed to organic chemistry, so I filled in carbon and the rest of the p-block elements first.

2. I dallied a bit on the transition metals, and really struggled on the elements after copernicium. (You'll notice I got the order wrong in places.) A lot of the transactinides got named after I had already left school, so they didn't have time to completely seep into my "internal store". As for the transition metals: I quickly filled in iron, cobalt, nickel, copper, silver, gold, zinc, cadmium, and mercury, while struggling with the rest. I don't have any excuses. :)

3. Regarding my mistakes in transactinide placement, see #2. As for my placement of La, Ac, Lu, and Lr (cf. https://doi.org/10.1021/ed059p634 and https://doi.org/10.1021/ed085p1482): I don't have a horse in the race, so I just tend to stick them all together at the bottom.

This BlueSky post on drawing a horse from your mind (https://bsky.app/profile/gorizord.bsky.social/post/3mlkh7y4ges2d) made me think of an analogous challenge for chemists (or people with chemical training):

> Draw a periodic table from memory.

Don't worry, this is just a toot for fun, and nobody's grading you, or giving you a time limit, or anything. Just draw one as you remember it, WITHOUT looking stuff up. (Please put your responses under a CW, so you don't spoil it for other respondents.)

Once you're done, consider the following questions:

1. Which parts did you fill in first? (Optionally, why?)

2. Which parts took you the longest to fill in?

3. Can you explain/rationalize any differences between the periodic table you drew, and the one you looked up after doing this exercise?

#chemistry

While I'm not yet ready to create new #mathart, I'll be posting some old ones I produced ~10-15 years ago, and may have already been seen in other places. The following one is my conception of an "ouroboros torus", a donut eating itself:

Finally, as a nod to my origins as a chemist, I thought I'd try the same transformation on the truncated icosahedron, which is the polyhedron that the usual football/soccer ball is based on, as well as the most frequent form of the justly famous allotropes of carbon, the fullerenes.

As noted by Hosoya and Maruyama (https://doi.org/10.1016/S1093-3263(00)00112-1), one can obtain the vertices of a truncated icosahedron by trisecting the edges of the icosahedron, and taking the trisection points as the new vertices. Applying the transformation \(\phi\mapsto-\tfrac1{\phi}\) to these points apparently gives the truncated great icosahedron (https://en.wikipedia.org/wiki/Truncated_great_icosahedron).

I thought I'd finally stop here, and leave further experiments for other people to do. ;) I'll just note that in all the cartoons in this thread, although I linearly interpolated between the coordinates of the initial and final polyhedron, I applied an easing function (https://easings.net/) near the beginning, and near the end, to show off the initial and final states better.

That's it for now, hope y'all enjoyed this excursion.

(5/n, n=5)

@johncarlosbaez also asked if I could do something similar for the great dodecahedron (https://en.wikipedia.org/wiki/Great_dodecahedron), which shares the same vertices as the icosahedron, but with slightly different connectivity. It also got a little messy, but you can see from the attached cartoon, this polyhedron transforms into the small stellated dodecahedron (https://en.wikipedia.org/wiki/Small_stellated_dodecahedron).

(4/n)

@johncarlosbaez then asked if I could make a version of those cartoons that included the faces of the polyhedra. I was a little more skeptical about these succeeding, because Mathematica often gets weird about self-intersecting polygons when displaying them, and with the (comparatively coarse) tweening method I used, I wasn't sure I could avoid self-intersections.

As you can see from the attached cartoons, the one for the dodecahedron got a little tangled near the end, but the one for the icosahedron surprisingly worked nicely.

(3/n)

As is well-known, the coordinates of the icosahedron, as well as its dual, the dodecahedron, can be obtained through signed permutations of a particular point; in both of these cases, the coordinates involve the golden ratio. @EdPeggJr and I wrote a nice utility for this, so it was easy to make the requisite cartoons, using Mathematica (actually, Wolfram Cloud, since I'm working off a mere smartphone right now).

It can be seen in the attached cartoons that the icosahedron gets transformed to the great icosahedron, while the dodecahedron is transformed to the great stellated dodecahedron (https://en.wikipedia.org/wiki/Great_stellated_dodecahedron), the dual of the great icosahedron.

(2/n)

RE: https://mathstodon.xyz/@johncarlosbaez/116589957362204579

@johncarlosbaez'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

(I don't know if @AmenZwa has already seen this, but I think he'll appreciate it.)

Some artwork I did a few years ago.

#mathart

If I abruptly stop writing here, I very likely may have fallen ill again. For now, I'll keep doing what I can.

I gotta say... trying to do stuff at 42°C (~108°F) heat and 80% humidity is sorta kinda cramping my creativity. I can't afford to live in a place with air conditioning, and bathing every 30 minutes can only cool you so much. (https://newsinfo.inquirer.net/2230472/heat-index-watch-danger-levels-expected-in-metro-manila-36-others)