5 added to My Authors

Sep 18 • 15 tweets • 10 min read

Sep 17 • 8 tweets • 6 min read

Sep 13 • 4 tweets • 1 min read

Yesterday I learned that “∀α. 2^(ℵ_α) = ℵ_(α+γ)” is demonstrably false if γ is infinite, and consistent with ZFC — under large cardinal assumptions — when γ is a concrete natural number ≥1 (when γ=1 this is the GCH and requires no large cardinals). mathoverflow.net/a/430293/17064
I don't know what intuitive sense to make of this. I find it puzzling that even though Easton's theorem allows us to more or less arbitrarily prescribe the continuum function at regular cardinals, singular cardinals are a whole different ball game.

Sep 13 • 7 tweets • 2 min read

Le problème avec les débats télé par petites phrases où on ferme la bouche des autres, c'est que ça incite les gens à retenir des petites phrases et à reproduire ce type de débat (voir le fil du tweet cité pour plein d'exemples) plutôt que construire une réflexion approfondie.

Quand on veut convaincre, il faut du temps, et la sérénité d'une écoute sincère.

Il y a une raison pour laquelle les controverses scientifiques ne se tranchent pas par petites phrases. Pour laquelle les procès d'assises ne se règlent pas par petites phrases.Quand on veut convaincre, il faut du temps, et la sérénité d'une écoute sincère.

Sep 11 • 15 tweets • 4 min read

[Image below: ‘Utopia City 2080’ by Damian Krzywonos source deviantart.com/damiankrzywono… — CC BY-NC-SA 3.0] We've all seen a million of them: post-apocalyptic movies with that characteristic dreary post-apocalyptic “let's make the sky so gloomy that real-life London looks like an ad for the Maldives in comparison” color palette that has become an utter cliché. I hate it. •2/15

Sep 11 • 4 tweets • 1 min read

Maybe worth retweeting this 🔽 with an English summary: many people (esp. in France…) have been claiming that the hymn ‘God Save the King’ was originally written around 1686 by Lully (to celebrate Louis XIV's successful anal fistula operation), … •1/4

… and then stolen by Händel before becoming the British anthem. Well, while it's not possible to definitely claim this DIDN'T happen, the sources are more than flaky and the claim is otherwise suspicious. The main source is the memoirs of the Marquise de Créquy, … •2/4
Sep 11 • 4 tweets • 1 min read

Sep 3 • 29 tweets • 21 min read

Sep 2 • 5 tweets • 2 min read

Ok, now I have a problem: I found a contradiction in mathematics / physics, so either the Universe does not exist or — as is somewhat more likely — I made a stupid mistake somewhere. 😭 Here's the problem: ⤵️ •1/5
Consider a refracting medium whose index of refraction doesn't depend on the x coordinate (say), only on y and z. We let various rays of light go through it with initial directions having the same projection on the (y,z) plane but differing x components. •2/5

Sep 2 • 10 tweets • 2 min read

A remark/clarification on the “ray equation” (differential form of the Snell-Descartes law / of the Fermat principle) that defines the trajectory of a ray of light in a medium of (differentially) varying index of refraction n(x,y,z). 🧵⤵️ •1/10
I won't comment on how to derive this equation (there are many possible ways, depending on what starting point you want: Fermat's principle, Maxwell's equations, or whatever: see the answers to physics.stackexchange.com/q/43711/39931 for that; … •2/10

Sep 1 • 16 tweets • 3 min read

One more (and hopefully last) thread about the question below🔽. In this situation (we take a plane P₁ of incident rays and refract them wrt a plane Π), I now claim that the refracted rays form a quadratic cone. A little algebraic geometry ahead! 🧵⤵️ •1/16

So, how can we parametrize the direction of a ray of light (either incident or outgoing)? We can use coordinates (x,y,z) of a vector along the ray. We could normalize by x²+y²+z²=1 but I choose NOT to do this. Also, I'll be adding a fourth coordinate, as I will explain. •2/16
Sep 1 • 5 tweets • 2 min read

OK, here's another interesting thing to say about the question below🔽: as I point out later in the thread, the answer is “no”, the refracted rays don't stay in a plane. BUT if we modify the Snell-Descartes law to use tan(θ) instead of sin(θ), then they they DO. Why? •1/5

Well, let Π have equation z=0. Let (x,y,z) be a (not normalized) vector collinear to an incident ray: the usual Snell-Descartes law (with sin(θ)) says “multiply √(x²+y²)/√(x²+y²+z²) (that's sin(θ)) by n₁/n₂ while keeping (x:y) constant” to compute the refracted ray, … •2/5
Sep 1 • 6 tweets • 2 min read

Sep 1 • 7 tweets • 2 min read

A question in geometrical optics:

Let a plane Π separate two indices of refraction n₁ and n₂, and O a point on Π. Consider rays of light from the n₁ side all going through O and all lying in one same plane P₁ (not orthogonal to Π). Do the refracted rays lie in one plane P₂? This is a very basic question and somehow I missed either the statement that it is true or the caveat that it is not, when I learned about optics.

I could do the computation from the Snell-Descartes law, but maybe someone has an enlightening argument why this is true — or false.

Let a plane Π separate two indices of refraction n₁ and n₂, and O a point on Π. Consider rays of light from the n₁ side all going through O and all lying in one same plane P₁ (not orthogonal to Π). Do the refracted rays lie in one plane P₂? This is a very basic question and somehow I missed either the statement that it is true or the caveat that it is not, when I learned about optics.

I could do the computation from the Snell-Descartes law, but maybe someone has an enlightening argument why this is true — or false.

Aug 31 • 4 tweets • 2 min read

And now for something completely different: a rotating icosahedron (ior=1.5). PoV-Ray source: gist.github.com/Gro-Tsen/e9799…
Previous videos:

Aug 31 • 15 tweets • 5 min read

I posted a few animations of a rotating dodecahedron (see thread of quoted tweet⬇️), but I just realized I had hit on a PoV-Ray limitation in computing photon maps. Here🔽's a corrected version, and a few explanations. ⤵️ The difference is in the lighting on the “ground”. •1/15

Note the difference between the (corrected) version above and the version I had originally posted (below) is not in the dodecahedron itself, but in how the light going through it or bouncing on it illuminates the ground by reflection and refraction. •2/15
Aug 30 • 6 tweets • 2 min read

Since today I remembered that there is such a thing as PoV-Ray and that it's fun to play with, please enjoy this completely gratuitous rotating glassy dodecahedron (source: gist.github.com/Gro-Tsen/45015…), just because it's fun to make and pretty to watch.
Or is it perhaps prettier if the vertices of the dodecahedron are snubbed to a sphere, so that the 12 faces are circles tangent at the edges? [Just insert the line `sphere { <0,0,0>, 8*(-1+sqrt(5))/2 }` between lines 41 and 42 of previous source.]

Aug 27 • 12 tweets • 3 min read

OK, since I made an embarrassing mistake in another thread, I now need to explain how to solve the following combinatorial problem: you have k+n characters of which k use 1 unit of space and n use 2: how many words can you form with N units of space? •1/12

In other words, what is the number u_N of finite words over an alphabet of k+n characters such that a+2b=N where a is the number of characters in the word that are taken from the first set and b is the number taken from the second? •2/12
Aug 27 • 19 tweets • 4 min read

How many bits of information can you cram in a single pure-text tweet? The answer is surprisingly difficult to compute (and may depend on what precisely you allow), so I don't know exactly; but somewhere around 5729 bits seems right. Let me explain. 🧵🔽 •1/17
Many people think a tweet can be 280 characters long, but that's not quite right: it's 280 units, and a Unicode character can consume 1 or 2 units (very roughly, alphabetic scripts consume 1 unit per char, anything else, e,g, Chinese ideographs, consumes 2). •2/17

Aug 26 • 4 tweets • 1 min read

Today's news in the “we have no f●cking idea how AIs work (even though we keep putting them everywhere with little to no oversight)”, Meta's chatbot calmly giving Dutch politician Marietje Schaake, out of the blue, as an example of a ter—ahem—rorist.

[I intercalated “ahem” in the previous tweet in the middle of the t-word so as to avoid creating further association of the latter to her name, e.g., not to pollute search results if an attempt to understand the cause of the association is made.]
Aug 25 • 4 tweets • 1 min read

Brilliant!

(The whole thread is incoherent rambling AFAICT, but some bits are fun nonetheless.)