You know what's gross? Cracking open an egg and finding a fairly advanced embryo. The network of blood vessels is somewhat fascinating though.

"If the images have the same z-index then the behavior is undefined."

Does that mean my computer can explode, or does it merely mean I won't know which pixel ends up in front? Because in C++ compilers I think it means my computer might explode.

Interesting to see a family member apparently cited in a hallucinated reference. In a paper whose abstract talks about accuracy improvements.

https://arxiv.org/abs/2607.12111

Now I know that retcons are real. Wikipedia is claiming that from 2021 to 2025 some guy called Joe Biden was president of the US.

Profound words from the manual

I can't decide which topic is less interesting to me: football or the question of whether machines can think. Actually, I was kinda sorta half rooting for Belgium, so that answers that.

I hate that "forgot your password" button. So patronising.

Bicomplex numbers in the wild ℂ⊗ℂ

Complex numbers are sometimes very convenient as a coordinate for 2D fluid dynamics. Complex numbers are also useful for reasoning about oscillatory perturbations to a fluid. You can use both at the same time but you need to take care because you're now working with two completely distinct square roots of -1. You can simply give the two numbers different names i and j but what you're really doing is working in the tensor product ℂ⊗ℂ. I was interested to see a paper on exactly this topic [1].

I mention this because it's exactly the structure you get in many programming languages when you iterate the complex number type constructor, eg. Complex (Complex Double) or complex<complex<double>>. As I mentioned in [2] you don't just get the vector space tensor product but you get the correct algebraic structure "for free". In fact, it's entirely possible to write generic code to work separately with complex coordinates and complex perturbations that makes the compiler build the bicomplex algebra under the hood without the user even realising.

[1] https://arxiv.org/abs/2203.05857
[2] https://blog.sigfpe.com/2026/04/introduction-i-want-to-return-to.html

(I should have said ℂ⊗_ℝ ℂ but it's unwieldy in Unicode.)

I really enjoyed Hartnett's book on Lean - the Proof in the Code. It was fun having watched all this unfold from the sidelines but wanting a more detailed narrative about what was going on.

It has the usual issue in pop-sci that some statements inside my domain of expertise were false and I don't know how much to trust the rest. But I just rolled with it and found it a pretty exciting story.

I think that despite being recent it's already out of date. I think the LLMs can help formalize statements and discover proofs better than the book ended up saying.

Curry-Howard. Who knew it would turn out to be so important? (Asking the wrong audience here. :)

"I very much agree that mathematics is a human activity. But I also think that using a computer is a human activity."

Johan Commelin, director of mathlib

The cool thing about volume knobs in the old days is that you could turn the volume down before switching on the device.