There are perfectly good solutions for finding the angle using atan and the cross product. Calculating A x B will yield a vector which lies is along their normal with length tan(θ)(A · B) so we can straightforwardly say e.g.:

`θ = atan2(norm(A ⅹ B), (A · B))`

No thresholds, no branching, only ~1 bit of significance lost. As the original author says himself, when you're introducing arbitrary-feeling thresholds, it's likely you're missing a solution which would improve more than just the weirdness at those thresholds.

[0] A good analysis of this is on Page 46 here, including a more optimized (but less obviously true) alternative to the formula above: https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf

I'm now wondering the hausdorf dimension of the graph of alphabetical numbers ]]>Thu, 16 Apr 2026 12:55:37 +0000https://news.ycombinator.com/item?id=47792315unholinesshttps://news.ycombinator.com/item?id=47792315https://news.ycombinator.com/item?id=47792315I discovered them last year on Substack and they quickly became a priority read. A sort of Quanta for biology, taking time to explain enough for a popular audience but keeping technical rigor deep into some fascinating topics.

Some highlights:

https://open.substack.com/pub/cell/p/dna-sequencing?utm_sour...

https://open.substack.com/pub/cell/p/phi80?utm_source=share&...

https://open.substack.com/pub/cell/p/antibody-design?utm_sou...

https://open.substack.com/pub/cell/p/viral-capsids?utm_sourc...

https://open.substack.com/pub/cell/p/legibility-problem?utm_...

...All in the last month! At least they went out with a bang.

FeatureScript is a different beast. It actually runs as part of regeneration in Onshape. Standard features (extrude, loft...) are also defined in FeatureScript, so your custom features are the same first-class citizens with a interactive GUIs and stable updates to upstream changes. You can freely mix interactive CAD and custom code by adding standard features and custom features.

I think it's tempting to be equally dismissive of branding and fashion because they are both forms of virtue signaling and therefore have all its perversions. However, they're operating on different actors. Branding is virtue signaling by companies to its customers. Fashion is virtue signaling by customers to other people.

One big annoyance with the power-ups is that the failure condition is checked before you can use them. It's particularly painful since they all replace the current piece, so they seem tailor-made to get rid of a nasty piece that would cause you to lose... but then they have to be used before you get to that piece that would cause you to lose!

Anyway, both times I played I got in the 40s without any power-ups used, then saw that the next piece would cause me to lose but none of the powers could save me. Probably the ideal fix is just to not trigger a loss while you still have powers?

A random note in case Non-Zero-Sum James is looking: It's frustrating that reading footnotes[0] requires scrolling back and finding your previous place. A link from the footnote back to the original place in the text or something that reveals a footnote in-place (e.g. on hover) is fairly universal and very helpful!

[0] e.g. https://nonzerosum.games/emergencespirals.html#notes

The reasonable things that continue happening each day in our universe would be extremely unlikely if we are just Boltzman brains. Every bit of sensible reality would be coincidental. The very continuance of that reality is an experiment constant proving the falsehood of Boltzman brains, at a rate of oh maybe millions of sigmas of confidence per second.

Now, if you believe the universe came to an initial state due to pure thermodynamic coincidence, millions of sigmas per second is laughably small compared to the chance that a whole universe outside your brain popped into existence, so Boltzman brains are the most believable thing and you should believe in them.

This completes a pretty direct argument: Believing the initial state of the universe was a thermodynamic coincidence forces you to believe in Boltzman brains, Boltzman brains force you to believe reality should collapse immediately, and reality does not collapse immediately. Therefore you simply can't believe the first assumption, that initial state of the universe was a thermodynamic coincidence.

Accepting this is often called the "Past Hypothesis". It's spoken of in deferential terms and said that it can't ever be proven... But to me this is rock-solid proof, with more sigmas of evidence than any other scientific discovery and increasing by the second! Can't we just call it the Past Theorem already?

With mesh faces now supported in BREP, I'm more optimistic about a mixed modeling approach, where you can do the braindead find-the-cylinders conversion but keep the rest mesh, not needing to force it into some eldritch contortion of BSurfs.

This multiplies, so after a million days a clock on the moon will read 58 seconds behind a clock on the earth.

[0]https://chromewebstore.google.com/detail/unhook-remove-youtu...

A high school sophomore! I'm thoroughly impressed, not just by the clarity of this explanation, but by the tone, showing appreciation for where and how the elegance pops up, and empathy for a reader who's never heard of these concepts. Throw in the minimal web design I was sure I was reading some graybeard.

Excellent writing, you should be proud of this.