my neurons think about themselves a lot

Right! There’s something comforting in accepting that it’s just looks nice — even if it might be a wild goose chase :)

There’s more evidence for that perspective:
x.com/hippopedoid/...

*John Williamson, not Healy.

Check out johnhw’s notes! They plot where things like factors and sequences show up, and compute number-theoretic stats.

The (pink) image is colored by integer magnitude (bright = large). Some trajectories hint at sequences, but no formal classification was made.

johnhw.github.io/umap_primes/...

Euclid’s Elements is the second-most printed and studied book in history — after the Bible.

Written around 300 BC in ancient Greece, this edition is its first English translation (1570). It remained a core math textbook well into the 20th century.

Euclid, Newton & Turing

This weekend, at the John Rylands Library in Manchester, I finally found a historical copy of Euclid’s Elements.

And right next to it: Newton’s Principia and Turing’s handwritten notes. Completely overwhelming.

Our paper, videos, and 3D models are available here:
go.epfl.ch/smooth_rolling_knots

And they work in real life too :)

Smooth-rolling objects require virtually no force to start moving – even with low friction, they roll.

An object is “smooth-rolling” if its center of mass remains at a constant height while it rolls.

We created knots with this property by combining Morton’s knots with Two-Disk Rollers.

Smooth-rolling knots!

At @markpauly.bsky.social’s lab, we found a way to optimize knots for smooth rolling.

I presented our work at Bridges 2025 – the conference for #MathArt.

More below ↓

Thank you Keenan — really means a lot!

Your Repulsive Curves are actually what got me working on knots in the first place. :)

Each number becomes a binary vector representing its prime divisors:

1 → [0, 0, 0, …]
2 → [1, 0, 0, …]
3 → [0, 1, 0, …]
6 (2×3) → [1, 1, 0, …]
30 (2×3×5) → [1, 1, 1, 0, …]

Each bit = “is divisible by the n-th prime?”

Here are the first 8 million integers, rendered by John Healy.

johnhw.github.io/umap_primes/...

What is the hidden structure of the natural numbers?

In the UMAP paper, @lelandmcinnes.bsky.social et al. embedded the integers as binary vectors of their prime factors.

This UMAP visualization of 30 million integers reveals a fractal-like geometry.

#UMAP #mathart #dataviz #numbertheory