packings

He then suggested some generalizations, and proposed that circle packings would prove to be important to me.

To be a little nitpicky: why did you start with obscure high-dimensional visualizations such as alluvial plots, dendrograms, sphere packings?

That's another problem: you can re-label all you want, but the glasses, packings and tins are still the same size.

It has way more applications than sphere packings but it's hard to pick a good example.

Conclusion: The study of Apollonian circle packings led to the challenge and ultimate disproval of the previously accepted local-global conjecture.

There exist infinitely many primitive Apollonian circle packings for which the number of missing curvatures up to N is Ω(√N).

Many of those encodings and packings, particularly for the fixed-size messages, used structure bitfield magic.

For a while, mathematicians considered highly symmetric, lattice-based packings to be the best way to arrange spheres as densely as possible.

To construct these packings, shift horizontal layers of FCC horizontally with respect to each other.

ChatGPT v4 Technical Article Version of this long-form post: Apollonian Circle Packings and the Local-Global Conjecture Background: - Apollonian circle packings is the study of how circles can fit into a larger circle.

Testing the Conjecture: James Rickards created software to examine any desired arrangement of circle packings.

Here, can't you imagine that in a compressed sample of atoms, a small bounded region, things arrange themselves respecting minimum packings?

Apollonian circles are geometry, but the conjecture is about the integers that show up as the curvatures of packings, and specifically about the "certain numerical buckets" they happen to fall into.

The whole idea of "pack multiple (often variable bit-length) instructions into a word and rely on compiler writers to determine optimal instruction packings" to make implementation simpler and therefore hopefully faster, but woe betide thee if thy compiler dost not find such optimal instruction packings; for then thy CPU will actually run code more slowly than a conventional design like x86!

Pedantic nitpick, but I might as well bring it up first so we can have it out of the way: “There’s also an uncountable infinity of other patterns that all give you equally dense packings.” According to the author’s examples using only {a, b, c}, the set of possible packings would be countably infinite.