Nice example of how weird large-dimensional space is. Apparently, when smart minds were asked to put as many 100-dimensional oranges in a 100-dimensional crate as they could, so far, the best they managed to do was fill less than 1% of its space with oranges, and decades of searching couldn’t find a spot to put another one.

EDIT: groped -> grouped

Also, the toolkit one uses in convex geometry, especially some of the harmonic analysis tools are quite handy in the study of sphere packing.

So "unexpected"? Not quite.

Those numbers sound wild. For various comms systems does this mean several orders of magnitude bandwidth improvement or power reduction?

I agree with this and I'm not even a mathematician, I've seen convex hull algorithms pop up in unexpected places to solve problems I would never have thought of using convex hull algorithms for, like a paper on automatic palette decomposition of images.

https://www.rose-hulman.edu/class/cs/csse451/Papers/DILvGRB....

The comments pointed out that anthropologist did not know that boiling was possible before the invention of pottery. Another comment pointed out that science teachers knew that it was possible because that was something they would do in class.

Final comment was about how people ReDiscover things in different fields - - like the trapezoidal rule for integration being discovered by someone studying glucose.

This is just yet another example of how bringing expertise from a different area can help.

Someone can take this challenge to provide a more secure and reliable communication systems hopefully with more energy efficiency, very much an exciting research direction.

In context of packing problem, it's a bit meta to me...

An LLM contains a k-dimensional packing of known knowledge. This packing is highly inefficient because it has holes and unbridged dimensions. By injecting random seed (prompts) into the LLM probability space, it gets perturbed. Sometimes this perturbation fills a hole in the packing and/or connects two adjacent units in way nobody thought of before because it wasn't fashionable any more, or wasn't top of mind. Thus new knowledge is created within the same k-dimensional box through a novel joining-of of existing know-how.

From the article:

> Klartag had broken open a central problem in the world of lattices and sphere packing after just a few months of study and a few weeks of proof writing. “It feels almost unfair,” he said. But that’s often how mathematics works: Sometimes all a sticky problem needs is a few fresh ideas, and venturing outside one’s immediate field can be rewarding. Klartag’s familiarity with convex geometry, usually a separate area of study, turned out to be just what the problem required. “This idea was at the top of my mind because of my work,” he said. “It was obvious to me that this was something I could try.”

Sounds like the technique is for high-dimensional ellipsoids. It relies on putting them on a grid, shrinking, then expanding according to some rules. Evidently this can produce efficient packing arrangements.

I don't think there's any shocking result ("record") for literal sphere packing. I actually encountered this in research when dynamically constructing a codebook for an error-correcting code. The problem reduces to sphere packing in N-dim space. With less efficient, naive approaches, I was able to get results that were good enough and it didn't seem to matter for what I was doing. But it's cool that someone is working on it.

A better title would have been something like: "Shrink-and-grow technique for efficiently packing n-dimensional spheres"