Proof of Aperiodic Tiling of Infinite Plane Has Been Accomplished

[Boydstun]

 

Proof of Aperiodic MonoTile Has Been Accomplished

Edited March 29, 2023 by Boydstun

[Doug Morris]

The link seems not to work.

 

[Boydstun]

Thanks, Doug! Let me try it again: https://arxiv.org/abs/2303.10798 

The reason it was called the "einstein" was because ein stein means one stone. That single tile has now been shown to exist. We can sleep more sweetly now.

The full paper: https://arxiv.org/pdf/2303.10798.pdf

Edited March 29, 2023 by Boydstun

[Boydstun]

Einstein Tiling Is Getting Material

Physicists have made a theoretical examination of what sort of material results if one places an atom at each vertex of the tiles.* This had been done with the Penrose tiles (1970's – 2 shapes of tile) covering the plane. The physicists look at such a tiled plane they have conceived as vertex-embedded with atoms as a slice through a 3D material. In the case of the Penrose tiling, the material turned out to be a quasicrystal. That solid material of the Penrose tiles was later found in nature. The matter that would result from the Einstein tiling (also called the Hat tiling because the shape of that tile resembles silhouette of a hat) is a quasicrystal, but it shares properties of the crystal structure called graphene.

If I recall correctly, the pure-carbon crystal graphene was invented in the wake of the invention of fullerene (buckyballs), a sphere-like molecule purely carbon. Fullerene was first theoretically fashioned, then produced physically. Later it was found to also exist in nature. This coincidence of the artificial and the natural is perfectly fitted with Rand’s metaphysics in Galt’s Speech and in her essay "The Metaphysical versus the Man-Made” and in Peikoff’s “The Analytic-Synthetic Dichotomy.” Objectivists had emphasized that products of invention have determinate natures just as things occurring in the course of nature. That the former were inventions initiated by free human choices does not mean that the nature of the inventions are any less (or more) necessary than the necessities within the purely naturally given.

Edited March 25 by Boydstun

[tadmjones]

It's as if Penrose is a human LLM analog