How would crystals be different in hyperbolic space?

Let's say our world is set in a hyperbolic space with a curvature K of −1 per 9 square meters. How would crystals be different than those in our world (assuming that the atoms are the same as the ones in our universe)?

One thing to note that on small scales, hyperbolic space is approximately euclidean. But not exactly euclidean. Even an atomic sized triangle will have angles adding up to slightly less than 180 degrees. This probably wouldn't affect amorphous solids, or organic materials, but I have a feeling it would affect crystals.

In particular, euclidean crystallographic groups would not be applicable. Well, they may apply approximately locally, but for a large crystal, it would not work for the entire structure.

Now, there are probably hyperbolic crystallographic groups, related to the hyperbolic honeycombs. The problem is that, unlike euclidean honeycombs, they do not scale. The size of the cells of a given hyperbolic honeycomb is fixed, and can be scaled up or down to different sizes. Therefore, for a crystal to take advantage of a hyperbolic honeycomb, the distance between the particles would need to match the edge length of the cells of the honeycomb.

posted almost 6 years ago

PyRulez‭

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