How would crystals be different in hyperbolic space?

Question

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.

Answer 1

Poly-crystalline crystals care not for your hyperbolic spaces

If the difference between Euclidean and hyperbolic spaces on the sub-atomic scale are sufficiently small, it may not matter. Said another way, hyperbolic-isity is a macro property of this physical system instead a property of the quantum realm. The concept is similar to how Newtonian physics prevails at human scales but don't mean a thing on the quantum scale. Hyperbolic properties may only show up at human scales.

Very few crystals are true mono-crystals. Dislocations, voids, discontinuities and other artifacts are exceptionally common (and downright maddening when trying to get a high quality mono-crystal). For larger crystals, these discontinuities in the crystal structure would 'absorb' any spatial irregularities at the atomic scale. To a human observer, the crystal would appear like other crystals with perhaps a higher number of discontinuities. Detecting that higher flaw rate would require specialized equipment and the knowledge that the same crystal in Euclidean space has fewer flaws.

For example, a silicon-carbon lattice, shown below. Over small scales, dislocation stresses caused by the hyperbolic space won't matter since the atomic bonds can deal with some stress. Eventually though, these stresses accumulate to overcome the chemical bonds between the carbon and silicon atoms. On these boundaries, the crystal will break into a cleavage face. While this cleavage face is weaker than a normal monocrystal, it doesn't mean the overall crystal is weak in an absolute sense.

As an example of strong polycrystaline materials, take iron. Iron-carbon crystals have lots of discontinuities and these affect the behavior of the material but they aren't visible on normal human scales. Maybe there might be a greater tendency towards smaller continuous crystals in this hyperbolic space but it would difficult to tell.

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Also, this field is hard. I was able to find several papers on quantum physics in hyperbolic spaces but I'm not a quantum physicist so the papers don't mean much to me. Also, calculating crystal lattices is a specialized field. The software to simulate crystal lattices is very specialized. High quality simulations of a crystal lattice are extremely compute intensive. Determining the behavior of a specific crystal in hyperbolic geometry would require specific attention and lots of theoretical work to nail down. (Then you couldn't test it because we don't live in hyperbolic space.) I'd be very hesitant to make broad pronouncements about changes to the shapes of individual lattices.