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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.

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    $\begingroup$ "or organic materials" — your feeling may be wrong here. Protein folding is precise and hard to predict thing. I wouldn't bet on it staying unaffected. $\endgroup$ – Mołot Nov 18 '18 at 23:01
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    $\begingroup$ @kingledion "I tried to solve for the differences in a cubic crystal, but I keep getting that there are none?" Well for one, no crystals would have a cubic crystal system, since you can't tessellation hyperbolic space with cubes (at least in the traditional way). $\endgroup$ – PyRulez Nov 19 '18 at 1:50
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    $\begingroup$ Nice question! My intuition is that they would look like usual crystals in Euclidean space, but riddled with topological defects of codimension 1 (domain walls). The reason would be that the volume excess creates an effective strain in the crystal, analogously to why when you try to fit a flat sheet of paper into a saddle shape it tears up at the sides. I assume the final results would depend on the crystal's structure and the strength of its molecular bonds. $\endgroup$ – pregunton Nov 19 '18 at 7:36
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    $\begingroup$ This question seems to make the assumption that the direction of the atomic bonds will be affected by the spatial curvature, but their length will not. Consider a rubber sheet of graph paper: since the distance between the grid's corners can change- it remains a grid even in twisted space. $\endgroup$ – Glurth Nov 20 '18 at 6:41
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    $\begingroup$ @Glurth how would you embed a grid in 2D hyperbolic space without changing distances. $\endgroup$ – PyRulez Dec 13 '18 at 13:11
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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.

Silicon Carbide

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.

Iron carbon (source)

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.

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