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"The groups of atoms in crystals repeat at regular intervals, just like the five known patterns."

What is "five known patterns" a reference to? I found [1] but I don't see five of anything here.

[1]: https://en.wikipedia.org/wiki/Crystal_structure

I assumed that he was talking about the possible point groups in 2d (since his demo was a slide). In 2d lattices, the possible point groups are cyclic groups of order 2,3,4,6. Cyclic groups of order 5 cannot occur; one consequence is that regular pentagons cannot tile the plane. See

https://en.wikipedia.org/wiki/Crystallographic_point_group

and

https://en.wikipedia.org/wiki/Crystallographic_restriction_t....

I should add that I know nothing about crystallography, and am relying on decades-old knowledge from the last abstract algebra class I took.

The Platonic Solids: https://en.wikipedia.org/wiki/Platonic_solid

Not all of those correspond to patterns of crystal structure (although some do: tetrahedron, cube, and octahedron, IIRC).

Quasicrystals are based on the symmetry system of the icosahedron/dodecahedron, in which it it is impossible to periodically tile space.

“Proper” crystal structures are periodic, and are not necessarily related to any Platonic solid. See https://en.wikipedia.org/wiki/Crystal_system (but I’m not quite sure where the 5 comes from in TFA).

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