So just thinking.
Bravais lattices have symmetry, when translated, their symmetry is preserved at all lattice sites, i.e. each lattice is the same.
There is the unit cell (of some sort) and these packed together form our (infinite) lattice.
Now consider the Platonic solids (i.e. cube, tetrahedron, octahedron, dodecahedron, icosahedron).
We can see that our simple cubic structure is just cubes packed around each other.
Are the others related? Visually, I can't see so, e.g. hexagonal, but otherwise?
Also, when you find the reciprocal lattice of a Bravais lattice (at least in the examples we've seen thus far), you get another Bravais lattice. E.g. Cubic goes to cubic. Face-centred goes to body-centre and body-centred goes to face-centred.
With the Platonic solids, if you consider the dual of it (swaps vertices and faces), you get another Platonic solid,i.e. tetrahedron duals with tetrahedron, cube with octahedron and dodecahedron with icosahedron.
I'm finding this similarity a bit suspicious - is it that common or just a nice shared characteristic?
Maybe they're related in the way they're stacked around each other?
You're trying to fill a space (infinite space) with a repeated pattern with no gaps. This is in the Bravais lattice and with Platonic solids.
Can anyone else see a similarity?
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beat you to the comment!
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For a solid to correspond to a bravais lattic we must be able to pack space with the solid using only translation. This assues all points are equivalent. This cannot be done with a tiagonal shape, i.e. a diamond crystal.
ReplyDeleteHowever two triagonal shapes can be combined into a cube like bravais structure.
Anyway i think thats right.
I think the relationship is due to the strong symmetry requirements of both Platonic solids and Bravais lattices, although Bravais lattice cells must be able to stack up in 3D, whereas Platonic solids don't need to be.
ReplyDeleteI also wouldn't 'read' too much into the dual relationships and reciprocalising Bravais lattice structures: I think the dual relationships of Platonic solids are due to the regularity constraints so it wouldn't really make sense if they weren't 'invertible'. So it seems like a nice coincidence! Also, there are more Bravais lattices than Platonic solids... those less educated in maths might suggest that there are two 'hidden' Platonic solids out there....
Interestingly, the regularity of the sides of Platonic solids means that they are the only shapes that can make dice with equal outcomes (otherwise you get small sides, which could be interesting if you designed a system to take advantage of this). The regularity of sides constraint doesn't apply to Bravais lattices, but by analogy it does apply to the translational constraint. See, I found a similarity! :)
Coincidence perhaps, but it seems a bit suspicious...
ReplyDeleteBy the way, Andy and I have experimentally found that you can't pack dodecahedra by themselves.
Still, symmetry's nice.
I agree with Josh's opinion. The similarity between platonic solids and bravais lattice is due to symmetry requirements in their definition. with the Bravais lattice having a stricter definition requiring the arrangement and orientation of all the other points to be the same wherever its viewed. You can't pack dodecahedron since it has 5 fold symmetry.
ReplyDeleteBy the way, there's three dodecaedra now...
ReplyDeleteI don't exactly see any direct relation between a Bravis lattice and platonic solid. I think a Bravis lattice is a 3D equivalent to a tiling of of a 2D plane by a single shape (of which all 3 solutions are 'platonic' 2D shapes). The orders of symmetry in both cases are the most important thing. Admitting more than a single shape to the tiling will necessarily give you locally symmetric, but not globally symmetric shapes.
ReplyDeletehang on, with 4 dodecahedra, we can show that one can stack them...
ReplyDeletemaybe just imperfect dodecahedra, with 5, doesn't pack
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