Hyperbolic crystals

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Hyperbolic crystals

Postby lllllllllwith10ls » Sat Mar 09, 2019 2:20 am

There are ways to represent higher dimensions in a hyperbolic quotient space. For example, here is a 4d rubiks cube represented using that method http://www.roguetemple.com/z/hyper/magiccube.php?c=-ch+-cprob+0+-magic+8+-viz+-nohelp
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Re: Hyperbolic crystals

Postby Klitzing » Sat Mar 09, 2019 9:29 am

Could you provide details of that representation?

I neither can make a rhyme of what is provided there on that linked page
A 2D visualization of a higher dimensional space, using HyperRogue's "crystal" method.

In this method, we represent the vertices of a d-dimensional grid as the tiles of a {2d,4} tiling of a quotient space of the hyperbolic plane, in such a way that two tiles are adjacent iff they are adjacent in the grid.

--- rk
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Re: Hyperbolic crystals

Postby lllllllllwith10ls » Sat Mar 09, 2019 7:11 pm

So, in 4 dimensions, you can move in 8 directions, forwards and backwards along each axis. The hyperbolic tiling functions kinda like that, you can move in 8 directions, and they pair up into "axes".
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Re: Hyperbolic crystals

Postby Klitzing » Sun Mar 10, 2019 8:27 am

Thanx, got it.
So, in 4D all axes are completely perpendicular. Whereas in the {8,4} you have additional angular relations, which only come in via that "projection". :\
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