Isometries of hyperbolic 3-space Here is the first of a set of animated gifs showing basic isometries of hyperbolic 3-space. This one displays a loxodromic transformation. I'll post one gif a day for a bit, but the shader rendering them all is ready for your tinkering! https://www.shadertoy.com/view/MstcWr Update based on comments I should have said a little about the banana shape. It is explained some in the shader comments, which I'll copy here. Also, see the comment thread below! This shader shows basic isometries (length preserving transformations) of hyperbolic 3-space in the upper half space model. The z=0 plane is the boundary plane-at-infinity. There are four classes of transformations: parabolic, elliptic, hyperbolic, and loxodromic. These may fix 1 or 2 ideal points on the boundary plane. In general, any Mobius transformation applied to the boundary plane will extend to an isometry of hyperbolic 3-space, but all can be built by composition of the basic transformati...
I found it rather hard to predict how the topology of the board affects play. Hexagonal boards in the form of an 10*10 rhombus were interesting, some peices proved more or less useful, and needed some work to make them function.
ReplyDeleteWhat about hyperbolic go?
ReplyDeleteYou'd probably have to play go in the style of reversi, where you can only build onto an existing structure.
ReplyDeleteEduard Baumann, check out Jenn3D for Go played on spherical polytopes! See the very bottom of this page:
ReplyDeletehttp://www.math.cmu.edu/~fho/jenn/
wendy krieger, I believe you that it's difficult, based on my experience with my "global chess" board (spherical chess).
ReplyDeletehttp://www.pa-network.com/global-chess/indexf.html
That's quite hard to play, and I'm sure some of the hyperbolic topologies have to be even more mind-bending.