{3,3,inf}

Tesselation {3, 3, inf}

This is pushing the program and the output has a lot of weird artifacts. Doesn't look nice so I didn't upload it until it was mentioned in one of the comments. The blue edges aren't really there anymore, but the program draws a small neighborhood around them, so they are actually visible.

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Science & Technology

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• 33inf

[mgoerner]

Good observation, tamfang. If you think of the hyperboloid model in Minkowski space, then two lines in a hyperbolic plane that neither intersect nor meet in an ideal point can still intersect outside of hyperbolic space. The "edges" of {3,3,inf} are living outside of hyperbolic space and my program just thinkened them to get visible in hyperbolic space.

Admittedly, it is a bit hacky. Thanks for pointing it out.

[mgoerner]

For x>6, T is not finite-volume anymore, instead of the vertex v, we see that T touches the boundary of the Poincare ball in a triangle. One edge starting from v will be pushed closer and closer to the boundary of the Poincare ball. At x=inf,this edge will disappear.

[mgoerner]

Look at the limit process {3,3,x}. At x=3,4,5 the tesselation is spherical and T is just a compact simplex. {3,3,6} is a hyperbolic tesselation, T is non-compact but finite volume,i.e. "one vertex" v is at the boundary of the Poincare ball.

[mgoerner]

In the following T is the simplex generating the reflection group (see Coxeter group, connection with reflection groups on wikipedia). [In my videos, T is spanned by the center of the red structure, the center of a polygon formed by the blue edges, the middle of a blue edge and the center of the green structure.]

[tamfang]

The cell of {3,3,∞} should be a tetrahedron whose faces are pairwise asymptotic, that is, each pair of faces lies in a pair of asymptotic planes, meaning they meet in one ideal point. Thus the "edges" seen here must be something other than edges of the tetrahedra; what are they?