Talk:Order-7 tetrahedral honeycomb

"ultra-ideal"

I cannot seem to find a definition for what this means, precisely. That each tetrahedron's edges diverge instead of converging towards a vertex? --Trɔpʏliʊm • blah 00:57, 28 October 2016 (UTC)

I think so. It diverges in hyperbolic space, but converges to vertices seen in the projection outside of the ideal sphere. Tom Ruen (talk) 00:58, 28 October 2016 (UTC)

On some thinking, I guess divergence is implied by {7,3,3} being its dual. The component tilings of that tiling exist on hypercycles, and instead of a center that could correspond to a dual's vertex (an actual point for cells inscribable in a sphere, a point at infinity for cells inscribable in a horocycle), they have a guiding plane. If we drew both tessellations inside the same space, what we'd end up would seem to be that tetrahedron edges entering a particular hypercycle then end up parallel to this guiding plane.

This doesn't seem to lead to edges that "converse outside the ideal sphere", though. The Poincaré model maps lines to circles that are orthogonal to the ideal sphere, and no two such circles can intersect outside the ideal sphere.

…but of course, in the Beltrami-Klein model, the edges are indeed lines and they will be intersecting outside the ideal circle. --Trɔpʏliʊm • blah 12:00, 28 March 2018 (UTC)

And, of course, this would not be a tiling with tetrahedra in the sense of finite regions of space, it'd be a tiling with infinite sections of space that are bounded by six "radiation symbols" formed by three mutually equidistant perpendicular lines. (I do not think such figures are considered "hyperbolic triangles"?)

There is also an infinite amount of space left untiled, really: every "tetrahedron" (they will look more like Cayley's nodal cubic surface with the nodes smoothed into hyperboloids) has room in each of its four "ultra-ideal vertices" it for another copy of the entire "honeycomb" all over again, mirrored with respect to the hyperplane that guides the inscribing hypercycles of its dual. (And then every new "tetrahedron" in these has room for another copy, again, ad infinitum…) --Trɔpʏliʊm • blah 02:26, 28 October 2016 (UTC)