Order-infinite-3 triangular honeycomb: Difference between revisions

Revision as of 23:52, 6 April 2019

Revision as of 00:39, 8 December 2019

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-3 triangular honeycomb''' (or '''3,∞,3 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,∞,3}.

== Geometry==

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=== Order-infinite-4 triangular honeycomb===

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-4 triangular honeycomb''' (or '''3,∞,4 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,∞,4}.

It has four [[infinite-order triangular tiling]]s, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an [[order-4 apeirogonal tiling]] [[vertex figure]].

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The [[Schläfli symbol]] of the apeirogonal tiling honeycomb is {∞,∞,3}, with three ''infinite-order apeirogonal tilings'' meeting at each edge. The [[vertex figure]] of this honeycomb is an infinite-order apeirogonal tiling, {∞,3}.

The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows aan [[Apollonian gasket]] pattern of circles inside a largest circle.

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