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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-8-3 triangular honeycomb''' (or '''3,8,3 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,8,3}.
== Geometry==
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=== Order-8-4 triangular honeycomb===
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-8-4 triangular honeycomb''' (or '''3,8,4 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,8,4}.
It has four [[order-8 triangular tiling]]s, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an [[order-4 hexagonal tiling]] [[vertex arrangement]].
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=== Order-8-5 triangular honeycomb===
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=== Order-8-3 square honeycomb===
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-8-3 square honeycomb''' (or '''4,8,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of a [[octagonal tiling]] whose vertices lie on a [[Hypercycle (geometry)|2-hypercycle]], each of which has a limiting circle on the ideal sphere.
The [[Schläfli symbol]] of the ''order-8-3 square honeycomb'' is {4,8,3}, with three order-4 octagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-8-3 pentagonal honeycomb''' (or '''5,8,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of an [[order-8 pentagonal tiling]] whose vertices lie on a [[Hypercycle (geometry)|2-hypercycle]], each of which has a limiting circle on the ideal sphere.
The [[Schläfli symbol]] of the ''order-6-3 pentagonal honeycomb'' is {5,8,3}, with three ''order-8 pentagonal tilings'' meeting at each edge. The [[vertex figure]] of this honeycomb is
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-8-3 hexagonal honeycomb''' (or '''6,8,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of a [[order-6 hexagonal tiling]] whose vertices lie on a [[Hypercycle (geometry)|2-hypercycle]], each of which has a limiting circle on the ideal sphere.
The [[Schläfli symbol]] of the ''order-8-3 hexagonal honeycomb'' is {6,8,3}, with three order-5 hexagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is
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|bgcolor=#e8dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
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|bgcolor=#e8dcc3|[[Schläfli symbol]]||{
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|bgcolor=#e8dcc3|[[Coxeter diagram]]||{{CDD|node_1|infin|node|8|node|3|node}}
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|bgcolor=#e8dcc3|Cells||[[Order-8 apeirogonal tiling|{∞,8}]] [[File:H2_tiling_28i-1.png|80px]]
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|bgcolor=#e8dcc3|Faces||[[Apeirogon]] {
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|bgcolor=#e8dcc3|[[Vertex figure]]||[[octagonal tiling|{8,3}]]
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|bgcolor=#e8dcc3|Dual||[[Order-8-infinite triangular honeycomb|{3,8,∞}]]
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|bgcolor=#e8dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[
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|bgcolor=#e8dcc3|Properties||Regular
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-8-3 apeirogonal honeycomb''' (or '''
The [[Schläfli symbol]] of the apeirogonal tiling honeycomb is {
The "ideal surface" projection below is a plane-at-infinity, in the Poincare half-space model of H3. It shows a [[Apollonian gasket]] pattern of circles inside a largest circle.
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-8-infinite apeirogonal honeycomb''' (or '''
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* [https://www.youtube.com/watch?v=GRo_FQm2KRc Hyperbolic Catacombs Carousel: {3,7,3} honeycomb] [[YouTube]], Roice Nelson
*[[John Baez]], ''Visual insights'': [http://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/ {7,3,3} Honeycomb] (2014/08/01) [http://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/ {7,3,3} Honeycomb Meets Plane at Infinity] (2014/08/14)
* [[Danny Calegari]], [http://lamington.wordpress.com/2014/03/04/kleinian-a-tool-for-visualizing-kleinian-groups/Kleinian Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination] 4 March 2014. [http://math.uchicago.edu/~dannyc/papers/kleinian_mtf_Feb_2014.pdf]
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