Order-8-3 triangular honeycomb

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,3}.

Order-8-3 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,3}
Coxeter diagrams
Cells	{3,8}
Faces	{3}
Edge figure	{3}
Vertex figure	{8,3}
Dual	Self-dual
Coxeter group	[3,8,3]
Properties	Regular

Geometry

It has three order-8 triangular tiling {3,8} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an octagonal tiling vertex figure.

Poincaré disk model

Related polytopes and honeycombs

It is a part of a sequence of regular honeycombs with order-8 triangular tiling cells: {3,8,p}.

It is a part of a sequence of regular honeycombs with octagonal tiling vertex figures: {p,8,3}.

It is a part of a sequence of self-dual regular honeycombs: {p,8,p}.

Order-8-4 triangular honeycomb

Order-8-4 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,4}
Coxeter diagrams	=
Cells	{3,8}
Faces	{3}
Edge figure	{4}
Vertex figure	{8,4} r{8,8}
Dual	{4,8,3}
Coxeter group	[3,8,4]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-4 triangular honeycomb (or 3,8,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,4}.

It has four order-8 triangular tilings, {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.

Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {3,8^1,1}, Coxeter diagram, , with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,4,1⁺] = [3,8^1,1].

Order-8-5 triangular honeycomb

Order-8-5 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,5}
Coxeter diagrams
Cells	{3,8}
Faces	{3}
Edge figure	{5}
Vertex figure	{8,5}
Dual	{5,8,3}
Coxeter group	[3,8,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,5}. It has five order-8 triangular tiling, {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-5 octagonal tiling vertex figure.

Poincaré disk model

Order-8-6 triangular honeycomb

Order-8-6 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,6} {3,(8,3,8)}
Coxeter diagrams	=
Cells	{3,8}
Faces	{3}
Edge figure	{6}
Vertex figure	{8,6} {(8,3,8)}
Dual	{6,8,3}
Coxeter group	[3,8,6]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-6 triangular honeycomb (or 3,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,6}. It has infinitely many order-8 triangular tiling, {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-6 octagonal tiling, {8,6}, vertex figure.

Poincaré disk model

Order-8-infinite triangular honeycomb

Order-8-infinite triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,∞} {3,(8,∞,8)}
Coxeter diagrams	=
Cells	{3,8}
Faces	{3}
Edge figure	{∞}
Vertex figure	{8,∞} {(8,∞,8)}
Dual	{∞,8,3}
Coxeter group	[∞,8,3] [3,((8,∞,8))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-infinite triangular honeycomb (or 3,8,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,∞}. It has infinitely many order-8 triangular tiling, {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 infinite-order octagonal tiling, {8,∞}, vertex figure.

Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(8,∞,8)}, Coxeter diagram, = , with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,∞,1⁺] = [3,((8,∞,8))].

Order-8-3 square honeycomb

Order-8-3 square honeycomb
Type	Regular honeycomb
Schläfli symbol	{4,8,3}
Coxeter diagram
Cells	{4,8}
Faces	{4}
Vertex figure	{8,3}
Dual	{3,8,4}
Coxeter group	[4,8,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-3 square honeycomb (or 4,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an octagonal tiling whose vertices lie on a 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 an octagonal tiling, {8,3}.

Poincaré disk model

Order-8-3 pentagonal honeycomb

Order-8-3 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{5,8,3}
Coxeter diagram
Cells	{5,8}
Faces	{5}
Vertex figure	{8,3}
Dual	{3,8,5}
Coxeter group	[5,8,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-3 pentagonal honeycomb (or 5,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-8 pentagonal tiling whose vertices lie on a 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 an octagonal tiling, {8,3}.

Poincaré disk model

Order-8-3 hexagonal honeycomb

Order-8-3 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{6,8,3}
Coxeter diagram
Cells	{6,8}
Faces	{6}
Vertex figure	{8,3}
Dual	{3,8,6}
Coxeter group	[6,8,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-3 hexagonal honeycomb (or 6,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 hexagonal tiling whose vertices lie on a 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 an octagonal tiling, {8,3}.

Poincaré disk model

Order-8-3 apeirogonal honeycomb

Order-8-3 apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{∞,8,3}
Coxeter diagram
Cells	{∞,8}
Faces	Apeirogon {∞}
Vertex figure	{8,3}
Dual	{3,8,∞}
Coxeter group	[∞,8,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-3 apeirogonal honeycomb (or ∞,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-8 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,8,3}, with three order-8 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.

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

Poincaré disk model

Order-8-4 square honeycomb

Order-8-4 square honeycomb
Type	Regular honeycomb
Schläfli symbol	{4,8,4}
Coxeter diagrams	=
Cells	{4,8}
Faces	{4}
Edge figure	{4}
Vertex figure	{8,4}
Dual	self-dual
Coxeter group	[4,8,4]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-4 square honeycomb (or 4,8,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,8,4}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 octagonal tiling vertex figure.

Poincaré disk model

Order-8-5 pentagonal honeycomb

Order-8-5 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{5,8,5}
Coxeter diagrams
Cells	{5,8}
Faces	{5}
Edge figure	{5}
Vertex figure	{8,5}
Dual	self-dual
Coxeter group	[5,8,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-5 pentagonal honeycomb (or 5,8,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,8,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-8 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.

Poincaré disk model

Order-8-6 hexagonal honeycomb

Order-8-6 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{6,8,6} {6,(8,3,8)}
Coxeter diagrams	=
Cells	{6,8}
Faces	{6}
Edge figure	{6}
Vertex figure	{8,6} {(5,3,5)}
Dual	self-dual
Coxeter group	[6,8,6] [6,((8,3,8))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-6 hexagonal honeycomb (or 6,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,8,6}. It has six order-8 hexagonal tilings, {6,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-6 octagonal tiling vertex arrangement.

Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(8,3,8)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,8,6,1⁺] = [6,((8,3,8))].

Order-8-infinite apeirogonal honeycomb

Order-8-infinite apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{∞,8,∞} {∞,(8,∞,8)}
Coxeter diagrams	↔
Cells	{∞,8}
Faces	{∞}
Edge figure	{∞}
Vertex figure	{8,∞} {(8,∞,8)}
Dual	self-dual
Coxeter group	[∞,8,∞] [∞,((8,∞,8))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-infinite apeirogonal honeycomb (or ∞,8,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,8,∞}. It has infinitely many order-8 apeirogonal tiling {∞,8} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 apeirogonal tilings existing around each vertex in an infinite-order octagonal tiling vertex figure.

Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(8,∞,8)}, Coxeter diagram, , with alternating types or colors of cells.

gollark: Also, this function seems to have no valid reason to exist.

gollark: > def indIncreaseCounter(tickInstance):Python convention is to use `snake_case`, not `camelCase`.

gollark: Just looking at this file here: https://github.com/mHappah3019/Tick-Counter/blob/main/TickClass.py> # creates an attribute called identifier and assigns to it> # the value of the "identifier" parameter> # creates an attribute called macro and assigns to it the> # value of the "macro" parameterThese comments are not useful. It is generally assumed that whoever is reading your code is aware of the basics of how the language is used, so your comments should instead describe higher-level stuff like *why* it's doing what it does, what an entire function does, unusual things it might be doing, etc.

gollark: Increasing the version number would imply that it's actually a significant change.

gollark: Not really, it's a minor UI reskin.

References

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II)
George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982)
Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)

External links

Hyperbolic Catacombs Carousel: {3,7,3} honeycomb YouTube, Roice Nelson
John Baez, Visual insights: {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14)
Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014.

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