Order-4-3 pentagonal honeycomb
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In the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell is an order-4 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
Order-4-3 pentagonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {5,4,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {5,4} ![]() |
Faces | {5} |
Vertex figure | {4,3} |
Dual | {3,4,5} |
Coxeter group | [5,4,3] |
Properties | Regular |
Geometry
The Schläfli symbol of the order-4-3 pentagonal honeycomb is {5,4,3}, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.
![]() Poincaré disk model (Vertex centered) |
![]() Ideal surface |
Related polytopes and honeycombs
Summarize
Perspective
It is a part of a series of regular polytopes and honeycombs with {p,4,3} Schläfli symbol, and tetrahedral vertex figures:
Order-4-3 hexagonal honeycomb
Order-4-3 hexagonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {6,4,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {6,4} ![]() |
Faces | {6} |
Vertex figure | {4,3} |
Dual | {3,4,6} |
Coxeter group | [6,4,3] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-3 hexagonal honeycomb or 6,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 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-4-3 hexagonal honeycomb is {6,4,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.
![]() Poincaré disk model (Vertex centered) |
![]() Ideal surface |
Order-4-3 heptagonal honeycomb
Order-4-3 heptagonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {7,4,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {7,4} ![]() |
Faces | {7} |
Vertex figure | {4,3} |
Dual | {3,4,7} |
Coxeter group | [7,4,3] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-3 heptagonal honeycomb or 7,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 heptagonal 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-4-3 heptagonal honeycomb is {7,4,3}, with three order-4 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.
![]() Poincaré disk model (Vertex centered) |
![]() Ideal surface |
Order-4-3 octagonal honeycomb
Order-4-3 octagonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {8,4,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {8,4} ![]() |
Faces | {8} |
Vertex figure | {4,3} |
Dual | {3,4,8} |
Coxeter group | [8,4,3] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-3 octagonal honeycomb or 8,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 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-4-3 octagonal honeycomb is {8,4,3}, with three order-4 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.
![]() Poincaré disk model (Vertex centered) |
Order-4-3 apeirogonal honeycomb
Order-4-3 apeirogonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {∞,4,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {∞,4} ![]() |
Faces | Apeirogon {∞} |
Vertex figure | {4,3} |
Dual | {3,4,∞} |
Coxeter group | [∞,4,3] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-3 apeirogonal honeycomb or ∞,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 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 {∞,4,3}, with three apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.
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 (Vertex centered) |
![]() Ideal surface |
See also
References
External links
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