In the geometry of hyperbolic 3-space, the order-5-3 square honeycomb or 4,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
Geometry.
The Schläfli symbol of the "order-5-3 square honeycomb" is {4,5,3}, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
Related polytopes and honeycombs.
It is a part of a series of regular polytopes and honeycombs with {"p",5,3} Schläfli symbol, and dodecahedral vertex figures:
Order-5-3 pentagonal honeycomb.
In the geometry of hyperbolic 3-space, the order-5-3 pentagonal honeycomb or 5,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 pentagonal honeycomb" is {5,5,3}, with three "order-5 pentagonal tilings" meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
Order-5-3 hexagonal honeycomb.
In the geometry of hyperbolic 3-space, the order-5-3 hexagonal honeycomb or 6,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 hexagonal honeycomb" is {6,5,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
Order-5-3 heptagonal honeycomb.
In the geometry of hyperbolic 3-space, the order-5-3 heptagonal honeycomb or 7,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 heptagonal honeycomb" is {7,5,3}, with three order-5 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
Order-5-3 octagonal honeycomb.
In the geometry of hyperbolic 3-space, the order-5-3 octagonal honeycomb or 8,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 octagonal honeycomb" is {8,5,3}, with three order-5 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
Order-5-3 apeirogonal honeycomb.
In the geometry of hyperbolic 3-space, the order-5-3 apeirogonal honeycomb or ∞,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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 {∞,5,3}, with three "order-5 apeirogonal tilings" meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
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.
References.
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