In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a hexagonal 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-6-3 square honeycomb" is {4,6,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
Related polytopes and honeycombs.
It is a part of a series of regular polytopes and honeycombs with {"p",6,3} Schläfli symbol, and dodecahedral vertex figures:
Order-6-3 pentagonal honeycomb.
In the geometry of hyperbolic 3-space, the order-6-3 pentagonal honeycomb or 5,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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 {5,6,3}, with three "order-6 pentagonal tilings" meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
Order-6-3 hexagonal honeycomb.
In the geometry of hyperbolic 3-space, the order-6-3 hexagonal honeycomb or 6,6,3 honeycomb is 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-6-3 hexagonal honeycomb" is {6,6,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
Order-6-3 apeirogonal honeycomb.
In the geometry of hyperbolic 3-space, the order-6-3 apeirogonal honeycomb or ∞,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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 {∞,6,3}, with three "order-6 apeirogonal tilings" meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
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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