In the geometry of hyperbolic 3-space, the order-infinite-3 triangular honeycomb (or 3,∞,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,∞,3}.
Geometry
It has three Infinite-order triangular tiling { 3,∞} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-3 apeirogonal tiling vertex figure.
Related polytopes and honeycombs
It is a part of a sequence of regular honeycombs with Infinite-order triangular tiling cells: { 3,∞,p}.
It is a part of a sequence of regular honeycombs with order-3 apeirogonal tiling vertex figures: { p,∞,3}.
It is a part of a sequence of self-dual regular honeycombs: { p,∞,p}.
Order-infinite-4 triangular honeycomb
In the geometry of hyperbolic 3-space, the order-infinite-4 triangular honeycomb (or 3,∞,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,∞,4}.
It has four infinite-order triangular tilings, { 3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-4 apeirogonal tiling vertex figure.
It has a second construction as a uniform honeycomb, Schläfli symbol { 3,∞1,1}, Coxeter diagram, , with alternating types or colors of infinite-order triangular tiling cells. In Coxeter notation the half symmetry is [3,∞,4,1+] = [3,∞1,1].
Order-infinite-5 triangular honeycomb
In the geometry of hyperbolic 3-space, the order-infinite-3 triangular honeycomb (or 3,∞,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,∞,5}. It has five infinite-order triangular tiling, { 3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-5 apeirogonal tiling vertex figure.
Order-infinite-6 triangular honeycomb
In the geometry of hyperbolic 3-space, the order-infinite-6 triangular honeycomb (or 3,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,∞,6}. It has infinitely many infinite-order triangular tiling, { 3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-6 apeirogonal tiling, { ∞,6}, vertex figure.
Order-infinite-7 triangular honeycomb
In the geometry of hyperbolic 3-space, the order-infinite-7 triangular honeycomb (or 3,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,∞,7}. It has infinitely many infinite-order triangular tiling, { 3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-7 apeirogonal tiling, { ∞,7}, vertex figure.
Order-infinite-infinite triangular honeycomb
In the geometry of hyperbolic 3-space, the order-infinite-infinite triangular honeycomb (or 3,∞,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,∞,∞}. It has infinitely many infinite-order triangular tiling, { 3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an infinite-order apeirogonal tiling, { ∞,∞}, vertex figure.
It has a second construction as a uniform honeycomb, Schläfli symbol { 3,(∞,∞,∞)}, Coxeter diagram, = , with alternating types or colors of infinite-order triangular tiling cells. In Coxeter notation the half symmetry is [3,∞,∞,1+] = [3,((∞,∞,∞))].
Order-infinite-3 square honeycomb
In the geometry of hyperbolic 3-space, the order-infinite-3 square honeycomb (or 4,∞,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a 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-infinite-3 square honeycomb is { 4,∞,3}, with three infinite-order square tilings meeting at each edge. The vertex figure of this honeycomb is an order-3 apeirogonal tiling, { ∞,3}.