In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb (or 3,7,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,7,3}.
Geometry
It has three order-7 triangular tiling { 3,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in a heptagonal tiling vertex figure.
Related polytopes and honeycombs
It a part of a sequence of self-dual regular honeycombs: { p,7,p}.
It is a part of a sequence of regular honeycombs with order-7 triangular tiling cells: { 3,7,p}.
It isa part of a sequence of regular honeycombs with heptagonal tiling vertex figures: { p,7,3}.
Order-7-4 triangular honeycomb
In the geometry of hyperbolic 3-space, the order-7-4 triangular honeycomb (or 3,7,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,7,4}.
It has four order-7 triangular tilings, { 3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol { 3,71,1}, Coxeter diagram, , with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is [3,7,4,1+] = [3,71,1].
Order-7-5 triangular honeycomb
In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb (or 3,7,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,7,5}. It has five order-7 triangular tiling, { 3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-5 heptagonal tiling vertex figure.
Order-7-6 triangular honeycomb
In the geometry of hyperbolic 3-space, the order-7-6 triangular honeycomb (or 3,7,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,7,6}. It has infinitely many order-7 triangular tiling, { 3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-6 heptagonal tiling, { 7,6}, vertex figure.
Order-7-infinite triangular honeycomb
In the geometry of hyperbolic 3-space, the order-7-infinite triangular honeycomb (or 3,7,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 3,7,∞}. It has infinitely many order-7 triangular tiling, { 3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an infinite-order heptagonal tiling, { 7,∞}, vertex figure.
It has a second construction as a uniform honeycomb, Schläfli symbol { 3,(7,∞,7)}, Coxeter diagram, = , with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is [3,7,∞,1+] = [3,((7,∞,7))].
Order-7-3 square honeycomb
In the geometry of hyperbolic 3-space, the order-7-3 square honeycomb (or 4,7,3 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-7-3 square honeycomb is { 4,7,3}, with three order-4 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, { 7,3}.
Order-7-3 pentagonal honeycomb
In the geometry of hyperbolic 3-space, the order-7-3 pentagonal honeycomb (or 5,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-7 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 { 5,7,3}, with three order-7 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, { 7,3}.
Order-7-3 hexagonal honeycomb
In the geometry of hyperbolic 3-space, the order-7-3 hexagonal honeycomb (or 6,7,3 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.