# Order-7-3 triangular honeycomb

Order-7-3 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,3}
Coxeter diagrams
Cells {3,7}
Faces {3}
Edge figure {3}
Vertex figure {7,3}
Dual Self-dual
Coxeter group [3,7,3]
Properties Regular

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.

 Poincaré disk model Ideal surface Upper half space model with selective cells shown[1]

## 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[]

Order-7-4 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,4}
Coxeter diagrams
=
Cells {3,7}
Faces {3}
Edge figure {4}
Vertex figure {7,4}
r{7,7}
Dual {4,7,3}
Coxeter group [3,7,4]
Properties Regular

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.

 Poincaré disk model Ideal surface

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[]

Order-7-5 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,5}
Coxeter diagrams
Cells {3,7}
Faces {3}
Edge figure {5}
Vertex figure {7,5}
Dual {5,7,3}
Coxeter group [3,7,5]
Properties Regular

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.

 Poincaré disk model Ideal surface

### Order-7-6 triangular honeycomb[]

Order-7-6 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,6}
{3,(7,3,7)}
Coxeter diagrams
=
Cells {3,7}
Faces {3}
Edge figure {6}
Vertex figure {7,6}
{(7,3,7)}
Dual {6,7,3}
Coxeter group [3,7,6]
Properties Regular

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.

 Poincaré disk model Ideal surface

### Order-7-infinite triangular honeycomb[]

Order-7-infinite triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,∞}
{3,(7,∞,7)}
Coxeter diagrams
=
Cells {3,7}
Faces {3}
Edge figure {∞}
Vertex figure {7,∞}
{(7,∞,7)}
Dual {∞,7,3}
Coxeter group [∞,7,3]
[3,((7,∞,7))]
Properties Regular

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.

 Poincaré disk model Ideal surface

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[]

Order-7-3 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,7,3}
Coxeter diagram
Cells {4,7}
Faces {4}
Vertex figure {7,3}
Dual {3,7,4}
Coxeter group [4,7,3]
Properties Regular

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}.

 Poincaré disk model Ideal surface

### Order-7-3 pentagonal honeycomb[]

Order-7-3 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,7,3}
Coxeter diagram
Cells {5,7}
Faces {5}
Vertex figure {7,3}
Dual {3,7,5}
Coxeter group [5,7,3]
Properties Regular

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}.

 Poincaré disk model Ideal surface

### Order-7-3 hexagonal honeycomb[]

Order-7-3 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbol {6,7,3}
Coxeter diagram
Cells {6,7}
Faces {6}
Vertex figure {7,3}
Dual {3,7,6}
Coxeter group [6,7,3]
Properties Regular

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.

The Schläfli symbol of the order-7-3 hexagonal honeycomb is {6,7,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.

 Poincaré disk model Ideal surface

### Order-7-3 apeirogonal honeycomb[]

Order-7-3 apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbol {∞,7,3}
Coxeter diagram
Cells {∞,7}
Faces Apeirogon {∞}
Vertex figure {7,3}
Dual {3,7,∞}
Coxeter group [∞,7,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-3 apeirogonal honeycomb (or ∞,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-7 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 {∞,7,3}, with three order-7 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.

The "ideal surface" projection below is a plane-at-infinity, in the Poincare half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.

 Poincaré disk model Ideal surface

### Order-7-4 square honeycomb[]

Order-7-4 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,7,4}
Coxeter diagrams
=
Cells {4,7}
Faces {4}
Edge figure {4}
Vertex figure {7,4}
Dual self-dual
Coxeter group [4,7,4]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-4 square honeycomb (or 4,7,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,7,4}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 heptagonal tiling vertex figure.

 Poincaré disk model Ideal surface

### Order-7-5 pentagonal honeycomb[]

Order-7-5 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,7,5}
Coxeter diagrams
Cells {5,7}
Faces {5}
Edge figure {5}
Vertex figure {7,5}
Dual self-dual
Coxeter group [5,7,5]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-5 pentagonal honeycomb (or 5,7,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,7,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-7 pentagonal tilings existing around each edge and with an order-5 heptagonal tiling vertex figure.

 Poincaré disk model Ideal surface

### Order-7-6 hexagonal honeycomb[]

Order-7-6 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,7,6}
{6,(7,3,7)}
Coxeter diagrams
=
Cells {6,7}
Faces {6}
Edge figure {6}
Vertex figure {7,6}
{(5,3,5)}
Dual self-dual
Coxeter group [6,7,6]
[6,((7,3,7))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-6 hexagonal honeycomb (or 6,7,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,7,6}. It has six order-7 hexagonal tilings, {6,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-6 heptagonal tiling vertex arrangement.

 Poincaré disk model Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(7,3,7)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,7,6,1+] = [6,((7,3,7))].

### Order-7-infinite apeirogonal honeycomb[]

Order-7-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,7,∞}
{∞,(7,∞,7)}
Coxeter diagrams
Cells {∞,7}
Faces {∞}
Edge figure {∞}
Vertex figure {7,∞}
{(7,∞,7)}
Dual self-dual
Coxeter group [∞,7,∞]
[∞,((7,∞,7))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-infinite apeirogonal honeycomb (or ∞,7,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,7,∞}. It has infinitely many order-7 apeirogonal tiling {∞,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 apeirogonal tilings existing around each vertex in an infinite-order heptagonal tiling vertex figure.

 Poincaré disk model Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(7,∞,7)}, Coxeter diagram, , with alternating types or colors of cells.