Order-3-5 heptagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {7,3,5} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{7,3}
![]() |
Faces | Heptagon {7} |
Vertex figure | icosahedron {3,5} |
Dual | {5,3,7} |
Coxeter group | [7,3,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-5 heptagonal 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-3-5 heptagonal honeycomb is {7,3,5}, with five heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, {3,5}.
![]() Poincaré disk model (vertex centered) |
![]() Ideal surface |
It is a part of a series of regular polytopes and honeycombs with {p,3,5} Schläfli symbol, and icosahedral vertex figures.
{p,3,5} polytopes | |||||||
---|---|---|---|---|---|---|---|
Space | S3 | H3 | |||||
Form | Finite | Compact | Paracompact | Noncompact | |||
Name |
{3,3,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{4,3,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{5,3,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{6,3,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{7,3,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{8,3,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
...
{∞,3,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Image |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Cells |
![]() {3,3} ![]() ![]() ![]() ![]() ![]() |
![]() {4,3} ![]() ![]() ![]() ![]() ![]() |
![]() {5,3} ![]() ![]() ![]() ![]() ![]() |
![]() {6,3} ![]() ![]() ![]() ![]() ![]() |
![]() {7,3} ![]() ![]() ![]() ![]() ![]() |
![]() {8,3} ![]() ![]() ![]() ![]() ![]() |
![]() {∞,3} ![]() ![]() ![]() ![]() ![]() |
Order-3-5 octagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {8,3,5} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{8,3}
![]() |
Faces | Octagon {8} |
Vertex figure | icosahedron {3,5} |
Dual | {5,3,8} |
Coxeter group | [8,3,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-5 octagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 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 3-5 heptagonal honeycomb is {8,3,5}, with five octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, {3,5}.
![]() Poincaré disk model (vertex centered) |
Order-3-5 apeirogonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {∞,3,5} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{∞,3}
![]() |
Faces | Apeirogon {∞} |
Vertex figure | icosahedron {3,5} |
Dual | {5,3,∞} |
Coxeter group | [∞,3,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-5 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-3 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 order-3-5 apeirogonal honeycomb is {∞,3,5}, with five order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, {3,5}.
![]() Poincaré disk model (vertex centered) |
![]() Ideal surface |