Order-4 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {4} |
Vertex figure |
{5,4}
![]() |
Dual | {4,5,3} |
Coxeter group | [3,5,4] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,4}.
It has four icosahedra {3,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-4 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {3,51,1}, Coxeter diagram, , with alternating types or colors of icosahedral cells. In
Coxeter notation the half symmetry is [3,5,4,1+] = [3,51,1].
It a part of a sequence of regular polychora and honeycombs with icosahedral cells: {3,5,p}
{3,5,p} polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | H3 | ||||||||||
Form | Compact | Noncompact | |||||||||
Name |
{3,5,3}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,6}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,7}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,8}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
...
{3,5,∞}![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
Image |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | ||||
Vertex figure |
![]() {5,3} ![]() ![]() ![]() ![]() ![]() |
![]() {5,4} ![]() ![]() ![]() ![]() ![]() |
![]() {5,5} ![]() ![]() ![]() ![]() ![]() |
![]() {5,6} ![]() ![]() ![]() ![]() ![]() |
![]() {5,7} ![]() ![]() ![]() ![]() ![]() |
![]() {5,8} ![]() ![]() ![]() ![]() ![]() |
![]() {5,∞} ![]() ![]() ![]() ![]() ![]() |
Order-5 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,5} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {5} |
Vertex figure |
{5,5}
![]() |
Dual | {5,5,3} |
Coxeter group | [3,5,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,5}. It has five icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-5 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
Order-6 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,6} {3,(5,∞,5)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {6} |
Vertex figure |
{5,6}
![]() |
Dual | {6,5,3} |
Coxeter group | [3,5,6] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,6}. It has six icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-6 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
Order-7 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,7} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {7} |
Vertex figure |
{5,7}
![]() |
Dual | {7,5,3} |
Coxeter group | [3,5,7] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,7}. It has seven icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-7 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
Order-8 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,8} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {8} |
Vertex figure |
{5,8}
![]() |
Dual | {8,5,3} |
Coxeter group | [3,5,8] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,8}. It has eight icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-8 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
Infinite-order icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,∞} {3,(5,∞,5)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {∞} |
Vertex figure |
{5,∞}
![]() {(5,∞,5)} ![]() |
Dual | {∞,5,3} |
Coxeter group | [∞,5,3] [3,((5,∞,5))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,∞}. It has infinitely many icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {3,(5,∞,5)}, Coxeter diagram, =
, with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,∞,1+] = [3,((5,∞,5))].
Order-4 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {4} |
Vertex figure |
{5,4}
![]() |
Dual | {4,5,3} |
Coxeter group | [3,5,4] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,4}.
It has four icosahedra {3,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-4 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {3,51,1}, Coxeter diagram, , with alternating types or colors of icosahedral cells. In
Coxeter notation the half symmetry is [3,5,4,1+] = [3,51,1].
It a part of a sequence of regular polychora and honeycombs with icosahedral cells: {3,5,p}
{3,5,p} polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | H3 | ||||||||||
Form | Compact | Noncompact | |||||||||
Name |
{3,5,3}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,6}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,7}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,5,8}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
...
{3,5,∞}![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
Image |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | ||||
Vertex figure |
![]() {5,3} ![]() ![]() ![]() ![]() ![]() |
![]() {5,4} ![]() ![]() ![]() ![]() ![]() |
![]() {5,5} ![]() ![]() ![]() ![]() ![]() |
![]() {5,6} ![]() ![]() ![]() ![]() ![]() |
![]() {5,7} ![]() ![]() ![]() ![]() ![]() |
![]() {5,8} ![]() ![]() ![]() ![]() ![]() |
![]() {5,∞} ![]() ![]() ![]() ![]() ![]() |
Order-5 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,5} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {5} |
Vertex figure |
{5,5}
![]() |
Dual | {5,5,3} |
Coxeter group | [3,5,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,5}. It has five icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-5 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
Order-6 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,6} {3,(5,∞,5)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {6} |
Vertex figure |
{5,6}
![]() |
Dual | {6,5,3} |
Coxeter group | [3,5,6] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,6}. It has six icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-6 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
Order-7 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,7} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {7} |
Vertex figure |
{5,7}
![]() |
Dual | {7,5,3} |
Coxeter group | [3,5,7] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,7}. It has seven icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-7 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
Order-8 icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,8} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {8} |
Vertex figure |
{5,8}
![]() |
Dual | {8,5,3} |
Coxeter group | [3,5,8] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,8}. It has eight icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-8 pentagonal tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
Infinite-order icosahedral honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {3,5,∞} {3,(5,∞,5)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,5}
![]() |
Faces | {3} |
Edge figure | {∞} |
Vertex figure |
{5,∞}
![]() {(5,∞,5)} ![]() |
Dual | {∞,5,3} |
Coxeter group | [∞,5,3] [3,((5,∞,5))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,∞}. It has infinitely many icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
![]() Poincaré disk model (Cell centered) |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {3,(5,∞,5)}, Coxeter diagram, =
, with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,∞,1+] = [3,((5,∞,5))].