Order-7 tetrahedral honeycomb | |
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Type | Hyperbolic regular honeycomb |
Schläfli symbols | {3,3,7} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,3}
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Faces | {3} |
Edge figure | {7} |
Vertex figure |
{3,7}
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Dual | {7,3,3} |
Coxeter group | [7,3,3] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.
![]() Poincaré disk model (cell-centered) |
![]() Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model |
It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, {3,3,p}.
{3,3,p} polytopes | |||||||||||
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Space | S3 | H3 | |||||||||
Form | Finite | Paracompact | Noncompact | ||||||||
Name |
{3,3,3}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,5}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,6}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,7}![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,8}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
...
{3,3,∞}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
Image |
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Vertex figure |
![]() {3,3} ![]() ![]() ![]() ![]() ![]() |
![]() {3,4} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {3,5} ![]() ![]() ![]() ![]() ![]() |
![]() {3,6} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {3,7} ![]() ![]() ![]() ![]() ![]() |
![]() {3,8} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {3,∞} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}.
{3,3,7} | {4,3,7} | {5,3,7} | {6,3,7} | {7,3,7} | {8,3,7} | {∞,3,7} |
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It is a part of a sequence of hyperbolic honeycombs, {3,p,7}.
Order-8 tetrahedral honeycomb | |
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Type | Hyperbolic regular honeycomb |
Schläfli symbols | {3,3,8} {3,(3,4,3)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,3}
![]() |
Faces | {3} |
Edge figure | {8} |
Vertex figure |
{3,8}
![]() {(3,4,3)} ![]() |
Dual | {8,3,3} |
Coxeter group | [3,3,8] [3,((3,4,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,8}. It has eight tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.
![]() Poincaré disk model (cell-centered) |
![]() Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model |
It has a second construction as a uniform honeycomb,
Schläfli symbol {3,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In
Coxeter notation the half symmetry is [3,3,8,1+] = [3,((3,4,3))].
Infinite-order tetrahedral honeycomb | |
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Type | Hyperbolic regular honeycomb |
Schläfli symbols | {3,3,∞} {3,(3,∞,3)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,3}
![]() |
Faces | {3} |
Edge figure | {∞} |
Vertex figure |
{3,∞}
![]() {(3,∞,3)} ![]() |
Dual | {∞,3,3} |
Coxeter group | [∞,3,3] [3,((3,∞,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,∞}. It has infinitely many tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
![]() Poincaré disk model (cell-centered) |
![]() Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model |
It has a second construction as a uniform honeycomb,
Schläfli symbol {3,(3,∞,3)}, Coxeter diagram, =
, with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,∞,1+] = [3,((3,∞,3))].