Langbahn Team – Weltmeisterschaft

Order-3-7 heptagonal honeycomb

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

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

Geometry

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


Poincaré disk model
Ideal surface

It a part of a sequence of regular polychora and honeycombs {p,3,p}:

{p,3,p} regular honeycombs
Space S3 Euclidean E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,3} {4,3,4} {5,3,5} {6,3,6} {7,3,7} {8,3,8} ...{∞,3,∞}
Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}

Order-3-8 octagonal honeycomb

Order-3-8 octagonal honeycomb
Type Regular honeycomb
Schläfli symbols {8,3,8}
{8,(3,4,3)}
Coxeter diagrams
=
Cells {8,3}
Faces {8}
Edge figure {8}
Vertex figure {3,8}
{(3,8,3)}
Dual self-dual
Coxeter group [8,3,8]
[8,((3,4,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-3-8 octagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {8,3,8}. It has eight octagonal tilings, {8,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octagonal tilings existing around each vertex in an order-8 triangular tiling vertex arrangement.


Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {8,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1+] = [8,((3,4,3))].

Order-3-infinite apeirogonal honeycomb

Order-3-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,3,∞}
{∞,(3,∞,3)}
Coxeter diagrams
↔
Cells {∞,3}
Faces {∞}
Edge figure {∞}
Vertex figure {3,∞}
{(3,∞,3)}
Dual self-dual
Coxeter group [∞,3,∞]
[∞,((3,∞,3))]
Properties Regular

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


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of apeirogonal tiling cells.

See also

References

Quelle: https://en.wikipedia.org/wiki/Order-3-infinite_apeirogonal_honeycomb