Order-7-3 triangular honeycomb

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}.

Order-7-3 triangular honeycomb
TypeRegular honeycomb
Schläfli symbols{3,7,3}
Coxeter diagrams
Cells{3,7}
Faces{3}
Edge figure{3}
Vertex figure{7,3}
DualSelf-dual
Coxeter group[3,7,3]
PropertiesRegular

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]

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
TypeRegular 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]
PropertiesRegular

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
TypeRegular 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]
PropertiesRegular

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
TypeRegular 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]
PropertiesRegular

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
TypeRegular 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))]
PropertiesRegular

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
TypeRegular 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]
PropertiesRegular

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
TypeRegular 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]
PropertiesRegular

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
TypeRegular 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]
PropertiesRegular

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
TypeRegular honeycomb
Schläfli symbol{∞,7,3}
Coxeter diagram
Cells{,7}
FacesApeirogon {∞}
Vertex figure{7,3}
Dual{3,7,}
Coxeter group[∞,7,3]
PropertiesRegular

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 Poincaré 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
TypeRegular honeycomb
Schläfli symbol{4,7,4}
Coxeter diagrams
=
Cells{4,7}
Faces{4}
Edge figure{4}
Vertex figure{7,4}
Dualself-dual
Coxeter group[4,7,4]
PropertiesRegular

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
TypeRegular honeycomb
Schläfli symbol{5,7,5}
Coxeter diagrams
Cells{5,7}
Faces{5}
Edge figure{5}
Vertex figure{7,5}
Dualself-dual
Coxeter group[5,7,5]
PropertiesRegular

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
TypeRegular 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)}
Dualself-dual
Coxeter group[6,7,6]
[6,((7,3,7))]
PropertiesRegular

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
TypeRegular honeycomb
Schläfli symbols{∞,7,∞}
{∞,(7,∞,7)}
Coxeter diagrams
Cells{,7}
Faces{∞}
Edge figure{∞}
Vertex figure {7,∞}
{(7,∞,7)}
Dualself-dual
Coxeter group[∞,7,∞]
[∞,((7,∞,7))]
PropertiesRegular

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.

See also

References

  1. Hyperbolic Catacombs Roice Nelson and Henry Segerman, 2014
  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II)
  • George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982)
  • Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
  • Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
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