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

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

Geometry

It has three Infinite-order triangular tiling {3,∞} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-3 apeirogonal tiling vertex figure.


Poincaré disk model

Ideal surface

It is a part of a sequence of regular honeycombs with Infinite-order triangular tiling cells: {3,∞,p}.

It is a part of a sequence of regular honeycombs with order-3 apeirogonal tiling vertex figures: {p,∞,3}.

It is a part of a sequence of self-dual regular honeycombs: {p,∞,p}.

Order-infinite-4 triangular honeycomb

Order-infinite-4 triangular honeycomb
TypeRegular honeycomb
Schläfli symbols{3,∞,4}
{3,∞1,1}
Coxeter diagrams
=
Cells{3,∞}
Faces{3}
Edge figure{4}
Vertex figure{∞,4}
r{∞,∞}
Dual{4,∞,3}
Coxeter group[3,∞,4]
[3,∞1,1]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-infinite-4 triangular honeycomb (or 3,∞,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,4}.

It has four infinite-order triangular tilings, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-4 apeirogonal tiling vertex figure.


Poincaré disk model

Ideal surface

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

Order-infinite-5 triangular honeycomb

Order-infinite-5 triangular honeycomb
TypeRegular honeycomb
Schläfli symbols{3,∞,5}
Coxeter diagrams
Cells{3,∞}
Faces{3}
Edge figure{5}
Vertex figure{∞,5}
Dual{5,∞,3}
Coxeter group[3,∞,5]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-infinite-3 triangular honeycomb (or 3,∞,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,5}. It has five infinite-order triangular tiling, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-5 apeirogonal tiling vertex figure.


Poincaré disk model

Ideal surface

Order-infinite-6 triangular honeycomb

Order-infinite-6 triangular honeycomb
TypeRegular honeycomb
Schläfli symbols{3,∞,6}
{3,(∞,3,∞)}
Coxeter diagrams
=
Cells{3,∞}
Faces{3}
Edge figure{6}
Vertex figure{∞,6}
{(∞,3,∞)}
Dual{6,∞,3}
Coxeter group[3,∞,6]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-infinite-6 triangular honeycomb (or 3,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,6}. It has infinitely many infinite-order triangular tiling, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-6 apeirogonal tiling, {∞,6}, vertex figure.


Poincaré disk model

Ideal surface

Order-infinite-7 triangular honeycomb

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

In the geometry of hyperbolic 3-space, the order-infinite-7 triangular honeycomb (or 3,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,7}. It has infinitely many infinite-order triangular tiling, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-7 apeirogonal tiling, {∞,7}, vertex figure.


Ideal surface

Order-infinite-infinite triangular honeycomb

Order-infinite-infinite triangular honeycomb
TypeRegular honeycomb
Schläfli symbols{3,∞,∞}
{3,(∞,∞,∞)}
Coxeter diagrams
=
Cells{3,∞}
Faces{3}
Edge figure{∞}
Vertex figure{∞,∞}
{(∞,∞,∞)}
Dual{∞,∞,3}
Coxeter group[∞,∞,3]
[3,((∞,∞,∞))]
PropertiesRegular

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


Poincaré disk model

Ideal surface

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

Order-infinite-3 square honeycomb

Order-infinite-3 square honeycomb
TypeRegular honeycomb
Schläfli symbol{4,∞,3}
Coxeter diagram
Cells{4,∞}
Faces{4}
Vertex figure{∞,3}
Dual{3,∞,4}
Coxeter group[4,∞,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-infinite-3 square honeycomb (or 4,∞,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-infinite-3 square honeycomb is {4,∞,3}, with three infinite-order square tilings meeting at each edge. The vertex figure of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.


Poincaré disk model

Ideal surface

Order-infinite-3 pentagonal honeycomb

Order-infinite-3 pentagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{5,∞,3}
Coxeter diagram
Cells{5,∞}
Faces{5}
Vertex figure{∞,3}
Dual{3,∞,5}
Coxeter group[5,∞,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-infinite-3 pentagonal honeycomb (or 5,∞,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an infinite-order 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,∞,3}, with three infinite-order pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {∞,3}.


Poincaré disk model

Ideal surface

Order-infinite-3 hexagonal honeycomb

Order-infinite-3 hexagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{6,∞,3}
Coxeter diagram
Cells{6,∞}
Faces{6}
Vertex figure{∞,3}
Dual{3,∞,6}
Coxeter group[6,∞,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-infinite-3 hexagonal honeycomb (or 6,∞,3 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-infinite-3 hexagonal honeycomb is {6,∞,3}, with three infinite-order hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.


Poincaré disk model

Ideal surface

Order-infinite-3 heptagonal honeycomb

Order-infinite-3 heptagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{7,∞,3}
Coxeter diagram
Cells{7,∞}
Faces{7}
Vertex figure{∞,3}
Dual{3,∞,7}
Coxeter group[7,∞,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-infinite-3 heptagonal honeycomb (or 7,∞,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an infinite-order 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-infinite-3 heptagonal honeycomb is {7,∞,3}, with three infinite-order heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.


Ideal surface

Order-infinite-3 apeirogonal honeycomb

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

In the geometry of hyperbolic 3-space, the order-infinite-3 apeirogonal honeycomb (or ∞,∞,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an infinite-order 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 {∞,∞,3}, with three infinite-order apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an infinite-order apeirogonal tiling, {∞,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-infinite-4 square honeycomb

Order-infinite-4 square honeycomb
TypeRegular honeycomb
Schläfli symbol{4,∞,4}
Coxeter diagrams
=
Cells{4,∞}
Faces{4}
Edge figure{4}
Vertex figure{∞,4}
{∞,∞}
Dualself-dual
Coxeter group[4,∞,4]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-infinite-4 square honeycomb (or 4,∞,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,∞,4}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with four infinite-order square tilings existing around each edge and with an order-4 apeirogonal tiling vertex figure.


Poincaré disk model

Ideal surface

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

Order-infinite-5 pentagonal honeycomb

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

In the geometry of hyperbolic 3-space, the order-infinite-5 pentagonal honeycomb (or 5,∞,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,∞,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five infinite-order pentagonal tilings existing around each edge and with an order-5 apeirogonal tiling vertex figure.


Poincaré disk model

Ideal surface

Order-infinite-6 hexagonal honeycomb

Order-infinite-6 hexagonal honeycomb
TypeRegular honeycomb
Schläfli symbols{6,∞,6}
{6,(∞,3,∞)}
Coxeter diagrams
=
Cells{6,∞}
Faces{6}
Edge figure{6}
Vertex figure{∞,6}
{(5,3,5)}
Dualself-dual
Coxeter group[6,∞,6]
[6,((∞,3,∞))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-infinite-6 hexagonal honeycomb (or 6,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,∞,6}. It has six infinite-order hexagonal tilings, {6,∞}, 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 apeirogonal tiling vertex figure.


Poincaré disk model

Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(∞,3,∞)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,∞,6,1+] = [6,((∞,3,∞))].

Order-infinite-7 heptagonal honeycomb

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

In the geometry of hyperbolic 3-space, the order-infinite-7 heptagonal honeycomb (or 7,∞,7 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,∞,7}. It has seven infinite-order heptagonal tilings, {7,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 apeirogonal tiling vertex figure.


Ideal surface

Order-infinite-infinite apeirogonal honeycomb

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

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


Poincaré disk model

Ideal surface

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

See also

References

    • 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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