# Order-3 apeirogonal tiling

Last updated
Order-3 apeirogonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 3
Schläfli symbol {,3}
t{,}
t(,,)
Wythoff symbol 2
2 |
|
Coxeter diagram

Symmetry group [,3], (*32)
[,], (*2)
[(,,)], (*)
Dual Infinite-order triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.

## Contents

The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as {∞,2}.

## Images

Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary.

## Uniform colorings

Like the Euclidean hexagonal tiling, there are 3 uniform colorings of the order-3 apeirogonal tiling, each from different reflective triangle group domains:

Regular Truncations

{∞,3}

t0,1{∞,∞}

t1,2{∞,∞}

t{∞[3]}
Hyperbolic triangle groups

[∞,3]

[∞,∞]

[(∞,∞,∞)]

### Symmetry

The dual to this tiling represents the fundamental domains of [(∞,∞,∞)] (*∞∞∞) symmetry. There are 15 small index subgroups (7 unique) constructed from [(∞,∞,∞)] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled as ∞∞2 symmetry by adding a mirror bisecting the fundamental domain. Dividing a fundamental domain by 3 mirrors creates a ∞32 symmetry.

A larger subgroup is constructed [(∞,∞,∞*)], index 8, as (∞*∞) with gyration points removed, becomes (*∞).

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}.

*n32 symmetry mutation of regular tilings: {n,3}
Spherical Euclidean Compact hyperb.Paraco.Noncompact hyperbolic
{2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3}{9i,3}{6i,3}{3i,3}
Paracompact uniform tilings in [,3] family
Symmetry: [,3], (*32) [,3]+
(32)
[1+,,3]
(*33)
[,3+]
(3*)

=

=

=
=
or
=
or

=
{,3} t{,3} r{,3} t{3,} {3,} rr{,3} tr{,3} sr{,3} h{,3}h2{,3} s{3,}
Uniform duals
V3 V3..V(3.)2V6.6. V3 V4.3.4.V4.6.V3.3.3.3.V(3.)3V3.3.3.3.3.
Paracompact uniform tilings in [,] family

=
=

=
=

=
=

=
=

=
=

=

=
{,} t{,} r{,} 2t{,}=t{,} 2r{,}={,} rr{,} tr{,}
Dual tilings
VV..V(.)2V..VV4..4.V4.4.
Alternations
[1+,,]
(*2)
[+,]
(*)
[,1+,]
(*)
[,+]
(*)
[,,1+]
(*2)
[(,,2+)]
(2*)
[,]+
(2)
h{,} s{,}hr{,}s{,} h2{,} hrr{,} sr{,}
Alternation duals
V(.)V(3.)3V(.4)4V(3.)3VV(4..4)2V3.3..3.
Paracompact uniform tilings in [(,,)] family
(,,)
h{,}
r(,,)
h2{,}
(,,)
h{,}
r(,,)
h2{,}
(,,)
h{,}
r(,,)
r{,}
t(,,)
t{,}
Dual tilings
VV...VV...VV...V..
Alternations
[(1+,,,)]
(*)
[+,,)]
(*)
[,1+,,)]
(*)
[,+,)]
(*)
[(,,,1+)]
(*)
[(,,+)]
(*)
[,,)]+
()
Alternation duals
V(.)V(.4)4V(.)V(.4)4V(.)V(.4)4V3..3..3.

## Related Research Articles

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}.

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{6,4}.

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

In geometry, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex. It has Schläfli symbol of t{3,8}.

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}.

In geometry, the tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,4}.

In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}.

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.

In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.

In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices.

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}.

In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}.

In geometry, the order-6 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}.

## References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN   978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN   0-486-40919-8. LCCN   99035678.