In geometry, the order-3 apeirogonal tiling is a regular tiling of the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

. It is represented by the

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...

, having three regular

apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to ...

s around each vertex. Each apeirogon is inscribed in a horocycle. The

order-2 apeirogonal tiling In geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedronConway (2008), p. 263 is a tiling of the plane consisting of two apeirogons. It may be considered an improper regular tiling of the Euclidean plane, with Sc ...

represents an infinite

dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with fla ...

in the Euclidean plane as .

Images

Each

face is

circumscribe In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

d by a horocycle, which looks like a circle in a

Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk th ...

, internally tangent to the projective circle boundary. : Order-3 apeirogonal tiling one cell horocycle

Uniform colorings

Like the Euclidean

hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathema ...

, there are 3 uniform colorings of the ''order-3 apeirogonal tiling'', each from different reflective

triangle group In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic trian ...

domains:

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 (*∞^∞).

Related polyhedra and tilings

This tiling is topologically related as a part of sequence of regular polyhedra with

References

* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) *

External links