geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the order-3 apeirogonal tiling is a

regular tiling Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his ''Harmonices Mundi'' (Latin langua ...

of the hyperbolic plane. It is represented by the Schläfli symbol , 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 th ...

s around each vertex. Each apeirogon is inscribed in a

horocycle In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horospher ...

. 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 Sch ...

represents an infinite dihedron 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

, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary. : Order-3 apeirogonal tiling one cell horocycle

Order-3 apeirogonal tiling one cell horocycle

Uniform colorings

Like the Euclidean hexagonal tiling, 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 triangl ...

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 Schläfli symbol .

References

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

External links