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 c ...

, the order-8 triangular tiling is a

regular tiling Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his '' Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619). Notation of Euc ...

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

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 ...

of ', having eight regular

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...

s around each vertex.

Uniform colorings

The half symmetry ⁺,8,3= 4,3,3)can be shown with alternating two colors of triangles: : H2_tiling_334-4

Symmetry

From 4,4,4)symmetry, there are 15 small index subgroups (7 unique) by mirror removal and alternation operators. 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. Adding 3 bisecting mirrors across each fundamental domains creates

832 symmetry In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of ''tr''. Symmetry The dual of this tili ...

. The

subgroup index In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the number of left cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or :H/math> or ( ...

-8 group, 1⁺,4,1⁺,4,1⁺,4)(222222) is the

commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

of 4,4,4) A larger subgroup is constructed 4,4,4^*) index 8, as (2*2222) with gyration points removed, becomes (*22222222). The symmetry can be doubled to

842 symmetry 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. Dual tiling Symmetry There are 15 sub ...

by adding a bisecting mirror across the fundamental domains. The symmetry can be extended by 6, as

, by 3 bisecting mirrors per domain.

Related polyhedra and tilings

From a

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

there are ten hyperbolic

uniform tilings A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, s ...

that can be based from the regular octagonal and order-8 triangular tilings. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms. It can also be generated from the (4 3 3) hyperbolic tilings:

References

John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branch ...

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

External links

* *
Hyperbolic and Spherical Tiling Gallery

* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch {{Tessellation Hyperbolic tilings Isogonal tilings Isohedral tilings Order-8 tilings Regular tilings Triangular tilings

Uniform colorings

Symmetry

Related polyhedra and tilings

See also

References

External links