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The dual to this tiling represent the fundamental domains of 8,8,3)(*883) symmetry. There are 3 small index subgroup symmetries constructed from 8,8,3)by mirror removal and alternation. 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
Hyperbolic and Spherical Tiling Gallery
* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch {{Tessellation Hyperbolic tilings Isogonal tilings Order-6 tilings Truncated tilings Uniform tilings Octagonal tilings
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the truncated order-6 octagonal tiling is a uniform 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 has Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
of t.
Uniform colorings
A secondary construction t is called a truncated trioctaoctagonal tiling: :
Symmetry
The dual to this tiling represent the fundamental domains of 8,8,3)(*883) symmetry. There are 3 small index subgroup symmetries constructed from 8,8,3)by mirror removal and alternation. 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 862 symmetry
In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one Square (geometry), square, one dodecagon, and one hexakaidecagon on each vertex (geometry), vertex. It has Schläfli symbol of tr.
Du ...
by adding a mirror bisecting the fundamental domain.
Related polyhedra and tiling
References
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) *See also
*Tilings of regular polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his (Latin: ''The Harmony of the World'', 1619).
Notation of Euclidean tilings
Eucl ...
*List of uniform planar tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their du ...
*List of regular polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
Overview
This table shows a summary of regular polytope counts by rank.
There are no Euclidean regular star tessellations in any number of dimensions.
...
External links
* *Hyperbolic and Spherical Tiling Gallery
* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch {{Tessellation Hyperbolic tilings Isogonal tilings Order-6 tilings Truncated tilings Uniform tilings Octagonal tilings