# Trioctagonal tiling

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Trioctagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (3.8)2
Schläfli symbol r{8,3} or ${\displaystyle {\begin{Bmatrix}8\\3\end{Bmatrix}}}$
Wythoff symbol 2 | 8 3|
3 3 | 4
Coxeter diagram or
Symmetry group [8,3], (*832)
[(4,3,3)], (*433)
Dual Order-8-3 rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the trioctagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 octagonal tiling. There are two triangles and two octagons alternating on each vertex. It has Schläfli symbol of r{8,3}.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

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 .

## Symmetry

 The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles, by Coxeter diagram . Dual tiling

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular octagonal tiling.

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.

In hyperbolic geometry, a uniformhyperbolic tiling is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

It can also be generated from the (4 3 3) hyperbolic tilings:

The trioctagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:

In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence a step closer to regular polyhedra than the semiregular which are merely vertex-transitive. The dual figures is also sometimes considered quasiregular, except that they are edge-transitive, face-transitive, and alternate between two regular vertex figures.

In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of equilateral triangles and regular hexagons, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular hexagonal tiling and a regular triangular tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an infinite arrangement of lines. Its dual is the rhombille tiling.

In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles and sets of six rhombi meet at their 60° angles.

## Related Research Articles

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

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{8,3}.

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 order-3 snub octagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one octagon on each vertex. It has Schläfli symbol of sr{8,3}.

In geometry, the Truncated octagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two hexakaidecagons on each vertex. It has Schläfli symbol of t{8,3}.

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 rhombitrioctagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.

In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.

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 order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} and is self-dual.

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 snub octaoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,8}.

In geometry, the tritetratrigonal tiling or shieldotritetragonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2(4,3,3). It can also be named as a cantic octagonal tiling, h2{8,3}.

In geometry, the alternated order-4 hexagonal tiling or ditetragonal tritetratrigonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}.

In geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,6}.

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-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}.

In geometry, the rhombihexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,6}.

In geometry, the snub hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are three triangles, one hexagon, and one octagon on each vertex. It has Schläfli symbol of sr{8,6}.

## 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.

John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge, in England, and the second half at Princeton University in New Jersey, where he now holds the title Professor Emeritus.

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

Eric Wolfgang Weisstein is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He is the author of the CRC Concise Encyclopedia of Mathematics. He currently works for Wolfram Research, Inc.

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.