Tetraheptagonal tiling

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Tetraheptagonal tiling
H2 tiling 247-2.png
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.7)2
Schläfli symbol r{7,4} or
rr{7,7}
Wythoff symbol 2 | 7 4
7 7 | 2
Coxeter diagram CDel node.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node 1.png
Symmetry group [7,4], (*742)
[7,7], (*772)
Dual Order-7-4 rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{4,7}.

Geometry branch of mathematics that measures the shape, size and position of objects

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.

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

Schläfli symbol notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.

Contents

Symmetry

Uniform tiling 77-t02.png
A half symmetry [1+,4,7] = [7,7] construction exists, which can be seen as two colors of heptagons. This coloring can be called a rhombiheptaheptagonal tiling.
Ord74 qreg rhombic til.png
The dual tiling is made of rhombic faces and has a face configuration V4.7.4.7.

See also

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.

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Snub tetrapentagonal tiling

In geometry, the snub tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,4}.

Rhombitetrapentagonal tiling

In geometry, the rhombitetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{4,5}.

Truncated order-5 square tiling

In geometry, the truncated order-5 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,5}.

Truncated order-5 pentagonal tiling

In geometry, the truncated order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,5}, constructed from one pentagons and two decagons around every vertex.

Truncated order-7 square tiling A uniform tiling of the hyperbolic plane

In geometry, the truncated order-7 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,7}.

Order-7 square tiling

In geometry, the order-7 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,7}.

Rhombitetraheptagonal tiling

In geometry, the rhombitetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{4,7}. It can be seen as constructed as a rectified tetraheptagonal tiling, r{7,4}, as well as an expanded order-4 heptagonal tiling or expanded order-7 square tiling.

Order-7 heptagonal tiling

In geometry, the order-7 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,7}, constructed from seven heptagons around every vertex. As such, it is self-dual.

Truncated order-7 heptagonal tiling

In geometry, the truncated order-7 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{7,7}, constructed from one heptagons and two tetrakaidecagons around every vertex.

Cantic octagonal tiling

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}.

Alternated order-4 hexagonal tiling

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}.

Cantic order-4 hexagonal tiling

In geometry, the tritetratrigonal tiling or cantic order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{(4,4,3)} or h2{6,4}.

Snub order-6 square tiling

In geometry, the snub tetratritetragonal tiling or snub order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{(4,4,3)} or s{4,6}.

Rhombitetraapeirogonal tiling

In geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{∞,4}.

Truncated order-5 hexagonal tiling

In geometry, the truncated order-5 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{6,5}.

Pentahexagonal tiling

In geometry, the pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{6,5} or t1{6,5}.

Rhombipentahexagonal tiling

In geometry, the rhombipentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{6,5}.

Snub pentahexagonal tiling

In geometry, the snub pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,5}.

Snub triapeirotrigonal tiling

In geometry, the snub triapeirotrigonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of s{3,∞}.

Order-8 pentagonal tiling

In geometry, the order-8 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,8}.

References

John Horton Conway British mathematician

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.

International Standard Book Number Unique numeric book identifier

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.