Truncated order-4 pentagonal tiling

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Truncated pentagonal tiling
H2 tiling 245-3.png
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.10.10
Schläfli symbol t{5,4}
Wythoff symbol 2 4 | 5
2 5 5 |
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png or CDel node 1.pngCDel split1-55.pngCDel nodes 11.png
Symmetry group [5,4], (*542)
[5,5], (*552)
Dual Order-5 tetrakis square tiling
Properties Vertex-transitive

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

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 Non-Euclidean 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

Uniform colorings

A half symmetry [1+,4,5] = [5,5] coloring can be constructed with two colors of decagons. This coloring is called a truncated pentapentagonal tiling.

Uniform tiling 552-t012.png

Symmetry

There is only one subgroup of [5,5], [5,5]+, removing all the mirrors. This symmetry can be doubled to 542 symmetry by adding a bisecting mirror.

Small index subgroups of [5,5]
TypeReflective domainsRotational symmetry
Index 12
Diagram 552 symmetry 000.png 552 symmetry aaa.png
Coxeter
(orbifold)
[5,5] = CDel node c1.pngCDel 5.pngCDel node c1.pngCDel 5.pngCDel node c1.png = CDel node c1.pngCDel split1-55.pngCDel branch c1.pngCDel label2.png
(*552)
[5,5]+ = CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 5.pngCDel node h2.png = CDel node h2.pngCDel split1-55.pngCDel branch h2h2.pngCDel label2.png
(552)

Related Research Articles

Truncated triheptagonal tiling

In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of tr{7,3}.

Order-4 pentagonal tiling

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal tiling in a bicolored quasiregular form.

Octagonal tiling

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}.

Truncated trioctagonal tiling

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}.

Order-4 hexagonal tiling

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

Rhombitetrahexagonal tiling

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

Truncated order-4 hexagonal tiling

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.

Truncated tetrapentagonal tiling

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{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-8 triangular tiling

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}.

Rhombitrioctagonal tiling

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.

Snub pentapentagonal tiling

In geometry, the snub pentapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,5}, constructed from two regular pentagons and three equilateral triangles around every vertex.

Order-4 heptagonal tiling

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

Truncated order-4 heptagonal tiling

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

Truncated tetraheptagonal tiling

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

Rhombitetraoctagonal tiling

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.

Truncated order-8 octagonal tiling

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}.

Order-4 apeirogonal tiling

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

Truncated order-6 pentagonal tiling

In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2{6,5}.

Truncated order-6 octagonal tiling

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

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.

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.

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.