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.

Order-4 pentagonal tiling
Order-4 pentagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 54
Schläfli symbol {5,4}
r{5,5} or
Wythoff symbol 4 | 5 2
2 | 5 5
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node.png or CDel node 1.pngCDel split1-55.pngCDel nodes.png
Symmetry group [5,4], (*542)
[5,5], (*552)
Dual Order-5 square tiling
Properties Vertex-transitive, edge-transitive, face-transitive

Symmetry

This tiling represents a hyperbolic kaleidoscope of 5 mirrors meeting as edges of a regular pentagon. This symmetry by orbifold notation is called *22222 with 5 order-2 mirror intersections. In Coxeter notation can be represented as [5*,4], removing two of three mirrors (passing through the pentagon center) in the [5,4] symmetry.

The kaleidoscopic domains can be seen as bicolored pentagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1{5,5} and as a quasiregular tiling is called a pentapentagonal tiling.

Uniform tiling 552-t1
Uniform tiling 552-t1

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with pentagonal faces, starting with the dodecahedron, with Schläfli symbol {5,n}, and Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel n.pngCDel node.png, progressing to infinity.

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram CDel node 1.pngCDel n.pngCDel node.pngCDel 4.pngCDel node.png, with n progressing to infinity.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

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)
  • Coxeter, H. S. M. (1999), Chapter 10: Regular honeycombs in hyperbolic space (PDF), The Beauty of Geometry: Twelve Essays, Dover Publications, ISBN 0-486-40919-8, LCCN 99035678, invited lecture, ICM, Amsterdam, 1954.

See also

External links

Dodecadodecahedron

In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882).

The edges of this model form 10 central hexagons, and projected onto a sphere, represent 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes.

Medial rhombic triacontahedron


In geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.

Its 24 vertices are all on the 12 axes with 5-fold symmetry (i.e. each corresponds to one of the 12 vertices of the icosahedron). This means that on each axis there is an inner and an outer vertex. The ratio of outer to inner vertex radius is , the golden ratio.

It has 30 intersecting rhombic faces, which correspond to the faces of the convex rhombic triacontahedron. The diagonals in the rhombs of the convex solid have a ratio of 1 to . The medial solid can be generated from the convex one by stretching the shorter diagonal from length 1 to . So the ratio of rhomb diagonals in the medial solid is 1 to .

This solid is to the compound of small stellated dodecahedron and great dodecahedron what the convex one is to the compound of dodecahedron and icosahedron: The crossing edges in the dual compound are the diagonals of the rhombs.

Order-4-3 pentagonal honeycomb

In the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell is an order-4 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

Order-4 120-cell honeycomb

In the geometry of hyperbolic 4-space, the order-4 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {5,3,3,4}, it has four 120-cells around each face. Its dual is the order-5 tesseractic honeycomb, {4,3,3,5}.

Order-4 dodecahedral honeycomb

In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

Order-4 icosahedral honeycomb

In the geometry of hyperbolic 3-space, the order-5 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,4}.

Order-5-4 square honeycomb

In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb (or 4,5,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,5,4}.

Order-5 dodecahedral honeycomb

The order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is a regular icosahedron.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

Order-5 square tiling

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

Pentagonal tiling

In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.

A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane and the sphere; the latter produces a tiling topologically equivalent to the dodecahedron.

Rectification (geometry)

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 rectification operator is sometimes denoted by the letter r with a Schläfli symbol. For example, r{4,3} is the rectified cube, also called a cuboctahedron, and also represented as . And a rectified cuboctahedron rr{4,3} is a rhombicuboctahedron, and also represented as .

Conway polyhedron notation uses a for ambo as this operator. In graph theory this operation creates a medial graph.

The rectification of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example the tetrahedron {3,3} becoming an octahedron {3,4}. As a special case, a square tiling {4,4} will turn into another square tiling {4,4} under a rectification operation.

Truncated order-4 pentagonal tiling

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}.

Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node h.png
Uniform tiling 54-t0 Uniform tiling 54-t01 Uniform tiling 54-t1 Uniform tiling 54-t12 Uniform tiling 54-t2 Uniform tiling 54-t02 Uniform tiling 54-t012 Uniform tiling 54-snub Uniform tiling 542-h01 Uniform tiling 552-t0
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
CDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node fh.png
Uniform tiling 54-t2 Order-5 tetrakis square tiling Order-5-4 quasiregular rhombic tiling Order-4 pentakis pentagonal tiling Uniform tiling 54-t0 Deltoidal tetrapentagonal tiling Order-4 bisected pentagonal tiling Order-5-4 floret pentagonal tiling Uniform tiling 552-t2
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55
Uniform pentapentagonal tilings
Symmetry: [5,5], (*552) [5,5]+, (552)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node 1.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node 1.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node 1.png
CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node 1.png
CDel node h.pngCDel 5.pngCDel node h.pngCDel 5.pngCDel node h.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 5.pngCDel node h.png
Uniform tiling 552-t0 Uniform tiling 552-t01 Uniform tiling 552-t1 Uniform tiling 552-t12 Uniform tiling 552-t2 Uniform tiling 552-t02 Uniform tiling 552-t012 Uniform tiling 552-snub
{5,5} t{5,5}
r{5,5} 2t{5,5}=t{5,5} 2r{5,5}={5,5} rr{5,5} tr{5,5} sr{5,5}
Uniform duals
CDel node f1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 5.pngCDel node fh.png
Uniform tiling 552-t2 Order5 pentakis pentagonal til Uniform tiling 54-t2 Order5 pentakis pentagonal til Uniform tiling 552-t0 Order-5-4 quasiregular rhombic tiling Order-5 tetrakis square tiling
V5.5.5.5.5 V5.10.10 V5.5.5.5 V5.10.10 V5.5.5.5.5 V4.5.4.5 V4.10.10 V3.3.5.3.5
{5,n} tilings
Uniform polyhedron-53-t0
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 54-t0
{5,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 55-t0
{5,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 56-t0
{5,6}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 57-t0
{5,7}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 7.pngCDel node.png
*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings
Spherical square hosohedron.png Spherical square bipyramid.png Uniform tiling 44-t0.svg H2 tiling 245-1.png H2 tiling 246-1.png H2 tiling 247-1.png H2 tiling 248-1.png H2 tiling 24i-1.png
24 34 44 54 64 74 84 ...4
*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact
Uniform tiling 432-t0
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 44-t0
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 245-4
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 246-4
{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 247-4
{4,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 248-4
{4,8}...
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 24i-4
{4,∞}
CDel node 1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png
*5n2 symmetry mutations of quasiregular tilings: (5.n)2
Symmetry
*5n2
[n,5]
Spherical Hyperbolic Paracompact Noncompact
*352
[3,5]
*452
[4,5]
*552
[5,5]
*652
[6,5]
*752
[7,5]
*852
[8,5]...
*∞52
[∞,5]
 
[ni,5]
Figures Uniform tiling 532-t1 Uniform tiling 54-t1 H2 tiling 255-2 H2 tiling 256-2 H2 tiling 257-2 H2 tiling 258-2 H2 tiling 25i-2
Config. (5.3)2 (5.4)2 (5.5)2 (5.6)2 (5.7)2 (5.8)2 (5.∞)2 (5.ni)2
Rhombic
figures
Rhombictriacontahedron Order-5-4 quasiregular rhombic tiling H2 tiling 245-4 Order-6-5 quasiregular rhombic tiling
Config. V(5.3)2 V(5.4)2 V(5.5)2 V(5.6)2 V(5.7)2 V(5.8)2 V(5.∞)2 V(5.∞)2

This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.