uniform, tilings, hyperbolic, plane, symmetric, subdivision, hyperbolic, geometry, examples, uniform, tilings, spherical, euclidean, hyperbolic, regular, tilings, sphere, euclidean, plane, hyperbolic, plane, using, regular, pentagonal, hexagonal, heptagonal, a. Symmetric subdivision in hyperbolic geometry Examples of uniform tilings Spherical Euclidean Hyperbolic 5 3 5 5 5 6 3 6 6 6 7 3 7 7 7 3 Regular tilings p q of the sphere Euclidean plane and hyperbolic plane using regular pentagonal hexagonal and heptagonal and apeirogonal faces t 5 3 10 10 3 t 6 3 12 12 3 t 7 3 14 14 3 t 3 3 Truncated tilings have 2p 2p q vertex figures from regular p q r 5 3 3 5 3 5 r 6 3 3 6 3 6 r 7 3 3 7 3 7 r 3 3 3 Quasiregular tilings are similar to regular tilings but alternate two types of regular polygon around each vertex rr 5 3 3 4 5 4 rr 6 3 3 4 6 4 rr 7 3 3 4 7 4 rr 3 3 4 4 Semiregular tilings have more than one type of regular polygon tr 5 3 4 6 10 tr 6 3 4 6 12 tr 7 3 4 6 14 tr 3 4 6 Omnitruncated tilings have three or more even sided regular polygons Construction of Archimedean Solids and Tessellations Symmetry Triangular dihedral symmetry Tetrahedral Octahedral Icosahedral p6m symmetry 3 7 symmetry 3 8 symmetry Starting solidOperation Symbol p q Triangular hosohedron 2 3 Triangular dihedron 3 2 Tetrahedron 3 3 Cube 4 3 Octahedron 3 4 Dodecahedron 5 3 Icosahedron 3 5 Hexagonal tiling 6 3 Triangular tiling 3 6 Heptagonal tiling 7 3 Order 7 triangular tiling 3 7 Octagonal tiling 8 3 Order 8 triangular tiling 3 8 Truncation t t p q triangular prism truncated triangular dihedron Half of the edges count as degenerate digon faces The other half are normal edges truncated tetrahedron truncated cube truncated octahedron truncated dodecahedron truncated icosahedron Truncated hexagonal tiling Truncated triangular tiling Truncated heptagonal tiling Truncated order 7 triangular tiling Truncated octagonal tiling Truncated order 8 triangular tiling Rectification r Ambo a r p q tridihedron All of the edges count as degenerate digon faces tetratetrahedron cuboctahedron icosidodecahedron Trihexagonal tiling Triheptagonal tiling Trioctagonal tiling Bitruncation 2t Dual kis dk 2t p q truncated triangular dihedron Half of the edges count as degenerate digon faces The other half are normal edges triangular prism truncated tetrahedron truncated octahedron truncated cube truncated icosahedron truncated dodecahedron truncated triangular tiling truncated hexagonal tiling Truncated order 7 triangular tiling Truncated heptagonal tiling Truncated order 8 triangular tiling Truncated octagonal tiling Birectification 2r Dual d 2r p q triangular dihedron 3 2 triangular hosohedron 2 3 tetrahedron octahedron cube icosahedron dodecahedron triangular tiling hexagonal tiling Order 7 triangular tiling Heptagonal tiling Order 8 triangular tiling Octagonal tiling Cantellation rr Expansion e rr p q triangular prism The edge between each pair of tetragons counts as a degenerate digon face The other edges the ones between a trigon and a tetragon are normal edges rhombitetratetrahedron rhombicuboctahedron rhombicosidodecahedron rhombitrihexagonal tiling Rhombitriheptagonal tiling Rhombitrioctagonal tiling Snub rectified sr Snub s sr p q triangular antiprism Three yellow yellow edges no two of which share any vertices count as degenerate digon faces The other edges are normal edges snub tetratetrahedron snub cuboctahedron snub icosidodecahedron snub trihexagonal tiling Snub triheptagonal tiling Snub trioctagonal tiling Cantitruncation tr Bevel b tr p q hexagonal prism truncated tetratetrahedron truncated cuboctahedron truncated icosidodecahedron truncated trihexagonal tiling Truncated triheptagonal tiling Truncated trioctagonal tiling In hyperbolic geometry a uniform hyperbolic tiling or regular quasiregular or semiregular hyperbolic tiling is an edge to edge filling of the hyperbolic plane which has regular polygons as faces and is vertex transitive transitive on its vertices isogonal i e there is an isometry mapping any vertex onto any other It follows that all vertices are congruent and the tiling has a high degree of rotational and translational symmetry Uniform tilings can be identified by their vertex configuration a sequence of numbers representing the number of sides of the polygons around each vertex For example 7 7 7 represents the heptagonal tiling which has 3 heptagons around each vertex It is also regular since all the polygons are the same size so it can also be given the Schlafli symbol 7 3 Uniform tilings may be regular if also face and edge transitive quasi regular if edge transitive but not face transitive or semi regular if neither edge nor face transitive For right triangles p 160 q 160 2 there are two regular tilings represented by Schlafli symbol p q and q p Contents 1 Wythoff construction 2 Right triangle domains 2 1 Regular hyperbolic tilings 2 2 7 160 3 160 2 2 3 8 160 3 160 2 2 4 5 160 4 160 2 2 5 6 160 4 160 2 2 6 7 160 4 160 2 2 7 8 160 4 160 2 2 8 5 160 5 160 2 2 9 6 160 5 160 2 2 10 6 6 2 2 11 8 6 2 2 12 7 7 2 2 13 8 8 2 3 General triangle domains 3 1 4 160 3 160 3 3 2 4 160 4 160 3 3 3 4 160 4 160 4 3 4 5 160 3 160 3 3 5 5 160 4 160 3 3 6 5 160 4 160 4 3 7 6 160 3 160 3 3 8 6 160 4 160 3 3 9 6 160 4 160 4 4 Summary of tilings with finite triangular fundamental domains 5 Quadrilateral domains 5 1 3 160 2 160 2 160 2 5 2 3 160 2 160 3 160 2 6 Ideal triangle domains 6 1 160 3 160 2 6 2 160 4 160 2 6 3 160 5 160 2 6 4 2 6 5 160 3 160 3 6 6 160 4 160 3 6 7 160 4 160 4 6 8 3 6 9 4 6 10 7 Summary of tilings with infinite triangular fundamental domains 8 References 9 External links Wythoff construction edit Example Wythoff construction with right triangles r 160 160 2 and the 7 generator points Lines to the active mirrors are colored red yellow and blue with the 160 3 160 nodes opposite them as associated by the Wythoff symbol There are an infinite number of uniform tilings based on the Schwarz triangles p 160 q 160 r where 1 p 160 160 1 q 160 160 1 r 160 lt 160 1 where p q r are each orders of reflection symmetry at three points of the fundamental domain triangle the symmetry group is a hyperbolic triangle group Each symmetry family contains 7 uniform tilings defined by a Wythoff symbol or Coxeter Dynkin diagram 7 representing combinations of 160 3 160 active mirrors An 8th represents an alternation operation deleting alternate vertices from the highest form with all mirrors active Families with r 160 160 2 contain regular hyperbolic tilings defined by a Coxeter group such as 7 3 8 3 9 3 5 4 6 4 Hyperbolic families with r 160 160 3 or higher are given by p 160 q 160 r and include 4 160 3 160 3 5 160 3 160 3 6 160 3 160 3 4 160 4 160 3 5 160 4 160 3 4 160 4 160 4 Hyperbolic triangles p 160 q 160 r define compact uniform hyperbolic tilings In the limit any of p q or r can be replaced by which defines a paracompact hyperbolic triangle and creates uniform tilings with either infinite faces called apeirogons that converge to a single ideal point or infinite vertex figure with infinitely many edges diverging from the same ideal point More symmetry families can be constructed from fundamental domains that are not triangles Selected families of uniform tilings are shown below using the Poincare disk model for the hyperbolic plane Three of them 7 160 3 160 2 5 160 4 160 2 and 4 160 3 160 3 and no others are minimal in the sense that if any of their defining numbers is replaced by a smaller integer the resulting pattern is either Euclidean or spherical rather than hyperbolic conversely any of the numbers can be increased even to infinity to generate other hyperbolic patterns Each uniform tiling generates a dual uniform tiling with many of them also given below Right triangle domains edit There are infinitely many p 160 q 160 2 triangle group families This article shows the regular tiling up to p q 160 160 8 and uniform tilings in 12 families 7 160 3 160 2 8 160 3 160 2 5 160 4 160 2 6 160 4 160 2 7 160 4 160 2 8 160 4 160 2 5 160 5 160 2 6 160 5 160 2 6 160 6 160 2 7 160 7 160 2 8 160 6 160 2 and 8 160 8 160 2 Regular hyperbolic tilings edit Wikimedia Commons has media related to Regular hyperbolic tilings The simplest set of hyperbolic tilings are regular tilings p q which exist in a matrix with the regular polyhedra and Euclidean tilings The regular tiling p q has a dual tiling q p across the diagonal axis of the table Self dual tilings 2 2 3 3 4 4 5 5 etc pass down the diagonal of the table Regular hyperbolic tiling table vte Spherical improper Platonic Euclidean hyperbolic Poincare disc compact paracompact noncompact tessellations with their Schlafli symbol p q 2 3 4 5 6 7 8 8734 ip l 2 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 8734 2 ip l 3 3 2 tetrahedron 3 3 octahedron 3 4 icosahedron 3 5 deltille 3 6 3 7 3 8 3 8734 3 ip l 4 4 2 cube 4 3 quadrille 4 4 4 5 4 6 4 7 4 8 4 8734 4 ip l 5 5 2 dodecahedron 5 3 5 4 5 5 5 6 5 7 5 8 5 8734 5 ip l 6 6 2 hextille 6 3 6 4 6 5 6 6 6 7 6 8 6 8734 6 ip l 7 7 2 7 3 7 4 7 5 7 6 7 7 7 8 7 8734 7 ip l 8 8 2 8 3 8 4 8 5 8 6 8 7 8 8 8 8734 8 ip l 8734 8734 2 8734 3 8734 4 8734 5 8734 6 8734 7 8734 8 8734 8734 8734 ip l ip l ip l 2 ip l 3 ip l 4 ip l 5 ip l 6 ip l 7 ip l 8 ip l 8734 ip l ip l 7 160 3 160 2 edit The 7 160 3 160 2 triangle group Coxeter group 7 3 orbifold 732 contains these uniform tilings Uniform heptagonal triangular tilings vte Symmetry 7 3 732 7 3 732 7 3 t 7 3 r 7 3 t 3 7 3 7 rr 7 3 tr 7 3 sr 7 3 Uniform duals V73 V3 14 14 V3 7 3 7 V6 6 7 V37 V3 4 7 4 V4 6 14 V3 3 3 3 7 8 160 3 160 2 edit The 8 160 3 160 2 triangle group Coxeter group 8 3 orbifold 832 contains these uniform tilings Uniform octagonal triangular tilings vte Symmetry 8 3 832 8 3 832 1 8 3 443 8 3 3 4 8 3 t 8 3 r 8 3 t 3 8 3 8 rr 8 3 s2 3 8 tr 8 3 sr 8 3 h 8 3 h2 8 3 s 3 8 or or Uniform duals V83 V3 16 16 V3 8 3 8 V6 6 8 V38 V3 4 8 4 V4 6 16 V34 8 V 3 4 3 V8 6 6 V35 4 5 160 4 160 2 edit The 5 160 4 160 2 triangle group Coxeter group 5 4 orbifold 542 contains these uniform tilings Uniform pentagonal square tilings vte Symmetry 5 4 542 5 4 542 5 4 5 2 5 4 1 552 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 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 6 160 4 160 2 edit The 6 160 4 160 2 triangle group Coxeter group 6 4 orbifold 642 contains these uniform tilings Because all the elements are even each uniform dual tiling one represents the fundamental domain of a reflective symmetry 3333 662 3232 443 222222 3222 and 642 respectively As well all 7 uniform tiling can be alternated and those have duals as well Uniform tetrahexagonal tilings vte Symmetry 6 4 642 with 6 6 662 4 3 3 443 8734 3 8734 3222 index 2 subsymmetries And 8734 3 8734 3 3232 index 4 subsymmetry 6 4 t 6 4 r 6 4 t 4 6 4 6 rr 6 4 tr 6 4 Uniform duals V64 V4 12 12 V 4 6 2 V6 8 8 V46 V4 4 4 6 V4 8 12 Alternations 1 6 4 443 6 4 6 2 6 1 4 3222 6 4 4 3 6 4 1 662 6 4 2 2 32 6 4 642 h 6 4 s 6 4 hr 6 4 s 4 6 h 4 6 hrr 6 4 sr 6 4 7 160 4 160 2 edit The 7 160 4 160 2 triangle group Coxeter group 7 4 orbifold 742 contains these uniform tilings Uniform heptagonal square tilings vte Symmetry 7 4 742 7 4 742 7 4 7 2 7 4 1 772 7 4 t 7 4 r 7 4 2t 7 4 t 4 7 2r 7 4 4 7 rr 7 4 tr 7 4 sr 7 4 s 7 4 h 4 7 Uniform duals V74 V4 14 14 V4 7 4 7 V7 8 8 V47 V4 4 7 4 V4 8 14 V3 3 4 3 7 V3 3 7 3 7 V77 8 160 4 160 2 edit The 8 160 4 160 2 triangle group Coxeter group 8 4 orbifold 842 contains these uniform tilings Because all the elements are even each uniform dual tiling one represents the fundamental domain of a reflective symmetry 4444 882 4242 444 22222222 4222 and 842 respectively As well all 7 uniform tiling can be alternated and those have duals as well Uniform octagonal square tilings vte 8 4 842 with 8 8 882 4 4 4 444 8734 4 8734 4222 index 2 subsymmetries And 8734 4 8734 4 4242 index 4 subsymmetry 8 4 t 8 4 r 8 4 2t 8 4 t 4 8 2r 8 4 4 8 rr 8 4 tr 8 4 Uniform duals img, wikipedia, wiki, book, books, library,

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