fbpx
Wikipedia

Infinite-order square tiling

May 07, 2024


In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.

Uniform colorings edit

There is a half symmetry form,     , seen with alternating colors:

 

Symmetry edit

This tiling represents the mirror lines of *∞∞∞∞ symmetry. The dual to this tiling defines the fundamental domains of (*2∞) orbifold symmetry.

 

Related polyhedra and tiling edit

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

*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact
 
{4,3}
      		 
{4,4}
      		 
{4,5}
      		 
{4,6}
      		 
{4,7}
      		 
{4,8}...
      		 
{4,∞}
     
Paracompact uniform tilings in [∞,4] family
                                         
             
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
                                         
             
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4∞ V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞) 		[∞+,4]
(∞*2) 		[∞,1+,4]
(*2∞2∞) 		[∞,4+]
(4*∞) 		[∞,4,1+]
(*∞∞2) 		[(∞,4,2+)]
(2*2∞) 		[∞,4]+
(∞42)
     
=     		                       
=     		           
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
       
Alternation duals
                                         
   
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞

See also edit

 
Wikimedia Commons has media related to Infinite-order square tiling.

References edit

  • John H. Conway; Heidi Burgiel; Chaim Goodman-Strauss (2008). "Chapter 19, The Hyperbolic Archimedean Tessellations". The Symmetries of Things. ISBN 978-1-56881-220-5.
  • H. S. M. Coxeter (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.

External links edit

May 07, 2024
infinite, order, square, tiling, this, article, includes, list, references, related, reading, external, links, sources, remain, unclear, because, lacks, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, march, 2014. This article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations Please help improve this article by introducing more precise citations March 2014 Learn how and when to remove this message Infinite order square tiling Poincare disk model of the hyperbolic plane Type Hyperbolic regular tiling Vertex configuration 4 Schlafli symbol 4 Wythoff symbol 4 2 Coxeter diagram Symmetry group 4 42 Dual Order 4 apeirogonal tiling Properties Vertex transitive edge transitive face transitive In geometry the infinite order square tiling is a regular tiling of the hyperbolic plane It has Schlafli symbol of 4 All vertices are ideal located at infinity seen on the boundary of the Poincare hyperbolic disk projection Contents 1 Uniform colorings 2 Symmetry 3 Related polyhedra and tiling 4 See also 5 References 6 External linksUniform colorings editThere is a half symmetry form nbsp nbsp nbsp nbsp seen with alternating colors nbsp Symmetry editThis tiling represents the mirror lines of symmetry The dual to this tiling defines the fundamental domains of 2 orbifold symmetry nbsp Related polyhedra and tiling editThis tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure 4n n42 symmetry mutation of regular tilings 4 n vte Spherical Euclidean Compact hyperbolic Paracompact nbsp 4 3 nbsp nbsp nbsp nbsp nbsp nbsp 4 4 nbsp nbsp nbsp nbsp nbsp nbsp 4 5 nbsp nbsp nbsp nbsp nbsp nbsp 4 6 nbsp nbsp nbsp nbsp nbsp nbsp 4 7 nbsp nbsp nbsp nbsp nbsp nbsp 4 8 nbsp nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp Paracompact uniform tilings in 4 family vte nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 t 4 r 4 2t 4 t 4 2r 4 4 rr 4 tr 4 Dual figures nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V 4 V4 V 4 2 V8 8 V4 V43 V4 8 Alternations 1 4 44 4 2 1 4 2 2 4 4 4 1 2 4 2 2 2 4 42 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp h 4 s 4 hr 4 s 4 h 4 hrr 4 s 4 nbsp nbsp nbsp nbsp Alternation duals nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V 4 4 V3 3 2 V 4 4 2 V3 3 4 2 V V 44 V3 3 4 3 See also edit nbsp Wikimedia Commons has media related to Infinite order square tiling Square tiling Uniform tilings in hyperbolic plane List of regular polytopesReferences editJohn H Conway Heidi Burgiel Chaim Goodman Strauss 2008 Chapter 19 The Hyperbolic Archimedean Tessellations The Symmetries of Things ISBN 978 1 56881 220 5 H S M Coxeter 1999 Chapter 10 Regular honeycombs in hyperbolic space The Beauty of Geometry Twelve Essays Dover Publications ISBN 0 486 40919 8 LCCN 99035678 External links editWeisstein Eric W Hyperbolic tiling MathWorld Weisstein Eric W Poincare hyperbolic disk MathWorld Hyperbolic and Spherical Tiling Gallery Retrieved from https en wikipedia org w index php title Infinite order square tiling amp oldid 1189603358, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.