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Alternated octagonal tiling

January 01, 1970

In geometry, the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of {(4,3,3)} or h{8,3}.

Geometry edit

Although a sequence of edges seem to represent straight lines (projected into curves), careful attention will show they are not straight, as can be seen by looking at it from different projective centers.

 
Triangle-centered
hyperbolic straight edges 		 
Edge-centered
projective straight edges 		 
Point-centered
projective straight edges

Dual tiling edit

 

In art edit

Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came". White curves within the figure, through the middle of each line of fish, divide the plane into squares and triangles in the pattern of the tritetragonal tiling. However, in the tritetragonal tiling, the corresponding curves are chains of hyperbolic line segments, with a slight angle at each vertex, while in Escher's woodcut they appear to be smooth hypercycles.

Related polyhedra and tiling edit

Uniform (4,3,3) tilings
Symmetry: [(4,3,3)], (*433) [(4,3,3)]+, (433)
                                       
                                               
               
h{8,3}
t0(4,3,3) 		r{3,8}1/2
t0,1(4,3,3) 		h{8,3}
t1(4,3,3) 		h2{8,3}
t1,2(4,3,3) 		{3,8}1/2
t2(4,3,3) 		h2{8,3}
t0,2(4,3,3) 		t{3,8}1/2
t0,1,2(4,3,3) 		s{3,8}1/2
s(4,3,3)
Uniform duals
               
V(3.4)3 V3.8.3.8 V(3.4)3 V3.6.4.6 V(3.3)4 V3.6.4.6 V6.6.8 V3.3.3.3.3.4
Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444) 		[(1+,4,4,4)]
(*4242) 		[(4+,4,4)]
(4*22)
    
      		    
      		    
      		    
      		    
      		    
      		    
      		    
      		    
      		    
     
                   
t0(4,4,4)
h{8,4} 		t0,1(4,4,4)
h2{8,4} 		t1(4,4,4)
{4,8}1/2 		t1,2(4,4,4)
h2{8,4} 		t2(4,4,4)
h{8,4} 		t0,2(4,4,4)
r{4,8}1/2 		t0,1,2(4,4,4)
t{4,8}1/2 		s(4,4,4)
s{4,8}1/2 		h(4,4,4)
h{4,8}1/2 		hr(4,4,4)
hr{4,8}1/2
Uniform duals
                   
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3

See also edit

References edit

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

External links edit

 
Wikimedia Commons has media related to Uniform tiling 3-4-3-4-3-4.
  • Douglas Dunham Department of Computer Science University of Minnesota, Duluth
    • Examples Based on Circle Limits III and IV, 2006:More “Circle Limit III” Patterns, 2007:A “Circle Limit III” Calculation, 2008:A “Circle Limit III” Backbone Arc Formula
  • Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
  • Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
  • Hyperbolic and Spherical Tiling Gallery 2013-03-24 at the Wayback Machine
  • KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
  • Hyperbolic Planar Tessellations, Don Hatch

January 01, 1970
alternated, octagonal, tiling, poincaré, disk, model, hyperbolic, plane, type, hyperbolic, uniform, tiling, vertex, configuration, schläfli, symbol, wythoff, symbol, coxeter, diagram, symmetry, group, dual, dual, tiling, properties, vertex, transitive, geometr. Alternated octagonal tiling Poincare disk model of the hyperbolic plane Type Hyperbolic uniform tiling Vertex configuration 3 4 3 Schlafli symbol 4 3 3 s 4 4 4 Wythoff symbol 3 3 4 Coxeter diagram Symmetry group 4 3 3 433 4 4 4 444 Dual Alternated octagonal tiling Dual tiling Properties Vertex transitive In geometry the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane It has Schlafli symbols of 4 3 3 or h 8 3 Contents 1 Geometry 2 Dual tiling 3 In art 4 Related polyhedra and tiling 5 See also 6 References 7 External linksGeometry editAlthough a sequence of edges seem to represent straight lines projected into curves careful attention will show they are not straight as can be seen by looking at it from different projective centers nbsp Triangle centeredhyperbolic straight edges nbsp Edge centeredprojective straight edges nbsp Point centeredprojective straight edgesDual tiling edit nbsp In art editCircle Limit III is a woodcut made in 1959 by Dutch artist M C Escher in which strings of fish shoot up like rockets from infinitely far away and then fall back again whence they came White curves within the figure through the middle of each line of fish divide the plane into squares and triangles in the pattern of the tritetragonal tiling However in the tritetragonal tiling the corresponding curves are chains of hyperbolic line segments with a slight angle at each vertex while in Escher s woodcut they appear to be smooth hypercycles Related polyhedra and tiling editUniform 4 3 3 tilings vte Symmetry 4 3 3 433 4 3 3 433 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 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 8 3 t0 4 3 3 r 3 8 1 2t0 1 4 3 3 h 8 3 t1 4 3 3 h2 8 3 t1 2 4 3 3 3 8 1 2t2 4 3 3 h2 8 3 t0 2 4 3 3 t 3 8 1 2t0 1 2 4 3 3 s 3 8 1 2s 4 3 3 Uniform duals nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V 3 4 3 V3 8 3 8 V 3 4 3 V3 6 4 6 V 3 3 4 V3 6 4 6 V6 6 8 V3 3 3 3 3 4 Uniform 4 4 4 tilings vte Symmetry 4 4 4 444 4 4 4 444 1 4 4 4 4242 4 4 4 4 22 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 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp t0 4 4 4 h 8 4 t0 1 4 4 4 h2 8 4 t1 4 4 4 4 8 1 2 t1 2 4 4 4 h2 8 4 t2 4 4 4 h 8 4 t0 2 4 4 4 r 4 8 1 2 t0 1 2 4 4 4 t 4 8 1 2 s 4 4 4 s 4 8 1 2 h 4 4 4 h 4 8 1 2 hr 4 4 4 hr 4 8 1 2 Uniform duals nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V 4 4 4 V4 8 4 8 V 4 4 4 V4 8 4 8 V 4 4 4 V4 8 4 8 V8 8 8 V3 4 3 4 3 4 V88 V 4 4 3See also editCircle Limit III Square tiling Uniform tilings in hyperbolic plane List of regular polytopesReferences editJohn H Conway Heidi Burgiel Chaim Goodman Strauss The Symmetries of Things 2008 ISBN 978 1 56881 220 5 Chapter 19 The Hyperbolic Archimedean Tessellations Chapter 10 Regular honeycombs in hyperbolic space The Beauty of Geometry Twelve Essays Dover Publications 1999 ISBN 0 486 40919 8 LCCN 99035678 External links edit nbsp Wikimedia Commons has media related to Uniform tiling 3 4 3 4 3 4 Douglas Dunham Department of Computer Science University of Minnesota Duluth Examples Based on Circle Limits III and IV 2006 More Circle Limit III Patterns 2007 A Circle Limit III Calculation 2008 A Circle Limit III Backbone Arc Formula Weisstein Eric W Hyperbolic tiling MathWorld Weisstein Eric W Poincare hyperbolic disk MathWorld Hyperbolic and Spherical Tiling Gallery Archived 2013 03 24 at the Wayback Machine KaleidoTile 3 Educational software to create spherical planar and hyperbolic tilings Hyperbolic Planar Tessellations Don Hatch Retrieved from https en wikipedia org w index php title Alternated octagonal tiling amp oldid 1189601778, wikipedia, wiki, book, books, library,

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