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Title: Math/Recreations/Tessellations - Catalogue of Isohedral Tilings From the article "One Corona is Enough for the Euclidean Plane" by Doris Schattschneider and Nikolai Dolbilin.
Chaos_Tiles This chaotic tiling consists of two equilateral pentagons, with angles (80, 160, 60, 140, 100) and (40, 200, 60, 100, 140). Sets are for sale.

The_14_Different_Types_of_Convex_Pentagons_that_Tile_the_Plane Graphics and references.

Discrete_Plane_Symmetry_Groups Web text by Ana Kozomara.

Escher_Web_Sketch Java applet drawing repeated patterns with selected symmetry.

Frieze_Patterns Definitions and images.

Gallery_of_Symmetric_Chaos Includes: Symmetric icons and fractals; Quilts and planar repeating patterns; Colored wallpaper patterns.


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The Math Forum: Catalog of Isohedral Tilings by Symmetric Polygonal Tiles 

Catalog of Isohedral Tilingsby Symmetric Polygonal Tiles

From the article "One Corona is Enough for the Euclidean Plane."Authors: Doris SchattschneiderMoravian College1200 Main St.Bethlehem, PA 18018-6650 USA<schattdo@moravian.edu> Nikolai DolbilinSteklov Mathematical InstituteGubkin 8, 117966 MoscowGSP-1 Russia<nikolai@dolbilin.mian.su>Published in "Quasicrystals and Discrete Geometry," J. Patera, editor,TheFields Institute for Research in Mathematical Sciences MonographSeries,Vol. 10, AMS, Providence, RI, 1998, pp. 207-246.In the article cited above, tilings* of the Euclidean plane by asingle polygon are considered, and it is shown that such a tiling isisohedral if and only if each polygon is "surrounded in the same way,"or, more technically, the centered coronas of tiles are pairwisecongruent.(*In these tilings, at each vertex of each polygon, three or morepolygons must meet.)When the polygon tile is asymmetric, the proof of the result is aconsequence of the local theorem for tilings, a general result thatholds for tilings in d-dimensional space (see N. Dolbilin and D.Schattschneider, "The Local Theorem for Tilings," in the volumementioned above, pp. 193-199). Therefore the result needs to beverified only for symmetric polygons. The proof is case-by-case,constructing all possible tilings with a single symmetric polygon tilethat satisfies the condition on pairwise congruent coronas. All areshown to be isohedral. One consequence of the proof is the productionof this complete catalog of isohedral tilings by symmetric polygontiles.About one-quarter of the 42 different types of tilings are rigid; thatis, they are unique up to similarity. However, the remaining tilingsare highly flexible, with the shape of the tile able to be deformed ina continuous manner to assume a variety of distinct shapes. Thetilings were constructed using The Geometer's Sketchpad, and thedynamic action of that program allows you to explore the range ofshapes of these tiles and their associated tilings by simply draggingon a vertex of a tile. This web site allows you to view the whole catalog, and also to view the deformation of the flexible tilings interactively, using theJavaSketchpad version of the tilings. You can also download theSketchpad files to your computer if you wish to explore them further. Each sketch shows a patch of a tiling with a central tile surroundedby shaded copies of the tile. The set consisting of thecentral tile and the shaded surrounding tiles is called a centered corona of the tiling. (Formally, the corona of a tile is the set of all tiles that havenonempty intersection with the tile.) The Laves net of the tiling isgiven, and any constraints on the tile are described. When the tile(and hence its tiling) is flexible, a "play" button is shown.This link leads to the interactive JavaSketchpad version ofthe sketch in which one or more vertices is highlighted. Drag on ahighlighted vertex to deform the tile and its tilingand view the wide range of shapes that the tile can assume. In many cases,one degree offreedom has been fixed, so the range of shapes does not show all possiblesizes. Furtherdetails on each of the tilings (as well as their construction) can be found in the article cited at the top of this page.Catalog of TilingsPage 1:- 3.12.12- 4.8.8- 6.6.6 (1)- 6.6.6 (2)- 6.6.6 (3)Page 2:- 3.4.6.4 (1)- 3.4.6.4 (2)- 3.6.3.6- 4.4.4.4 (1)- 4.4.4.4 (2)- 4.4.4.4 (3)Page 3:- 4.4.4.4 (4)- 4.4.4.4 (5)- 4.4.4.4 (6)- 4.4.4.4 (7)- 4.4.4.4 (8)- 4.4.4.4 (9)Page 4:- 4.4.4.4 (10)- 4.4.4.4 (11)- 3.3.4.3.4 (1)- 3.3.4.3.4 (2)- 3.3.4.3.4 (3)Page 5:- 3.3.4.3.4 (4)- 3.3.3.4.4 (1)- 3.3.3.4.4 (2)- 3.3.3.4.4 (3)- 3.3.3.4.4 (4)- 3.3.3.4.4 (5)Page 6:- 3.3.3.3.6 (1)- 3.3.3.3.6 (2)- 36 (1)- 36 (2)- 36 (3)- 36 (4)Page 7:- 36 (5)- 36 (6)- 36 (7)- 36 (8)- 36 (9)- 36 (10)Page 8:- 36 (11)- 36 (12)Hosted by The Math ForumAugust 1998
 

From

the

article

"One

Corona

is

Enough

for

the

Euclidean

Plane"

by

Doris

Schattschneider

and

Nikolai

Dolbilin.

http://mathforum.org/dynamic/one-corona/

Catalogue of Isohedral Tilings 2009 January

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From the article "One Corona is Enough for the Euclidean Plane" by Doris Schattschneider and Nikolai Dolbilin.

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