MOSAICOSREGULARES

8/9/2019 MOSAICOSREGULARES

1/7

http://en.wikipedia.org/wiki/Tilings_of_regular_polygons

Tiling by regular polygons

From Wikipedia, the free encyclopedia

(Redirected from Tilings of regular polygons)

Planetilings by regular polygons have been widely used since antiquity. The first

systematic mathematical treatment was that ofKeplerinHarmonices Mundi.

Contents

[hide]

1 Regular tilings

2 Archimedean, uniform or semiregular tilings

3 Combinations of regular polygons that can meet at a vertex

4 Other edge-to-edge tilings

5 Tilings that are not edge-to-edge

6 The hyperbolic plane

7 See also

8 References

9 External links

Regular tilings

Following Grnbaum and Shephard (section 1.3), a tiling is said to be regularif the

symmetry group of the tiling acts transitively on theflags of the tiling, where a flag is atriple consisting of a mutually incident vertex, edge and tile of the tiling. This means that

for every pair of flags there is a symmetry operation mapping the first flag to the second.

This is equivalent to the tiling being an edge-to-edge tiling bycongruent regular polygons.There must be six equilateral triangles, foursquares or three regularhexagons at a vertex,

yielding the three regular tessellations.
http://en.wikipedia.org/w/index.php?title=Tilings_of_regular_polygons&redirect=nohttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Tessellationhttp://en.wikipedia.org/wiki/Regular_polygonhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Harmonices_Mundihttp://en.wikipedia.org/wiki/Harmonices_Mundihttp://toggletoc%28%29/http://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Regular_tilingshttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Archimedean.2C_uniform_or_semiregular_tilingshttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Combinations_of_regular_polygons_that_can_meet_at_a_vertexhttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Other_edge-to-edge_tilingshttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Tilings_that_are_not_edge-to-edgehttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#The_hyperbolic_planehttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#See_alsohttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Referenceshttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#External_linkshttp://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaumhttp://en.wikipedia.org/wiki/Symmetry_grouphttp://en.wikipedia.org/wiki/Group_actionhttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Edge-to-edge_tilinghttp://en.wikipedia.org/wiki/Congruence_(geometry)http://en.wikipedia.org/wiki/Congruence_(geometry)http://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Square_(geometry)http://en.wikipedia.org/wiki/Hexagonhttp://en.wikipedia.org/wiki/Hexagonhttp://en.wikipedia.org/w/index.php?title=Tilings_of_regular_polygons&redirect=nohttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Tessellationhttp://en.wikipedia.org/wiki/Regular_polygonhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Harmonices_Mundihttp://toggletoc%28%29/http://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Regular_tilingshttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Archimedean.2C_uniform_or_semiregular_tilingshttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Combinations_of_regular_polygons_that_can_meet_at_a_vertexhttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Other_edge-to-edge_tilingshttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Tilings_that_are_not_edge-to-edgehttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#The_hyperbolic_planehttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#See_alsohttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#Referenceshttp://en.wikipedia.org/wiki/Tilings_of_regular_polygons#External_linkshttp://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaumhttp://en.wikipedia.org/wiki/Symmetry_grouphttp://en.wikipedia.org/wiki/Group_actionhttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Edge-to-edge_tilinghttp://en.wikipedia.org/wiki/Congruence_(geometry)http://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Square_(geometry)http://en.wikipedia.org/wiki/Hexagon


2/7

36

Triangular tiling

44

Square tiling

63

Hexagonal

Archimedean, uniform or semiregular tilings

Vertex-transitivity means that for every pair of vertices there is a symmetry operation

mapping the first vertex to the second.

If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while thecondition that the tiling is edge-to-edge is kept, there are eight additional tilings possible,

known asArchimedean, uniform orsemiregulartilings. Note that there are two mirror

image (enantiomorphic orchiral) forms of 34.6 (snub hexagonal) tiling, both of which are

shown in the following table. All other regular and semiregular tilings are achiral.

34.6

Snub hexagonal tiling

34.6

Snub hexagonal tilingreflection

3.6.3.6

Trihexagona
http://en.wikipedia.org/wiki/Triangular_tilinghttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Vertex-transitivehttp://en.wikipedia.org/wiki/Uniform_tessellationhttp://en.wikipedia.org/wiki/Mirror_imagehttp://en.wikipedia.org/wiki/Mirror_imagehttp://en.wikipedia.org/wiki/Chirality_(mathematics)http://en.wikipedia.org/wiki/Snub_hexagonal_tilinghttp://en.wikipedia.org/wiki/Snub_hexagonal_tilinghttp://en.wikipedia.org/wiki/Reflectionhttp://en.wikipedia.org/wiki/Reflectionhttp://en.wikipedia.org/wiki/Trihexagonal_tilinghttp://en.wikipedia.org/wiki/File:Tiling_Semiregular_3-6-3-6_Trihexagonal.svghttp://en.wikipedia.org/wiki/File:Tiling_Semiregular_3-3-3-3-6_Snub_Hexagonal_Mirror.svghttp://en.wikipedia.org/wiki/File:Tiling_Semiregular_3-3-3-3-6_Snub_Hexagonal.svghttp://en.wikipedia.org/wiki/File:Tiling_Regular_6-3_Hexagonal.svghttp://en.wikipedia.org/wiki/File:Tiling_Regular_4-4_Square.svghttp://en.wikipedia.org/wiki/File:Tiling_Regular_3-6_Triangular.svghttp://en.wikipedia.org/wiki/Triangular_tilinghttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Vertex-transitivehttp://en.wikipedia.org/wiki/Uniform_tessellationhttp://en.wikipedia.org/wiki/Mirror_imagehttp://en.wikipedia.org/wiki/Mirror_imagehttp://en.wikipedia.org/wiki/Chirality_(mathematics)http://en.wikipedia.org/wiki/Snub_hexagonal_tilinghttp://en.wikipedia.org/wiki/Snub_hexagonal_tilinghttp://en.wikipedia.org/wiki/Reflectionhttp://en.wikipedia.org/wiki/Trihexagonal_tiling


3/7


4/7

species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding

twenty-one types of vertex. Only eleven of these can occur in a uniform tiling of regular

polygons. In particular, if three polygons meet at a vertex and one has an odd number ofsides, the other two polygons must be the same size. If they are not, they would have to

alternate around the first polygon, which is impossible if its number of sides is odd.

With 3 polygons at a vertex:

3.7.42 (cannot appear in any tiling of regular polygons)




3.122 - semi-regular, truncated hexagonal tiling


4.6.12 - semi-regular, great rhombitrihexagonal tiling

4.82 - semi-regular, truncated square tiling

5

2

.10 (cannot appear in any tiling of regular polygons) 63 - regular,hexagonal tiling


32.4.12 - not uniform, has two different types of vertices 32.4.12 and 36

3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4

32.62 - not uniform, occurs in two patterns with vertices 32.62/36 and 32.62/3.6.3.6.

3.6.3.6 - semi-regular, trihexagonal tiling

44 - regular,square tiling

3.42.6 - not uniform, has vertices 3.42.6 and 3.6.3.6.

3.4.6.4 - semi-regular, small rhombitrihexagonal tiling


34.6 -snub hexagonal tiling 33.42 - semi-regular, Elongated triangular tiling

32.4.3.4 - semi-regular, Snub square tiling


36 - regular,Triangular tiling

Other edge-to-edge tilings

Any number of non-uniform (sometimes called demiregular) edge-to-edge tilings by

regular polygons may be drawn. Here are four examples:
http://en.wikipedia.org/wiki/Truncated_hexagonal_tilinghttp://en.wikipedia.org/wiki/Great_rhombitrihexagonal_tilinghttp://en.wikipedia.org/wiki/Truncated_square_tilinghttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Uniform_polyhedronhttp://en.wikipedia.org/wiki/Trihexagonal_tilinghttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/Small_rhombitrihexagonal_tilinghttp://en.wikipedia.org/wiki/Snub_hexagonal_tilinghttp://en.wikipedia.org/wiki/Snub_hexagonal_tilinghttp://en.wikipedia.org/wiki/Elongated_triangular_tilinghttp://en.wikipedia.org/wiki/Snub_square_tilinghttp://en.wikipedia.org/wiki/Triangular_tilinghttp://en.wikipedia.org/wiki/Triangular_tilinghttp://en.wikipedia.org/wiki/Truncated_hexagonal_tilinghttp://en.wikipedia.org/wiki/Great_rhombitrihexagonal_tilinghttp://en.wikipedia.org/wiki/Truncated_square_tilinghttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Uniform_polyhedronhttp://en.wikipedia.org/wiki/Trihexagonal_tilinghttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/Small_rhombitrihexagonal_tilinghttp://en.wikipedia.org/wiki/Snub_hexagonal_tilinghttp://en.wikipedia.org/wiki/Elongated_triangular_tilinghttp://en.wikipedia.org/wiki/Snub_square_tilinghttp://en.wikipedia.org/wiki/Triangular_tiling


5/7

32.62 and 36 32.62 and 3.6.3.6

32.4.12 and 36 3.42.6 and 3.6.3.6

Such periodic tilings may be classified by the number oforbits of vertices, edges and tiles.If there are n orbits of vertices, a tiling is known as n-uniform orn-isogonal; if there are n

orbits of tiles, as n-isohedral; if there are n orbits of edges, as n-isotoxal. The examples

above are four of the twenty 2-uniform tilings. Chavey lists all those edge-to-edge tilingsby regular polygons which are at most 3-uniform, 3-isohedral or 3-isotoxal.

Tilings that are not edge-to-edge
http://en.wikipedia.org/wiki/Group_actionhttp://en.wikipedia.org/wiki/File:Dem3446bc.gifhttp://en.wikipedia.org/wiki/File:Dem3343tbc.gifhttp://en.wikipedia.org/wiki/File:Dem3366rbc.gifhttp://en.wikipedia.org/wiki/File:Dem3366bc.pnghttp://en.wikipedia.org/wiki/Group_action


6/7

Regular polygons can also form plane tilings that are not edge-to-edge. Such tilings may

also be known as uniform if they are vertex-transitive; there are eight families of such

uniform tilings, each family having a real-valued parameter determining the overlapbetween sides of adjacent tiles or the ratio between the edge lengths of different tiles.

The hyperbolic plane

Main article: Uniform tilings in hyperbolic plane

These tessellations are also related to regular and semiregular polyhedra and tessellations ofthe hyperbolic plane. Semiregular polyhedra are made from regular polygon faces, but their

angles at a point add to less than 360 degrees. Regular polygons in hyperbolic geometry

have angles smaller than they do in the plane. In both these cases, that the arrangement ofpolygons is the same at each vertex does not mean that the polyhedron or tiling is vertex-

transitive.

Some regular tilings of the hyperbolic plane (Using Poincar disc model projection)

See also

List of uniform tilings Wythoff symbol

Tessellation

Wallpaper group

Regular polyhedron (the Platonic solids)

Semiregular polyhedron (including the Archimedean solids) Hyperbolic geometry

Penrose tiling
http://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_planehttp://en.wikipedia.org/wiki/Hyperbolic_geometryhttp://en.wikipedia.org/wiki/List_of_uniform_tilingshttp://en.wikipedia.org/wiki/Wythoff_symbolhttp://en.wikipedia.org/wiki/Tessellationhttp://en.wikipedia.org/wiki/Wallpaper_grouphttp://en.wikipedia.org/wiki/Regular_polyhedronhttp://en.wikipedia.org/wiki/Platonic_solidhttp://en.wikipedia.org/wiki/Semiregular_polyhedronhttp://en.wikipedia.org/wiki/Archimedean_solidhttp://en.wikipedia.org/wiki/Hyperbolic_geometryhttp://en.wikipedia.org/wiki/Penrose_tilinghttp://en.wikipedia.org/wiki/File:Hyperbolic_tiling_snub_3-7.pnghttp://en.wikipedia.org/wiki/File:Hyperbolic_tiling_omnitruncated_3-7.pnghttp://en.wikipedia.org/wiki/File:Hyperbolic_tiling_runcinated_3-7.pnghttp://en.wikipedia.org/wiki/File:Hyperbolic_tiling_truncated_7-3.pnghttp://en.wikipedia.org/wiki/File:Hyperbolic_tiling_rectified_3-7.pnghttp://en.wikipedia.org/wiki/File:Hyperbolic_tiling_truncated_3-7.pnghttp://en.wikipedia.org/wiki/File:Hyperbolic_tiling_7-3.pnghttp://en.wikipedia.org/wiki/File:Hyperbolic_tiling_3-7.pnghttp://en.wikipedia.org/wiki/File:Hyperspace_tiling_5-4.pnghttp://en.wikipedia.org/wiki/File:Hyperspace_tiling_4-5.pnghttp://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_planehttp://en.wikipedia.org/wiki/Hyperbolic_geometryhttp://en.wikipedia.org/wiki/List_of_uniform_tilingshttp://en.wikipedia.org/wiki/Wythoff_symbolhttp://en.wikipedia.org/wiki/Tessellationhttp://en.wikipedia.org/wiki/Wallpaper_grouphttp://en.wikipedia.org/wiki/Regular_polyhedronhttp://en.wikipedia.org/wiki/Platonic_solidhttp://en.wikipedia.org/wiki/Semiregular_polyhedronhttp://en.wikipedia.org/wiki/Archimedean_solidhttp://en.wikipedia.org/wiki/Hyperbolic_geometryhttp://en.wikipedia.org/wiki/Penrose_tiling


7/7

References

Grnbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman

and Company. ISBN 0-7167-1193-1. D. Chavey (1989). "Tilings by Regular PolygonsII: A Catalog of Tilings".

Computers & Mathematics with Applications17: 147165. doi:10.1016/0898-1221(89)90156-9.

D. M. Y. Sommerville,An Introduction to the Geometry ofn Dimensions. New

York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X:

The Regular Polytopes
http://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaumhttp://en.wikipedia.org/w/index.php?title=G.C._Shephard&action=edit&redlink=1http://en.wikipedia.org/wiki/Special:BookSources/0716711931http://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1016%2F0898-1221(89)90156-9http://dx.doi.org/10.1016%2F0898-1221(89)90156-9http://en.wikipedia.org/wiki/Duncan_MacLaren_Young_Sommervillehttp://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaumhttp://en.wikipedia.org/w/index.php?title=G.C._Shephard&action=edit&redlink=1http://en.wikipedia.org/wiki/Special:BookSources/0716711931http://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1016%2F0898-1221(89)90156-9http://dx.doi.org/10.1016%2F0898-1221(89)90156-9http://en.wikipedia.org/wiki/Duncan_MacLaren_Young_Sommerville

Documents

MOSAICOSREGULARES