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Mathematics book
tiling theory: colored patterns and tilings, polygonal tilings, aperiodic tilings, Wang tiles, and tilings with unusual kinds of tiles. Each chapter open
Tilings_and_patterns
Non-periodic tiling of the plane
symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger
Penrose_tiling
uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the
List of Euclidean uniform tilings
List_of_Euclidean_uniform_tilings
Subdivision of the plane into polygons that are all regular
tilings for n = 6; and 7 such tilings for n = 7. Below is an example of a 3-unifom tiling: There are twenty (20) 2-uniform tilings of the Euclidean plane
Euclidean tilings by convex regular polygons
Euclidean_tilings_by_convex_regular_polygons
Zigzagging chevron pattern
The herringbone pattern is an arrangement of rectangles used for floor tilings and road pavement, so named for a fancied resemblance to the bones of a
Herringbone_pattern
Form of plane tiling without repeats at scale
non-periodically. The tilings produced by one of these sets of prototiles may be called aperiodic tilings. The Penrose tilings are a well-known example
Aperiodic_tiling
Regular non-convex polygon
in a tessellation pattern. In his 1619 work Harmonice Mundi, among periodic tilings, Johannes Kepler includes nonperiodic tilings, like that with three
Star_polygon
Non-periodic tiling of the plane
tilings, which were found by Robinson in 1971. The A1 tiles are one of five sets of tiles discovered by Ammann and described in Tilings and patterns.
Ammann_A1_tilings
Regular tiling of a two-dimensional space
Monohedral tilings by convex polygons Tilings and patterns, from list of 107 isohedral tilings, pp. 473–481 Tilings and patterns, uniform tilings that are
Hexagonal_tiling
Regularity in sensory qualia or abstract ideas
spirals, meanders, waves, foams, tilings, cracks, and those created by symmetries of rotation and reflection. Patterns have an underlying mathematical
Pattern
Tiling of a plane by regular hexagons and equilateral triangles
Cyclotruncated simplectic honeycomb List of uniform tilings Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 978-0-7167-1193-3.
Trihexagonal_tiling
Semiregular tiling of the Euclidean plane
comparative overlay of this tiling and its dual) Tilings and patterns Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman
Rhombitrihexagonal_tiling
Polytope or tiling with one type of edge
Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (6.4 Isotoxal tilings, pp. 309–321) Coxeter, Harold Scott
Isotoxal_figure
Natural number
Krotenheerdt tilings, with no other such k-uniform tilings for k > 7, and it is also the only k for which the count of Krotenheerdt tilings agrees with
7
Regular tiling of the plane
triangular tiling) List of uniform tilings Simplectic honeycomb Tilings of regular polygons Triangular tiling honeycomb Tilings and patterns, p.102-107
Triangular_tiling
Regular tiling of the Euclidean plane
Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. p. 21, 29. Lorenzo, Sadun (2008). Topology of Tiling Spaces. American Mathematical
Square_tiling
Non-periodic tiling of the plane
and described in Tilings and patterns. The Ammann–Beenker tilings have many properties similar to the more famous Penrose tilings: They are nonperiodic,
Ammann–Beenker_tiling
Semiregular tiling of the plane
Klitzing, Richard. "2D Euclidean tilings s4s4s - snasquat - O10". Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1
Snub_square_tiling
Semiregular tiling of the Euclidean plane
Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65) Williams,
Snub_trihexagonal_tiling
Yugoslav American mathematician (1929-2018)
subject. His monograph Tilings and patterns, coauthored with G. C. Shephard, helped to rejuvenate interest in this classic field, and has proved popular with
Branko_Grünbaum
Vertex-transitive tiling of the plane by regular polygons
Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra; these can be considered uniform tilings of the sphere
Uniform_tiling
include 3 regular tilings, and 8 semiregular tilings. A 1-uniform tiling can be defined by its vertex configuration. Higher k-uniform tilings are listed by
List_of_k-uniform_tilings
Covering by shapes without overlaps or gaps
and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings
Tessellation
from spherical tilings to Euclidean tilings to hyperbolic tilings. Hyperbolic tilings can also be divided between compact, paracompact and divergent cases
Uniform tiling symmetry mutations
Uniform_tiling_symmetry_mutations
Square tiles with a color on each edge
(1987), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1193-1. Steven Dutch's page including many pictures of aperiodic tilings Animated
Wang_tile
Semiregular tiling of a plane
hexagonal tiling). Tilings of regular polygons List of uniform tilings Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers &
Truncated_hexagonal_tiling
Semiregular tiling of the plane
Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65) Williams,
Elongated_triangular_tiling
Tiling of the plane with 60° rhombi
monohedral tilings it is denoted [3.6.3.6]. It is also one of 56 possible isohedral tilings by quadrilaterals, and one of only eight tilings of the plane
Rhombille_tiling
Nine-pointed star polygon
Scott, A Greek-English Lexicon, on Perseus. Grünbaum, B. and G. C. Shephard; Tilings and patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1
Enneagram_(geometry)
Uniform tiling of the plane with regular polygons
the 3-4-3-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, and dodecagons, arranged
3-4-3-12_tiling
Manufactured pieces for covering surfaces
Latin tessella, 'tile') and such a tiling is called a tessellation. Geometric patterns of some Islamic polychrome decorative tilings are rather complicated
Tile
Semiregular tiling
Uniform tiling 4-8-8 (truncated square tiling). Euclidean tilings by convex regular polygons List of uniform tilings Percolation threshold Tilings of regular
Truncated_square_tiling
Polygon able to tessellate edge-to-edge, without rotation
Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. list of 107 isohedral tilings, p. 473-481 Fedorov's Five
Parallelogon
Notation for a polyhedron's vertex figure
6 (60) Regular tilings: Hexagonal tiling: 6.6.6 Semiregular tilings: Truncated hexagonal tiling: 3.12.12 Truncated trihexagonal tiling: 4.6.12 Truncated
Vertex_configuration
Geometric shape formed from seven squares
05.002. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman and Company. ISBN 0-7167-1193-1. "Polyominoes: Even more
Heptomino
American mathematician (1921–2006)
several new aperiodic tilings, each among the simplest known examples of aperiodic sets of tiles. He also showed how to generate tilings using lines in the
Robert_Ammann
Tiling of the plane by pentagons
and type 4 tiles, and 3-isohedral tilings, all edge-to-edge, by special cases of type 1 tiles. There is no upper bound on k for k-isohedral tilings by
Pentagonal_tiling
Star polygon
Science Focus Magazine. Retrieved 1 March 2023. Grünbaum, B. and G.C. Shephard; Tilings and patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1
Octagram
Geometric shape formed from squares
collections of cells and stack polyominoes”, Journal of Algebra 357 (2012), 279–303. Grünbaum, Branko; Shephard, G.C. (1987). Tilings and Patterns. New York: W
Polyomino
Symmetric subdivision in hyperbolic geometry
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}
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
Mathematical problem
the Fibonacci tiling by 110 times and replacing one of the 110-squares with Duijvestijn's perfects the tiling. In Tilings and patterns, published in 1987
Squaring_the_square
intersection of two truncated square tilings with offset positions. And its appearance is similar to a truncated square tiling, except only half of the vertices
Chamfered_square_tiling
Euclidean tilings using 2 or more regular polygon faces
Grünbaum and Shephard enumerated the full list of 20 2-uniform tilings in Tilings and patterns, 1987: Ghyka lists 10 of them with 2 or 3 vertex types, calling
Demiregular_tiling
Geoffrey C. (1986), Tilings and Patterns, New York: W. H. Freeman, ISBN 978-0-7167-1194-0, according to Dutch, Steven (2003), Aperiodic Tilings, University of
List of aperiodic sets of tiles
List_of_aperiodic_sets_of_tiles
Tiling of the plane by pentagons
pentagons can form this pattern, belonging to two of the 15 families of convex pentagons that can tile the plane. Their tilings have varying symmetries;
Cairo_pentagonal_tiling
Mathematical spiral tiling
Shephard in the 1970s. A spiral tiling is depicted on the cover of Grünbaum and Shephard's 1987 book Tilings and patterns. Wikimedia Commons has media related
Voderberg_tiling
Magical talisman
particularly among Wiccans. The term pentacle is used in Tilings and patterns by Branko Grünbaum and G. C. Shephard to indicate a five-pointed star composed
Pentacle
Uniform Tiling
3-4-6-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, hexagons and dodecagons
3-4-6-12_tiling
Geometric pattern characteristic of Muslim art
114 patterns including coloured designs for girih tilings and muqarnas quarter or semidomes. The mathematical properties of the decorative tile and stucco
Islamic_geometric_patterns
Natural number
C. (1987). "Section 2.9 Archimedean and uniform colorings". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 102–107. doi:10.2307/2323457
32_(number)
Set of tile shapes that can create nonrepeating patterns
of the tiles in the set can be fitted together to cover the entire space. A given set of tiles might admit periodic tilings — that is, tilings that remain
Aperiodic_set_of_prototiles
Uniform tiling of the Euclidean plane
trihexagonal tiling). Tilings of regular polygons List of uniform tilings Conway, 2008, Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288
Truncated_trihexagonal_tiling
On lattices and sphere packing in Euclidean space
ISSN 0025-5831, S2CID 119472023. Grünbaum, Branko; Shepherd, G. C. (2016), Tilings and Patterns (2nd ed.), Dover Publications, p. 517. Reinhardt, Karl (1928). Zur
Hilbert's_eighteenth_problem
Polytope or tiling whose vertices are identical
G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1. (p. 33 k-isogonal tiling, p. 65 k-uniform tilings) Weisstein, Eric
Isogonal_figure
Mathematical term in geometry
Pbk. (1999), ISBN 0-521-66405-5. p. 175 Grünbaum, B. and G.C. Shephard; Tilings and patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1
Polygram_(geometry)
Symmetry with three or more colours
Grünbaum and Shephard's Tilings and patterns (1987), by Senechal (1990) and by Thomas (2012). Late 1950s M.C. Escher's artworks based on dichromatic and polychromatic
Polychromatic_symmetry
Geometric shape formed from five squares
earliest tilings of rectangles with a complete set of pentominoes appeared in the Problemist Fairy Chess Supplement in 1935, and further tiling problems
Pentomino
Five tiles used in Islamic decorative art
Peter J. Lu and Paul J. Steinhardt suggested that girih tilings possess properties consistent with self-similar fractal quasicrystalline tilings such as Penrose
Girih_tile
Uniform tiling of the plane using regular polygons
the prismatic pentagonal tiling and Cairo pentagonal tilings. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1
33344-33434_tiling
Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (6.4 Isotoxal tilings, 309–321) Coxeter, Harold Scott
List of isotoxal polyhedra and tilings
List_of_isotoxal_polyhedra_and_tilings
Tiling by squares of two sizes
each tile is a regular polygon and in which every vertex can be mapped to every other vertex by a symmetry of the tiling. Usually, uniform tilings additionally
Pythagorean_tiling
Convex polygon which can tile the plane by itself
book, Tilings and patterns, Branko Grünbaum calls the vertex-uniform tilings Archimedean in parallel to the Archimedean solids. Their dual tilings are called
Planigon
Mumford, Caroline Series, and David Wright Regular Polytopes — H. S. M. Coxeter Tilings and patterns — Branko Grünbaum and G. C. Shephard Topology — James
List_of_mathematics_books
Generalisation of dice with identical faces
"Introductory Tiling Theory for Computer Graphics" Archived 2022-12-08 at the Wayback Machine, 2009, Chapter 5: "Isohedral Tilings", p. 35. Tilings and patterns, p
Isohedral_figure
Star polygon with 7 sides
Publishing House. ISBN 0835600025. Bibliography Grünbaum, B. and G.C. Shephard; Tilings and patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1
Heptagram
Star polygon with 12 vertices
Weisstein, Eric W. "Dodecagram". MathWorld. Grünbaum, B. and G.C. Shephard; Tilings and patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1
Dodecagram
Number, approximately 1.618
Gähler, F. "Robinson Triangle". Tilings Encyclopedia. Clason, Robert G (1994). "A family of golden triangle tile patterns". The Mathematical Gazette. 78
Golden_ratio
3} tilings while the {m, m/2} dual tilings are facetings of the {3, m} tilings and greatenings of the {m, 3} tilings. The patterns {m/2, m} and {m, m/2}
List_of_regular_polytopes
Tiling forced to use inequivalent tile placements
translation) and another with isohedral number 9 (occurring in 36 orbits under translation).[1] Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New
Anisohedral_tiling
British mathematician, artist and author
& Sciences (ARSC) and Mathematical Sciences (MASC). He does research in the Geometry of Tilings and Patterns, a branch of Convex and Discrete Geometry
Edmund_Harriss
Mathematics book
material, and of the use of some non-standard terminology. In 1987, Branko Grünbaum and Geoffrey Colin Shephard writing in Tilings and patterns criticised
Geometric_symmetry_(book)
1971 mathematics book by Arthur L. Loeb
Branko Grünbaum and G.C. Shephard in their book Tilings and patterns gave an assessment of previous work in the field. Commenting on Color and Symmetry they
Color_and_Symmetry
Classification of a two-dimensional repetitive pattern
repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tiles, and
Wallpaper_group
fundamental domain, colored by even and odd reflections. Selected tilings created by the Wythoff construction are given below. Tilings are shown as polyhedra. Some
Lists of uniform tilings on the sphere, plane, and hyperbolic plane
Lists_of_uniform_tilings_on_the_sphere,_plane,_and_hyperbolic_plane
Five-pointed star polygon
OCLC 65081051. Grünbaum, Branko; Shephard, Geoffrey Colin (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3. Grünbaum, Branko
Pentagram
Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65) Williams,
Tetrakis_square_tiling
Geometric patterns in Islamic architecture
Scroll explicitly shows girih patterns together with the tilings used to create them. A set of tiles consisting of a dart and a kite shape can be used to
Girih
Chinese tile-based game
rules are highly pattern-based. The rulebook contains 81 combinations, based on patterns and scoring elements popular in classic and modern regional Chinese
Mahjong
Ordered chemical structure with no repeating pattern
discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane. These tilings displayed instances of
Quasicrystal
Natural number
Shephard, G. C. (1987). "Section 2.1: Regular and uniform tilings". Tilings and Patterns. New York: W. H. Freeman and Company. p. 59. doi:10.2307/2323457. ISBN 0-7167-1193-1
12_(number)
Visible regularity of form found in the natural world
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled
Patterns_in_nature
1964 book by A.V. Shubnikov and N.V. Belov
Branko Grünbaum and G.C. Shephard in their book Tilings and patterns the work of the Russian color symmetry school led by A.V. Shubnikov and N.V. Belov was
Colored_Symmetry
Square tiles used in graphic design
visualization and graphic design, Truchet tiles are square tiles decorated with patterns that are not rotationally symmetric. When placed in a square tiling of the
Truchet_tiling
Two-colour symmetry (examples, history and dimensional counts)
Grünbaum and Shephard's Tilings and patterns (1987), and Brückler and Stilinović (2024) Late 1950s M.C. Escher's artworks based on dichromatic and polychromatic
Dichromatic_symmetry
1974 book by A.V. Shubnikov and V.A. Koptsik
G.C. Shephard in their book Tilings and patterns the work of the Russian color symmetry school led by A.V. Shubnikov and N.V. Belov was put into its proper
Symmetry_in_Science_and_Art
Semiregular tiling of the hyperbolic plane
below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. From a Wythoff construction
Truncated triheptagonal tiling
Truncated_triheptagonal_tiling
Regular tiling in geometry
diagram , and continues with larger tilings as n increases toward infinity. Wikimedia Commons has media related to Order-4 apeirogonal tiling. Tilings of regular
Order-4_apeirogonal_tiling
Pattern of intersecting vertical and horizontal stripes
cotton and show the prominence of the check pattern in traditional dress. Check and its variant patterns have been commonly employed as fabric and textile
Check_(pattern)
Tile-based rummy game, similar to mahjong
three types of tiles for each of the 21 patterns, with six tiles per pattern in total: Digging Flowers tile types 3 "plain" (白皮) pattern tiles with no border
Digging_Flowers
Book by mathematician Doris Schattschneider published in 1990
notebook patterns, answering the question "how did he do it?", and relating the patterns to his prints. For the person interested in tilings and patterns, Visions
M. C. Escher: Visions of Symmetry
M._C._Escher:_Visions_of_Symmetry
Tiles used in mahjong game
patterns and colours are similar to the Canton tiles. Taiwan style. The lines of the one bamboo are simpler, black paint is used instead of blue, and
Mahjong_tiles
American amateur mathematician (1923–2017)
American amateur mathematician most famous for her discoveries of pentagonal tilings. Rice was born February 16, 1923, in St. Petersburg, Florida. Marjorie
Marjorie_Rice
On surrounding polygons by layers of copies
Parkettierungsproblem. Cologne and Opladen: Westdeutscher Verlag. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. Eppstein,
Heesch's_problem
Uniform tiling of the hyperbolic plane
woodcut they appear to be smooth hypercycles. Circle Limit III Square tiling Uniform tilings in hyperbolic plane List of regular polytopes John Horton Conway
Alternated_octagonal_tiling
Six-pointed star polygon
Its Origin and Usage 4th ed. Toronto: The Free Press 777, 2001. ISBN 0-9689383-0-2 Grünbaum, B. and G. C. Shephard; Tilings and patterns, New York: W
Hexagram
Semiregular tiling of the hyperbolic plane
Uniform tiling 6-6-7. Triangular tiling Order-3 heptagonal tiling Order-7 triangular tiling Tilings of regular polygons List of uniform tilings HOW TO
Truncated order-7 triangular tiling
Truncated_order-7_triangular_tiling
Ceramic tile of different colours of clay
Encaustic or inlaid tiles are ceramic tiles in which the pattern or figure on the surface is not a product of the glaze but of different colors of clay
Encaustic_tile
Timurid dynasty scroll
indirectly and directly by architects to create the tiling patterns in many mosques around the world, including the quasicrystal Girih tilings from Darb-e
Topkapı_Scroll
Tiling of the hyperbolic plane
two-dimensional family of symmetries. There exist binary tilings with tiles of arbitrarily small area. Binary tilings were first studied mathematically in 1974 by
Binary_tiling
Tile used to keep out rain
As a result of this, flat tiles require more tiles to cover a certain area than other patterns of similar size. These tiles commonly feature a squared
Roof_tiles
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