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Plane tilings by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in Harmonices Mundi. Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ...
A tessellated plane seen in street pavement. ...
A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ...
Johannes Kepler (December 27, 1571 – November 15, 1630) was a German Lutheran mathematician, astronomer and astrologer, and a key figure in the 17th century astronomical revolution. ...
Harmonices Mundi (1619) is a book by Johannes Kepler. ...
Regular tilings
Following Grünbaum and Shephard (section 1.3), a tiling is said to be regular if the symmetry group of the tiling acts transitively on the flags of the tiling, where a flag is a triple 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 by congruent regular polygons. There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations. Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ...
The symmetry group of an object (e. ...
In mathematics, a symmetry group describes all symmetries of objects. ...
In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ...
An edge-to-edge tiling is a type of tiling where each tile is a polygon and adjacent tiles only share full sides, i. ...
An example of congruence. ...
For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
For other uses, see Square. ...
A regular hexagon. ...
Image File history File links Tile_3,6. ...
In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ...
Image File history File links Tile_4,4. ...
In geometry, the Square tiling is a regular tiling of the Euclidean plane. ...
Image File history File links Tile_6,3. ...
In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ...
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. In mathematics, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : G → G such that f ( v1 ) = v2. ...
If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as Archimedean, uniform or semiregular tilings. In mathematics, a uniform tessellation is a tessellation of a d-dimensional space, or a (hyper)surface, such that all its vertices are identical, i. ...
Grünbaum and Shephard distinguish the description of these tilings as Archimedean as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as uniform as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. Image File history File links Tile_33336. ...
In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. ...
Image File history File links Tile_3636. ...
In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
Image File history File links Tile_33344. ...
In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. ...
Image File history File links Tile_33434. ...
In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. ...
Image File history File links Tile_3464. ...
In geometry, the Small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
Image File history File links Tile_488. ...
In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ...
Image File history File links Tile_3bb. ...
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. ...
Image File history File links Tile_46b. ...
In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
Combinations of regular polygons that can meet at a vertex The internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular n-gon has internal angle degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a 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 of sides, 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. External angles law In geometry, an interior angle (or internal angle) is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon. ...
With 3 polygons at a vertex:
- 3.7.42 (cannot appear in any tiling of regular polygons)
- 3.8.24 (cannot appear in any tiling of regular polygons)
- 3.9.18 (cannot appear in any tiling of regular polygons)
- 3.10.15 (cannot appear in any tiling of regular polygons)
- 3.122 - semi-regular, truncated hexagonal tiling
- 4.5.20 (cannot appear in any tiling of regular polygons)
- 4.6.12 - semi-regular, great rhombitrihexagonal tiling
- 4.82 - semi-regular, truncated square tiling
- 52.10 (cannot appear in any tiling of regular polygons)
- 63 - regular, hexagonal tiling
With 4 polygons at a vertex: In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. ...
In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ...
In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ...
- 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, occures 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
With 5 polygons at a vertex: In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
In geometry, the Square tiling is a regular tiling of the Euclidean plane. ...
In geometry, the Small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
With 6 polygons at a vertex: In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. ...
In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. ...
In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. ...
In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ...
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:
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 of orbits of vertices, edges and tiles. If there are n orbits of vertices, a tiling is known as n-uniform or n-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 tilings by regular polygons which are at most 3-uniform, 3-isohedral or 3-isotoxal. Image File history File links Dem3366rbc. ...
Image File history File links Dem3343tbc. ...
Image File history File links Dem3446bc. ...
In mathematics, a symmetry group describes all symmetries of objects. ...
Tilings that are not edge-to-edge 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 overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles.
Beyond the plane These tessellations are also related to regular and semiregular polyhedra and tessellations of the 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 of polygons is the same at each vertex does not mean that the polyhedron or tiling is vertex-transitive. Lines through a given point P and hyperparallel to line l. ...
Some regular tilings of the hyperbolic plane (Using Poincaré disc model projection)
Image File history File links Download high-resolution version (947x939, 80 KB) View of regular hyperbolic tiling omnitruncated {3,7} generated by software: [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks I, the creator of this work, hereby release it into the public domain. ...
See also This table shows the 11 convex uniform tilings of the Euclidean plane, and their dual tilings. ...
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ...
A tessellated plane seen in street pavement. ...
Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ...
In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ...
In geometry, a Platonic solid is a convex regular polyhedron. ...
A semiregular polyhedron is a geometric shape constructed from a finite number of regular polygon faces with every face edge shared by one other face, and with every vertex containing the same sequence of faces, and, moreover, for every two vertices there is an isometry mapping one into the other. ...
In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...
Lines through a given point P and hyperparallel to line l. ...
A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ...
References - Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.
- D. Chavey (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications 17: 147–165.
Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ...
External links Euclidean and general tiling links: Hyperbolic tiling links: For the Manfred Mann album, see 2006 (album). ...
September 9 is the 252nd day of the year (253rd in leap years) in the Gregorian calendar. ...
For the Manfred Mann album, see 2006 (album). ...
September 9 is the 252nd day of the year (253rd in leap years) in the Gregorian calendar. ...
Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ...
For the Manfred Mann album, see 2006 (album). ...
September 9 is the 252nd day of the year (253rd in leap years) in the Gregorian calendar. ...
- Hatch, Don. Hyperbolic Planar Tessellations. Retrieved on 2006-09-09.
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