Plane tilings by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in Harmonices Mundi. In mathematics, a plane is the fundamental two-dimensional object. ... A tessellated plane A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ... Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German mathematician, astronomer and astrologer of famed brilliance. ... 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. The symmetry group of an object (e. ... This article is about the mathematical concept. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... An edge-to-edge tiling is a type of tiling where each tile is a polygon and adjacent tiles only share full sides, i. ... In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In plane geometry, a square is a polygon with four equal sides and equal angles. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ...

3.3.3.3.3.3 Triangular tiling

4.4.4.4 Square tiling

6.6.6 Hexagonal tiling

Image File history File links Tile333333bc. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... Image File history File links Tile4444bc. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... Image File history File links Tile666bc. ... 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.

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. ...

3.3.3.3.6 Snub hexagonal tiling	3.6.3.6 Trihexagonal tiling
3.3.3.4.4 Prismatic trisquare tiling	3.3.4.3.4 Snub square tiling
3.4.6.4 Small rhombitrihexagonal tiling	4.8.8 Truncated square tiling
3.12.12 Truncated hexagonal tiling	4.6.12 Great rhombitrihexagonal tiling

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 Tile33336bc. ... In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Example semiregular tiling - image made by me for Wikipedia File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Tile33344bc. ... In geometry, the prismatic trisquare tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Tile33434bc. ... In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Tile3464bc. ... In geometry, the Small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Tile488bc. ... In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Tile3ttbc. ... In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Tile46tbc. ... 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 fifteen of these can occur in any tiling of regular polygons. For example, 3.7.42 cannot occur because it is the only type involving a regular heptagon, meaning that triangles and 42-gons would need to alternate around the heptagon, which is impossible since it has an odd number of sides. In geometry, an internal angle is an angle that 2 sides of a polygon form by touching. ... A heptagon is a plane figure with seven sides and seven angles. ...

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)
• 3.12.12
• 4.5.20 (cannot appear in any tiling of regular polygons)
• 4.6.12
• 4.8.8 (can only appear in one tiling of regular polygons, which is a uniform tiling)
• 5.5.10 (cannot appear in any tiling of regular polygons)
• 6.6.6

With 4 polygons at a vertex:

• 3.3.4.12 or 3.4.3.12
• 3.3.6.6 or 3.6.3.6
• 4.4.4.4
• 3.4.4.6 or 3.4.6.4

With 5 polygons at a vertex:

• 3.3.3.3.6
• 3.3.3.4.4 or 3.3.4.3.4

With 6 polygons at a vertex:

• 3.3.3.3.3.3

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:

3.3.6.6 & 3.3.3.3.3.3	3.3.6.6 & 3.6.3.6
3.3.4.12 & 3.3.3.3.3.3	3.4.4.6 & 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 Dem3366bc. ... Image File history File links Dem3366rbc. ... Image File history File links Dem3343tbc. ... Image File history File links Dem3446bc. ... This article is about the mathematical concept. ...

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. A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ...

See also

• List of uniform planar tilings
• Tessellation
• Regular polyhedra
  • Platonic solid
• Semiregular polyhedra
  • Archimedean solid
• Hyperbolic geometry

This table shows the 11 uniform tilings of the plane, and their dual tilings. ... A tessellated plane A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. ... In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ... 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 in 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. ... A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ...

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

• http://www.uwgb.edu/dutchs/symmetry/uniftil.htm
• http://www.ics.uci.edu/~eppstein/junkyard/tiling.html
  • http://www.uwgb.edu/dutchs/symmetry/archpol.htm
• http://www.ics.uci.edu/~eppstein/junkyard/hypertile.html
  • http://www.hadron.org/~hatch/HyperbolicTesselations/
  • http://aleph0.clarku.edu/~djoyce/poincare/poincare.html
  • http://members.lycos.co.uk/tiletris/