Pentagonal tiling
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the intern
Conway criterion
In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of
33344-33434 tiling
In geometry of the Euclidean plane, a 33344-33434 tiling is one of two of 20 2-uniform tilings of the Euclidean plane by regular polygons. They contains regular triangle and square faces, arranged in
Corner-point grid
In geometry, a corner-point grid is a tessellation of a Euclidean 3D volume where the base cell has 6 faces (hexahedron). A set of straight lines defined by their end points define the pillars of the
List of k-uniform tilings
A k-uniform tiling is a tiling of tilings of the plane by convex regular polygons, connected edge-to-edge, with k types of vertices. The 1-uniform tiling include 3 regular tilings, and 8 semiregular t
Anisohedral tiling
In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalen
Prototile
In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation.
Regular grid
A regular grid is a tessellation of n-dimensional Euclidean space by congruent parallelotopes (e.g. bricks). Its opposite is irregular grid. Grids of this type appear on graph paper and may be used in
Tetrad (geometry puzzle)
In geometry, a tetrad is a set of four simply connected disjoint planar regions in the plane, each pair sharing a finite portion of common boundary. It was named by Michael R. W. Buckley in 1975 in th
Self-tiling tile set
A self-tiling tile set, or setiset, of order n is a set of n shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of n shapes. That is, the n shapes c
Hilbert's eighteenth problem
Hilbert's eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by mathematician David Hilbert. It asks three separate questions about lattices and sphere
Bravais lattice
In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite array of discrete points generated by a set of discrete translation operations described in three dimens
Algebra and Tiling
Algebra and Tiling: Homomorphisms in the Service of Geometry is a mathematics textbook on the use of group theory to answer questions about tessellations and higher dimensional honeycombs, partitions
Demiregular tiling
In geometry, the demiregular tilings are a set of Euclidean tessellations made from 2 or more regular polygon faces. Different authors have listed different sets of tilings. A more systematic approach
Euclidean tilings by convex regular polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi (Latin: The Harmony of
Spherical polyhedron
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of t
Edge tessellation
In geometry, an edge tessellation is a partition of the plane into non-overlapping polygons (a tessellation) with the property that the reflection of any of these polygons across any of its edges is a
Tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to
Squaring the square
Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.) The name was coined in a humorous a
Quasicrystal
A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. Wh
Substitution tiling
In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not ad
Hosohedron
In spherical geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläf
Tiling with rectangles
A tiling with rectangles is a tiling which uses rectangles as its parts. The domino tilings are tilings with rectangles of 1 × 2 sideratio. The tilings with straight polyominoes of shapes such as 1 ×
Holmium–magnesium–zinc quasicrystal
A holmium–magnesium–zinc (Ho–Mg–Zn) quasicrystal is a quasicrystal made of an alloy of the three metals holmium, magnesium and zinc that has the shape of a regular dodecahedron, a Platonic solid with
Haeckelites
Haeckelites are three-fold coordinated networks of carbon atoms generated by a periodic arrangement of pentagons, hexagons and heptagons. They were first proposed by Humberto and Mauricio Terrones and
3-4-6-12 tiling
In geometry of the Euclidean plane, the 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, arran
Voderberg tiling
The Voderberg tiling is a mathematical spiral tiling, invented in 1936 by mathematician (1911-1945). It is a monohedral tiling: it consists of only one shape that tessellates the plane with congruent
Quaquaversal tiling
The quaquaversal tiling is a nonperiodic tiling of the euclidean 3-space introduced by John Conway and Charles Radin. The basic solid tiles are half prisms arranged in a pattern that relies essentiall
Truchet tiles
In information 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 plane, they can form
Harmonious set
In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters. Equivalently, a suitably defined dual
Gilbert tessellation
In applied mathematics, a Gilbert tessellation or random crack network is a mathematical model for the formation of mudcracks, needle-like crystals, and similar structures. It is named after Edgar Gil
Rep-tile
In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathemat
Tiling puzzle
Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some ti
Darb-e Imam
The shrine of Darb-e Imam (Persian: امامزاده درب امام), located in the Dardasht quarter of Isfahan, Iran, is a funerary complex, with a cemetery, shrine structures, and courtyards belonging to differe
Keller's conjecture
In geometry, Keller's conjecture is the conjecture that in any tiling of n-dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire (n − 1)-dimensional face w
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangl
3-4-3-12 tiling
In geometry of the Euclidean plane, 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 in tw
Quasiperiodic tiling
A quasiperiodic tiling is a tiling of the plane that exhibits local periodicity under some transformations: every finite subset of its tiles reappears infinitely often throughout the tiling, but there
Girih tiles
Girih tiles are a set of five tiles that were used in the creation of Islamic geometric patterns using strapwork (girih) for decoration of buildings in Islamic architecture. They have been used since
Heesch's problem
In geometry, the Heesch number of a shape is the maximum number of layers of copies of the same shape that can surround it. Heesch's problem is the problem of determining the set of numbers that can b