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Regular tiling of the plane
geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the
Triangular_tiling
The cantitruncated triangular tiling honeycomb, , has truncated trihexagonal tiling, truncated hexagonal tiling, and triangular prism cells, with a mirrored
Triangular_tiling_honeycomb
Semiregular tiling of the plane
In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex
Elongated_triangular_tiling
Concept in geometry
the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}. The symmetry group of the tiling is the (2,3
Order-7_triangular_tiling
Concept in geometry
In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal
Infinite-order triangular tiling
Infinite-order_triangular_tiling
Semiregular tiling of the Euclidean plane
degenerate into edges, a triangular tiling results, constructed as a snub triangular tiling, . There is one related 2-uniform tiling, having hexagons dissected
Rhombitrihexagonal_tiling
Semiregular tiling of the hyperbolic plane
geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are
Truncated order-7 triangular tiling
Truncated_order-7_triangular_tiling
Uniform tiling of the Euclidean plane
truncated trihexagonal tiling has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling. The first dissects
Truncated_trihexagonal_tiling
Semiregular tiling of a plane
topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.) This tiling is topologically related as a part
Truncated_hexagonal_tiling
Regular tiling of a two-dimensional space
one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling. The hexagonal tiling has a structure consisting
Hexagonal_tiling
Regular paracompact honeycomb
the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each
Hexagonal_tiling_honeycomb
Polyhedron resembling a soccerball
each edge creates 12 pentagonal faces and transforms the original 20 triangular faces into regular hexagons. Therefore, the resulting polyhedron has 32
Truncated_icosahedron
order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling is {6,3}, this honeycomb has four such hexagonal tilings meeting at each
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
colorings of the 11 uniform tilings: Triangular tiling – 9 uniform colorings, 4 wythoffian, 5 nonwythoffian Square tiling – 9 colorings: 7 wythoffian
List of Euclidean uniform tilings
List_of_Euclidean_uniform_tilings
Solid with twenty equal triangular faces
regular faces again. Replacing bases of a pentagonal antiprism with ten triangular pyramids, the regular icosahedron is classified as a composite polyhedron
Regular_icosahedron
order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling is {6,3}, this honeycomb has five such hexagonal tilings meeting at each
Order-5 hexagonal tiling honeycomb
Order-5_hexagonal_tiling_honeycomb
Solid with four equal triangular faces
A regular tetrahedron is a polyhedron with four equilateral triangular faces. A regular tetrahedron is a tetrahedron (that is, a four-sided polyhedron)
Regular_tetrahedron
Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into prisms. It is one of 28 convex
Triangular prismatic honeycomb
Triangular_prismatic_honeycomb
Concept in geometry
In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular
Order-8_triangular_tiling
Solid with eight equal triangular faces
dihedral angle of a regular octahedron is the angle between its two adjacent triangular faces. The angle can be obtained from the dihedral angle of an equilateral
Regular_octahedron
Geometric operation which truncates the edges of polyhedra
an example of Goldberg polyhedron GPIII(2,0) or {3+,3}2,0, containing triangular and hexagonal faces. Its dual is the alternate-triakis tetratetrahedron
Chamfer_(geometry)
Polyhedron with two kinds of faces
continues as the trihexagonal tiling, vertex figure (3.6)2 - a quasiregular tiling based on the triangular tiling and hexagonal tiling. The checkerboard pattern
Quasiregular_polyhedron
honeycombs with square tiling cells: The rectified square tiling honeycomb, t1{4,4,3}, has cube and square tiling facets, with a triangular prism vertex figure
Square_tiling_honeycomb
Semiregular tiling of the plane
In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli
Snub_square_tiling
Tiling of the hyperbolic plane
In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons
Heptagonal_tiling
icosahedron Square tiling Triangular tiling Hexagonal tiling Apeirogon Dihedron Lobachevski plane Hyperbolic tiling Order-7 heptagrammic tiling Heptagrammic-order
List_of_mathematical_shapes
Polyhedron with 2 faces
called bihedra, flat polyhedra, or doubly covered polygons. As a spherical tiling, a dihedron can exist as nondegenerate form, with two n-sided faces covering
Dihedron
tiling-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, hexagonal tiling, and trihexagonal tiling cells
Hexagonal tiling-triangular tiling honeycomb
Hexagonal_tiling-triangular_tiling_honeycomb
Tiling of a plane by regular hexagons and equilateral triangles
derives from the fact that it combines a regular hexagonal tiling and a regular triangular tiling. Two hexagons and two triangles alternate around each vertex
Trihexagonal_tiling
Symmetric subdivision in hyperbolic geometry
hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
Covering by shapes without overlaps or gaps
wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form
Tessellation
Geometry term
infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}. The dual of this tiling represents the
Truncated infinite-order triangular tiling
Truncated_infinite-order_triangular_tiling
Semiregular tiling of the Euclidean plane
densest packing from the triangular tiling. This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3
Snub_trihexagonal_tiling
Semiregular tiling of the hyperbolic plane
In geometry, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex
Truncated order-8 triangular tiling
Truncated_order-8_triangular_tiling
Shape with three equal sides
tessellation's instances are the triangular tiling where six equilateral triangles surrounds a common vertex, and the sphinx tiling as a special case of the polyiamond
Equilateral_triangle
Regular tiling of the Euclidean plane
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex
Square_tiling
Kepler–Poinsot polyhedron with 20 faces
5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic
Great_icosahedron
hyperbolic 3-space, the cubic-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from cube, triangular tiling, and cuboctahedron cells
Cubic-triangular tiling honeycomb
Cubic-triangular_tiling_honeycomb
Concept in mathematics
In geometry, the snub tritetratrigonal tiling or snub order-8 triangular tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of
Snub order-8 triangular tiling
Snub_order-8_triangular_tiling
Regular geometrical object in hyperbolic space
and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure. A geometric honeycomb is a space-filling of polyhedral
Order-6 dodecahedral honeycomb
Order-6_dodecahedral_honeycomb
triangular tiling {3,8} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing
Order-8-3 triangular honeycomb
Order-8-3_triangular_honeycomb
Geometric tiling
an expanded heptagonal tiling or expanded order-7 triangular tiling. The dual tiling is called a deltoidal triheptagonal tiling, and consists of congruent
Rhombitriheptagonal_tiling
Non-periodic tiling in geometry
Conversely, the tiles of the pinwheel tiling can be grouped into groups of five that form a larger pinwheel tiling. In this tiling, isometric copies
Pinwheel_tiling
Type of tabletop game using tiles
tiles are rectangular, twice as long as they are wide and at least twice as wide as they are thick. Other games exist with square tiles, triangular tiles
Tile-based_game
Vertex-transitive tiling of the plane by regular polygons
regular triangular tiling). A tiling can also be self-dual. The square tiling, with Schläfli symbol {4,4}, is self-dual; shown here are two square tilings (red
Uniform_tiling
Method of describing higher-order polyhedra
Euclidean tilings can also be used as seeds: Q = Quadrille = Square tiling H = Hextille = Hexagonal tiling = dΔ Δ = Deltille = Triangular tiling = dH These
Conway_polyhedron_notation
Prism with a 3-sided base
honeycomb, elongated triangular prismatic honeycomb. John Horton Conway and Charles Radin in 1998 introduced the quaquaversal tiling, a three-dimensional
Triangular_prism
with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure. A geometric honeycomb is a space-filling of polyhedral
Order-6_tetrahedral_honeycomb
3-space, the tetrahedral-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, tetrahedron, and octahedron
Tetrahedral-triangular tiling honeycomb
Tetrahedral-triangular_tiling_honeycomb
honeycombs with order-7 triangular tiling cells: {3,7,p}. It is a part of a sequence of regular honeycombs with heptagonal tiling vertex figures: {p,7,3}
Order-7-3 triangular honeycomb
Order-7-3_triangular_honeycomb
Infinite regular skew polyhedron
Euclidean tilings and 3 skew honeycombs): Triangular tiling: {3, 6} Square tiling: {4, 4} Hexagonal tiling: {6, 3} Petrial triangular tiling: {3, 6}π Petrial
Regular_skew_apeirohedron
rhombitrihexagonal tiling, triangular tiling, and triangular prism facets, with a triangular cupola vertex figure. The runcicantic order-6 hexagonal tiling honeycomb
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_honeycomb
Semiregular tiling of the hyperbolic plane
has media related to Uniform dual tiling V 4-6-14. In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed
3-7_kisrhombille
the alternated hexagonal tiling honeycomb, h{6,3,3}, or , is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron
Alternated hexagonal tiling honeycomb
Alternated_hexagonal_tiling_honeycomb
uniform tilings Uniform tilings in hyperbolic plane Archimedean tiling Square tiling Triangular tiling Hexagonal tiling Truncated square tiling Snub square
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Automorphism group of the Klein quartic
Dually, it can be tiled with 56 equilateral triangles, with 24 vertices, each of degree 7, as a quotient of the order-7 triangular tiling. Klein's quartic
PSL(2,7)
Polytope whose facets are all simplices
theorem to a maximal planar graph. For a simplicial tiling, examples are triangular tiling and Laves tiling. Simplicial 4-polytopes include: convex regular
Simplicial_polytope
infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement. It is a part of a sequence of regular polychora
Order-7_tetrahedral_honeycomb
Classification of a two-dimensional repetitive pattern
Examples of group p3m1 Triangular tiling (ignoring colors: p6m) Hexagonal tiling (ignoring colors: p6m) Truncated hexagonal tiling (ignoring colors: p6m)
Wallpaper_group
Polyform whose base form is an equilateral triangle
attempt to tile a plane with as many polyiamonds as possible, subject to the game rules. Triangular tiling Rhombille tiling Sphinx tiling Weisstein, Eric
Polyiamond
Compact Riemann surface of genus 3
the 24 vertices of the regular triangular tiling, or equivalently the centers of the 24 heptagons in the heptagonal tiling, and can be realized as follows
Klein_quartic
Type of uniform tiling in geometry
In geometry, the snub infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of s{3,∞}. John H. Conway, Heidi
Snub infinite-order triangular tiling
Snub_infinite-order_triangular_tiling
not only the triangular tiling, but also the coloring, and hence are a proper subgroup of the full isometry group. The corresponding tiling of the hyperbolic
Small_cubicuboctahedron
Group realized geometrically by reflections across the sides of a triangle
centrally symmetric. Hence each of them determines a tiling of the real projective plane, an elliptic tiling. Its symmetry group is the quotient of the spherical
Triangle_group
Shape with six sides
Euclidean space Hexagonal crystal system Hexagonal number Hexagonal tiling: a regular tiling of hexagons in a plane Hexagram: six-sided star within a regular
Hexagon
Turing machine on a two-dimensional grid
Paterson's worms are a type of turmite defined on the edges of a triangular tiling. It has been shown that turmites in general are exactly equivalent
Turmite
Semiregular tiling of the hyperbolic plane
The tiling has a vertex configuration of 3.14.14. The dual tiling is called an order-7 triakis triangular tiling, seen as an order-7 triangular tiling with
Truncated_heptagonal_tiling
Notation for a polyhedron's vertex figure
3.5 (60) Semiregular tilings: Snub hexagonal tiling: 3.3.3.3.6 (chiral) Elongated triangular tiling: 3.3.3.4.4 Snub square tiling: 3.3.4.3.4 (note that
Vertex_configuration
Mathematics book
topics in tiling theory: colored patterns and tilings, polygonal tilings, aperiodic tilings, Wang tiles, and tilings with unusual kinds of tiles. Each chapter
Tilings_and_patterns
Board game consisting of triangular tiles
dominoes using triangular tiles published in 1965. A popular version of this game is marketed as Tri-Ominos by the Pressman Toy Corp. A triomino tile is in the
Triominoes
Tiling of the hyperbolic plane
order-7 heptagrammic tiling is a tiling of the hyperbolic plane by overlapping heptagrams. This tiling is a regular star-tiling, and has Schläfli symbol
Order-7_heptagrammic_tiling
flattened triangular tiling with half of the triangles as holes. These cut face-diagonally through the original cubes. There are also square tiling plane
Tetragonal disphenoid honeycomb
Tetragonal_disphenoid_honeycomb
ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement. It has a second
Order-6-4 triangular honeycomb
Order-6-4_triangular_honeycomb
N-dimensional gradient noise function
Simplex and 2D OpenSimplex both use different orientations of the triangular tiling, but whereas 3D Simplex uses the tetragonal disphenoid honeycomb,
OpenSimplex_noise
Three-dimensional geometric shape
are equilateral triangles, it can be constructed from a stretched triangular tiling net with four triangles in one direction and an even number in the
Kaleidocycle
Concept in mathematics
apeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{∞,3}. The dual tiling, the infinite-order triakis triangular tiling, has
Truncated order-3 apeirogonal tiling
Truncated_order-3_apeirogonal_tiling
Operation in Euclidean geometry
of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example the tetrahedron
Rectification_(geometry)
Regular tiling of the hyperbolic plane
In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons
Octagonal_tiling
Property of objects which appear unchanged after a partial rotation
for this symmetry); it is e.g. the rotation group of the regular triangular tiling with the equilateral triangles alternatingly colored. p4 (442): 2×4-fold
Rotational_symmetry
Triangle with at least two sides congruent
Adventitious Angles puzzle, and the 30-30-120 triangle of the triakis triangular tiling. Five Catalan solids, the triakis tetrahedron, triakis octahedron
Isosceles_triangle
Subdivision of the plane into polygons that are all regular
vertices with 2 different vertex types, so this tiling would be classed as a "3-uniform (2-vertex types)" tiling. Broken down, 36; 36 (both of different transitivity
Euclidean tilings by convex regular polygons
Euclidean_tilings_by_convex_regular_polygons
Complex structures in matter physics
be reduced to that of plane tilings with equilateral triangles. A well known solution is provided by the triangular tiling with a total compatibility between
Geometrical_frustration
tiling1p+1q<12:Hyperbolic plane tiling By enumerating the permutations, we find five convex forms, four star forms and three plane tilings, all with polygons {p}
List_of_regular_polytopes
Periodic tiling of the hyperbolic disk
In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular
Order-3_apeirogonal_tiling
is {7,3,6}, with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}. It has a quasiregular construction
Order-3-6 heptagonal honeycomb
Order-3-6_heptagonal_honeycomb
Set of tile shapes that can create nonrepeating patterns
problem asked for a single polyhedron tiling Euclidean 3-space, such that no tiling by it is isohedral (an anisohedral tile). The problem as stated was solved
Aperiodic_set_of_prototiles
Type of tessallation
non-periodic tiling of the plane. The sphinx is therefore a rep-tile (a self-replicating tessellation). It is one of few known pentagonal rep-tiles and is the
Sphinx_tiling
Polyhedron made from triangles that approximates a sphere
the original edges. The subgrids can be extracted by looking at a triangular tiling, positioning a large triangle on top of grid vertices and walking
Geodesic_polyhedron
Tessellation of convex uniform polyhedron cells
A radical nonsimplectic subgroup is ↔ , which can be doubled into a triangular prism domain as ↔ . There are 15 forms, generated by ring permutations
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
as the regular order-7 triangular tiling, {3,7}. The full set of edges coincide with the edges of a heptakis heptagonal tiling. It is related to a Kepler-Poinsot
Heptagrammic-order heptagonal tiling
Heptagrammic-order_heptagonal_tiling
Regular space-filling tessellation with Schläfli symbol (7,3,7)
ideal boundary) with seven heptagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure. It a part of a sequence
Order-3-7 heptagonal honeycomb
Order-3-7_heptagonal_honeycomb
Concept in mathematics
In geometry, the truncated octagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two hexakaidecagons on each vertex
Truncated_octagonal_tiling
Sporadic simple group
original triangular tiling which bisect the squares and octagons". This can be visualized by coloring the triangles – the corresponding tiling is topologically
Mathieu_group_M24
Spherical polyhedron composed of lunes
must have at least three sides. When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons) can be represented
Hosohedron
blue plastic icosahedron, and the pieces are 20 white equilateral-triangular snap-in tiles with black dots and 12 yellow pins for the corners.[citation needed]
Icosoku
the ideal boundary) with seven hexagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure. It a part of a sequence
Order-3-7_hexagonal_honeycomb
Installation artwork by feminist artist Judy Chicago
The Heritage Floor, made up of more than 2,000 white luster-glazed triangular tiles, each inscribed in gold scripts with the name of one of 998 women and
The_Dinner_Party
Operation that cuts polytope vertices, creating a new facet in place of each vertex
uniform truncation performed on the regular triangular tiling {3,6} results in the regular hexagonal tiling {6,3}. A truncated n-sided polygon will have
Truncation_(geometry)
Infinite polyhedron with non-planar faces
forms of the regular planar tessellations. Both the square tiling {4,4} and triangular tiling {3,6} can be curved into approximating infinite cylinders
Skew_apeirohedron
Geometric operation applied to a polyhedron
honeycomb is the snub hexagonal tiling honeycomb, as s{3,6,3} and , which can also be constructed as an alternated hexagonal tiling honeycomb, h{6,3,3}, . It
Snub_(geometry)
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