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Regular tiling of the hyperbolic plane
In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}. This tiling represents a hyperbolic
Order-4_hexagonal_tiling
In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
Uniform tiling of the hyperbolic plane
In geometry, the alternated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}
Alternated order-4 hexagonal tiling
Alternated_order-4_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
Uniform tiling of the hyperbolic plane
In geometry, the cantic order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{(4,4,3)} or h2{6,4}. John
Cantic order-4 hexagonal tiling
Cantic_order-4_hexagonal_tiling
Pattern in hyperbolic geometry
In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction
Truncated order-4 hexagonal tiling
Truncated_order-4_hexagonal_tiling
Semiregular tiling of a plane
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each
Truncated_hexagonal_tiling
truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure. The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has hexagonal tiling
Order-5 hexagonal tiling honeycomb
Order-5_hexagonal_tiling_honeycomb
Regular paracompact honeycomb
symbol of 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
Hexagonal_tiling_honeycomb
rectified order-4 hexagonal tiling honeycomb, t1{4,4,4}, has square tiling facets, with a cubic vertex figure. It is the same as the regular square tiling honeycomb
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
The truncated order-6 hexagonal tiling honeycomb, t0,1{6,3,6}, has triangular tiling and truncated hexagonal tiling facets, with a hexagonal pyramid vertex
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_honeycomb
3-space, the hexagonal tiling-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, hexagonal tiling, and trihexagonal
Hexagonal tiling-triangular tiling honeycomb
Hexagonal_tiling-triangular_tiling_honeycomb
boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement. It has a second construction
Order-8-3 triangular honeycomb
Order-8-3_triangular_honeycomb
Euclidean 3-space) 1 p + 1 q = 1 2 : Euclidean plane tiling 1 p + 1 q < 1 2 : Hyperbolic plane tiling {\displaystyle {\begin{aligned}&{\frac {1}{p}}+{\frac
List_of_regular_polytopes
Regular tiling of the hyperbolic plane
geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual. This tiling represents
Order-6_hexagonal_tiling
Semiregular tiling of the Euclidean plane
the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has
Rhombitrihexagonal_tiling
sphere. The Schläfli symbol of the order-4-3 hexagonal honeycomb is {6,4,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of
Order-4-3 pentagonal honeycomb
Order-4-3_pentagonal_honeycomb
boundary) with infinitely many triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement. It has a second construction
Order-6-4 triangular honeycomb
Order-6-4_triangular_honeycomb
Uniform tiling of the Euclidean plane
There are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling). Drawing the tiles colored as red on the
Truncated_trihexagonal_tiling
Semiregular tiling of the Euclidean plane
the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each
Snub_trihexagonal_tiling
In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically
Truncated order-6 hexagonal tiling
Truncated_order-6_hexagonal_tiling
Semiregular tiling of the hyperbolic plane
the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}. This tiling can also be constructed
Truncated order-8 hexagonal tiling
Truncated_order-8_hexagonal_tiling
the order-4-4 hexagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 hexagonal tiling whose
Order-4-4 pentagonal honeycomb
Order-4-4_pentagonal_honeycomb
Uniform tiling of the hyperbolic plane
tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling. There are two uniform constructions of this tiling
Rhombitetrahexagonal_tiling
Regular tiling of the hyperbolic plane
In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}. This tiling represents a hyperbolic
Order-6_square_tiling
Hexagonal tiling honeycomb Order-4 hexagonal tiling honeycomb Order-5 hexagonal tiling honeycomb Order-6 hexagonal tiling honeycomb 5-polytope Honeycomb
List_of_mathematical_shapes
In geometry, the truncated order-5 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{6,5}. John H. Conway, Heidi
Truncated order-5 hexagonal tiling
Truncated_order-5_hexagonal_tiling
Semiregular tiling of the hyperbolic plane
are fourteen hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling. Drawing the tiles colored as red on the original faces
Truncated tetrahexagonal tiling
Truncated_tetrahexagonal_tiling
runcitruncated triangular tiling honeycomb, , has hexagonal tiling, rhombitrihexagonal tiling, triangular prism, and hexagonal prism cells, with an isosceles-trapezoidal
Triangular_tiling_honeycomb
Isogonal polytope with regular facets
↔ Alternated order-5 hexagonal tiling honeycomb, ↔ Alternated order-6 hexagonal tiling honeycomb, ↔ Alternated square tiling honeycomb, ↔ (quasiregular)
Semiregular_polytope
regular polychora and honeycombs with hexagonal tiling cells. In the geometry of hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is
Order-3-7_hexagonal_honeycomb
Regular space-filling tessellation
beyond the ideal boundary) with four order-6 square tilings existing around each edge and with an order-4 hexagonal tiling vertex figure. It a part of a sequence
Order-6-4_square_honeycomb
Tiling of a plane by regular hexagons and equilateral triangles
hexadeltille, combining alternate elements from a hexagonal tiling (hextille) and triangular tiling (deltille). Kagome (Japanese: 籠目) is a traditional
Trihexagonal_tiling
Regular tiling of the hyperbolic plane
In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be
Order-4_octagonal_tiling
Regular tiling of hyperbolic 3-space
honeycomb is also related to the 16-cell, cubic honeycomb, and order-4 hexagonal tiling honeycomb all which have octahedral vertex figures: This honeycomb
Order-4 dodecahedral honeycomb
Order-4_dodecahedral_honeycomb
Shape with six sides
Hexagonal crystal system Hexagonal number Hexagonal tiling: a regular tiling of hexagons in a plane Hexagram: six-sided star within a regular hexagon
Hexagon
Tiling of the plane with 60° rhombi
represents a regular compound tiling. It can also be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi. The diagonals
Rhombille_tiling
spherical space. It is also seen as a rectified order-4 square tiling honeycomb, r{4,4,4}: The square tiling honeycomb has three reflective symmetry constructions:
Square_tiling_honeycomb
is {4,6,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}. It is a part of
Order-6-3_square_honeycomb
regular 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
Order-7-3 triangular honeycomb
Order-7-3_triangular_honeycomb
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
order-4 apeirogonal tiling {∞,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings
Order-4-5 pentagonal honeycomb
Order-4-5_pentagonal_honeycomb
Four-dimensional analog of the octahedron
16-cell is also related to the cubic honeycomb, order-4 dodecahedral honeycomb, and order-4 hexagonal tiling honeycomb which all have octahedral vertex figures
16-cell
Periodic tiling of the hyperbolic disk
edges of the tiling, shown in blue, form an order-3 Cayley tree. Like the Euclidean hexagonal tiling, there are 3 uniform colorings of the order-3 apeirogonal
Order-3_apeirogonal_tiling
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
Polyhedron; 2 hexagonal pyramids joined base-to-base
A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices
Hexagonal_bipyramid
Semiregular tiling of the plane
tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated
Elongated_triangular_tiling
Tiling of the plane by pentagons
Their tilings have varying symmetries; all are face-symmetric. One particular form of the tiling, dual to the snub square tiling, has tiles with the
Cairo_pentagonal_tiling
Regular tiling of the plane
regular tilings of the plane. The other two are the square tiling and the hexagonal tiling. There are 9 distinct uniform colorings of a triangular tiling. (Naming
Triangular_tiling
Semi-regular arrangement of squares, decagons, and dodecagons
In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon
Truncated pentahexagonal tiling
Truncated_pentahexagonal_tiling
Polygon able to tessellate edge-to-edge, without rotation
a distorted square tiling while a hexagonal parallelogon can tile the plane as a distorted regular hexagonal tiling. Aleksandr Danilovich Alexandrov (2005)
Parallelogon
Regular tiling of the hyperbolic plane
geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}. This regular tiling can also be constructed
Order-6_pentagonal_tiling
order-5 hexagonal tilings, {6,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings
Order-5-4_square_honeycomb
Aspect of topological graph theory
hexagonal faces of the hexagonal tiling as type {6,3}(2,0). The petrial octahedron, {3,4}π, has 6 vertices, 12 edges, and 4 skew hexagon faces. It has an Euler
Petrie_dual
Regular geometrical object in hyperbolic space
runcitruncated order-5 hexagonal tiling honeycomb. The omnitruncated order-6 dodecahedral honeycomb is the same as the omnitruncated order-5 hexagonal tiling honeycomb
Order-6 dodecahedral honeycomb
Order-6_dodecahedral_honeycomb
Regular tiling of the Euclidean plane
properties, the square tiling is categorized as one of three regular tilings; the remaining being triangular tiling and hexagonal tiling with its prototiles
Square_tiling
bitruncated hexagonal tiling honeycomb. The runcitruncated order-6 tetrahedral honeycomb is equivalent to the runcicantellated hexagonal tiling honeycomb
Order-6_tetrahedral_honeycomb
cuboctahedron It contains a subgroup H2 tiling, the alternated order-4 hexagonal tiling, , with vertex figure (3.4)4. A lower symmetry form, index 6, of this
Cubic-octahedral_honeycomb
geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}. The dual tiling represents the fundamental
Truncated order-6 square tiling
Truncated_order-6_square_tiling
Non-periodic tiling of the plane
Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is
Penrose_tiling
Tessellation Uniform tiling Convex uniform honeycombs List of k-uniform tilings List of Euclidean uniform tilings Uniform tilings in hyperbolic plane Weisstein
List_of_tessellations
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
Classification of a two-dimensional repetitive pattern
hexagonal tiling with one color for outlining the hexagons and one for the background. Examples of group p6m Computer generated Trihexagonal tiling Small
Wallpaper_group
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
Tiling of the plane by pentagons
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
Pentagonal_tiling
Polyform with a regular hexagon as the base form
a regular hexagonal tiling. The rules for joining hexagons together may vary. Generally, however, the following rules apply: Two hexagons may be joined
Polyhex
Uniform Tiling
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
3-4-6-12_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
Polyhedron made of 12 congruent kites
reduced to cyclic symmetry, C6 symmetry, order 6. The hexagonal trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating
Hexagonal_trapezohedron
Semiregular tiling of the hyperbolic plane
related to Uniform tiling 3-14-14. Truncated hexagonal tiling Heptagonal tiling Tilings of regular polygons List of uniform tilings John H. Conway, Heidi
Truncated_heptagonal_tiling
Isogonal honeycomb of uniform polytope facets
polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb articles were
Uniform_honeycomb
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
List_of_k-uniform_tilings
Tessellation of convex uniform polyhedron cells
graphs: [(3,3,4,1+,4)] = [((3,∞,3)),((3,∞,3))] or , [(3,4,4,1+,4)] = [((4,∞,3)),((3,∞,4))] or , [(4,4,4,1+,4)] = [((4,∞,4)),((4,∞,4))] or . = , = , =
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
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
Semiregular tiling of the hyperbolic plane
geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There is one square, one hexagon, and one tetradecagon (14-sides)
Truncated triheptagonal tiling
Truncated_triheptagonal_tiling
Rule from the theory of the tiling of the plane
periodic tiling of the plane—and does so using only 180-degree rotations. The Conway criterion is a sufficient condition to prove that a prototile tiles the
Conway_criterion
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
In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of q{4,6}. It is constructed from *3232
Quarter_order-6_square_tiling
Semiregular tiling of the hyperbolic plane
In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There
Truncated order-7 triangular tiling
Truncated_order-7_triangular_tiling
Polyform whose base form is an equilateral triangle
Triangular tiling Rhombille tiling Sphinx tiling Weisstein, Eric W. "Polyiamond". MathWorld. Polyiamonds at The Poly Pages. Polyiamond tilings. VERHEXT
Polyiamond
the tiles). A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself. Such a tiling is
List of aperiodic sets of tiles
List_of_aperiodic_sets_of_tiles
Convex polygon which can tile the plane by itself
Laves tilings are unique except for the square tiling (1 degree of freedom), barn pentagonal tiling (1 degree of freedom), and hexagonal tiling (2 degrees
Planigon
Geometric operation which truncates the edges of polyhedra
and hexagonal faces. Its dual is the alternate-triakis tetratetrahedron. chamfered cube: from a cube, the resulting polyhedron has twelve hexagonal and
Chamfer_(geometry)
Shape subdivided into copies of itself
shape necessarily forms the prototile for a tiling of the plane, in many cases a nonperiodic tiling. A rep-tile dissection using different sizes of the original
Rep-tile
tiling facets, with a square prism vertex figure. The truncated order-4 octahedral honeycomb, t0,1{3,4,4}, has truncated octahedron and square tiling
Order-4_octahedral_honeycomb
Regular tiling of the hyperbolic plane
truncated order-8 square tiling, t{4,8}. Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual
Octagonal_tiling
order-5 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli symbol of the order-5-3
Order-5-3_square_honeycomb
Semiregular tiling of the hyperbolic plane
In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles and one heptagon on each vertex
Snub_triheptagonal_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 nine sides
euclidean tiling with gaps. These gaps can be filled with regular hexagons and isosceles triangles. In the notation of symmetrohedron this tiling is called
Nonagon
Uniform tiling of the plane with regular polygons
4.3.12 and 3.12.12. The 3.12.12 vertex figure alone generates a truncated hexagonal tiling, while the 3.4.3.12 only exists in this 2-uniform tiling.
3-4-3-12_tiling
Tiling of euclidean or hyperbolic space of three or more dimensions
that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified
Honeycomb_(geometry)
Polygon with 18 edges
concave hexagonal gaps. And another tiling mixes in nonagons and octagonal gaps. The first tiling is related to a truncated hexagonal tiling, and the
Octadecagon
Natural number
number. A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. A hexagon also has 6 edges as well as 6 internal
6
Euclidean tiling makes another hyperbolic tiling.) Point groups: (p 2 2) dihedral symmetry, p = 2 , 3 , 4 … {\displaystyle p=2,3,4\dots } (order 4 p {\displaystyle
Lists of uniform tilings on the sphere, plane, and hyperbolic plane
Lists_of_uniform_tilings_on_the_sphere,_plane,_and_hyperbolic_plane
Polyhedron with eight triangular faces
convex polyhedra include: Hexagonal prism: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges. With all faces
Octahedron
Convex polyhedron whose faces are almost regular polygons
polygons. These cases use 4.4.4.4 vertex figures of the square tiling, 3.3.3.3.3.3 vertex figure of the triangular tiling, as well as 60 degree rhombi
Near-miss_Johnson_solid
Archimedean solid with 62 faces
below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. Wenninger Model Number
Truncated_icosidodecahedron
Notation for a polyhedron's vertex figure
Hexagonal tiling: 6.6.6 Semiregular tilings: Truncated hexagonal tiling: 3.12.12 Truncated trihexagonal tiling: 4.6.12 Truncated square tiling: 4.8.8 Quadruples
Vertex_configuration
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