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Index of articles associated with the same name
this type include: Order-2 apeirogonal tiling, Euclidean tiling of two half-spaces Order-3 apeirogonal tiling, hyperbolic tiling with 3 apeirogons around
Apeirogonal_tiling
Plane tiling with two infinite-sided polygons
apeirogonal tiling, the apeirogonal hosohedron, the apeirogonal prism, and the apeirogonal antiprism. Order-3 apeirogonal tiling - hyperbolic tiling Order-4
Order-2_apeirogonal_tiling
Regular tiling in geometry
In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It covers the hyperbolic plane, which is a non-Euclidean surface
Order-4_apeirogonal_tiling
Geometric tiling of the plane
In geometry, an apeirogonal hosohedron or infinite hosohedron is a tiling of the plane consisting of two vertices at infinity. It may be considered an
Apeirogonal_hosohedron
Prism with an infinite-sided polygon base
with alternate colored square faces. Its dual tiling is an apeirogonal bipyramid. The apeirogonal tiling is the arithmetic limit of the family of prisms
Apeirogonal_prism
geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}. The dual to this tiling represents the
Order-5_apeirogonal_tiling
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
The infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely
Infinite-order apeirogonal tiling
Infinite-order_apeirogonal_tiling
Antiprism with an infinite-sided polygon base
tiling. The apeirogonal antiprism can be constructed by applying an alternation operation to an apeirogonal prism. The dual tiling of an apeirogonal antiprism
Apeirogonal_antiprism
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
Tiling of the hyperbolic plane
geometry, the order-6 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}. The dual to this tiling represents the
Order-6_apeirogonal_tiling
In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}. A half symmetry coloring
Truncated order-4 apeirogonal tiling
Truncated_order-4_apeirogonal_tiling
Concept in mathematics
truncated order-3 apeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{∞,3}. The dual tiling, the infinite-order
Truncated order-3 apeirogonal tiling
Truncated_order-3_apeirogonal_tiling
contains and that tile 2-hypercycle surfaces, which are similar to these paracompact order-4 apeirogonal tilings : The order-4 square tiling honeycomb is a
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
7,3}, with three order-7 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {7,3}. The "ideal surface"
Order-7-3 triangular honeycomb
Order-7-3_triangular_honeycomb
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
honeycomb contains that tile 2-hypercycle surfaces, which are similar to the paracompact order-3 apeirogonal tiling : The square tiling honeycomb is a regular
Square_tiling_honeycomb
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
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
order-3-6 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-3 apeirogonal tiling whose vertices
Order-3-6 heptagonal honeycomb
Order-3-6_heptagonal_honeycomb
infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface. The triangular tiling honeycomb is
Triangular_tiling_honeycomb
Polygon with an infinite number of sides
apeirogons, and are the infinite analogues of n-polytopes. Apeirogonal tiling Apeirogonal prism Apeirogonal antiprism Teragon, a fractal generalized polygon that
Apeirogon
ideal sphere. The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,4,3}, with three apeirogonal tilings meeting at each edge. The vertex figure
Order-4-3 pentagonal honeycomb
Order-4-3_pentagonal_honeycomb
hexagonal tiling honeycomb is {6,3,6}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_honeycomb
order-3-5 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-3 apeirogonal tiling whose vertices
Order-3-5 heptagonal honeycomb
Order-3-5_heptagonal_honeycomb
Tiling of the hyperbolic plane
In geometry, a binary tiling (sometimes called a Böröczky tiling) is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane
Binary_tiling
8,3}, with three order-8 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}. The "ideal surface"
Order-8-3 triangular honeycomb
Order-8-3_triangular_honeycomb
sphere. The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,6,3}, with three order-6 apeirogonal tilings meeting at each edge. The vertex figure
Order-6-3_square_honeycomb
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
The Schläfli symbol of the order-3-4 apeirogonal honeycomb is {∞,3,4}, with four order-3 apeirogonal tilings meeting at each edge. The vertex figure
Order-3-4 heptagonal honeycomb
Order-3-4_heptagonal_honeycomb
tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli symbol of the apeirogonal tiling honeycomb
Order-5-3_square_honeycomb
Regular space-filling tessellation with Schläfli symbol (7,3,7)
apeirogonal tiling {∞,3} around each edge. All vertices are ultra-ideal (Existing beyond the ideal boundary) with infinitely many apeirogonal tilings existing
Order-3-7 heptagonal honeycomb
Order-3-7_heptagonal_honeycomb
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
Vertex-transitive tiling of the plane by regular polygons
prism and apeirogonal antiprism. The stacking of the finite faces of these two prismatic tilings constructs one non-Wythoffian uniform tiling of the plane
Uniform_tiling
Regular space-filling tessellation
apeirogonal tiling {∞,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-6 apeirogonal
Order-6-4_square_honeycomb
operations. They are higher-dimensional analogues of the order-2 apeirogonal tiling and apeirogonal hosohedron. There are 15 flat regular honeycombs of hyperbolic
List_of_regular_polytopes
120-cell Honeycomb Cubic honeycomb Hosohedron Dihedron Order-2 apeirogonal tiling Apeirogonal hosohedron Order-4 square hosohedral honeycomb Order-6 triangular
List_of_mathematical_shapes
sphere. The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,4,4}, with three order-4 apeirogonal tilings meeting at each edge. The vertex figure
Order-4-4 pentagonal honeycomb
Order-4-4_pentagonal_honeycomb
Spherical polyhedron composed of lunes
hosohedron is the n-gonal prism. In the limit, the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation: Multidimensional analogues
Hosohedron
Polyhedron with 2 faces
distance larger than zero, the faces are infinite polygons (a bit like the apeirogonal hosohedron's digon faces, having a width larger than zero, are infinite
Dihedron
Regular infinite tree structure used in statistical mechanics
Lie group. The vertices and edges of an order- k {\displaystyle k} apeirogonal tiling of the hyperbolic plane form a Bethe lattice of degree k {\displaystyle
Bethe_lattice
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
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
2-dimensional hyperbolic geometry, the infinite-order hexagonal tiling is a regular tiling. It has Schläfli symbol of {6,∞}. All vertices are ideal, located
Infinite-order hexagonal tiling
Infinite-order_hexagonal_tiling
boundary) with infinitely many order-5 apeirogonal tilings existing around each vertex in an infinite-order pentagonal tiling vertex arrangement. It has a second
Order-5-4_square_honeycomb
Polygon with 2 sides and 2 vertices
regular tessellation of the Euclidean plane, even when its dual order-2 apeirogonal tiling (infinite dihedron) is. A compound of two "line segment" digons, as
Digon
2-dimensional hyperbolic geometry, the infinite-order pentagonal tiling is a regular tiling. It has Schläfli symbol of {5,∞}. All vertices are ideal, located
Infinite-order pentagonal tiling
Infinite-order_pentagonal_tiling
Polyhedron associated with another by swapping vertices for faces
hyperbolic honeycombs are: Compact hyperbolic tilings: {5,5}, {6,6}, ... {p,p}. Paracompact hyperbolic tiling: {∞,∞} Compact hyperbolic honeycombs: {3,5
Dual_polyhedron
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 fundamental
Truncated infinite-order triangular tiling
Truncated_infinite-order_triangular_tiling
infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}. In (*∞44) symmetry this tiling has 3 colors. Bisecting
Truncated infinite-order square tiling
Truncated_infinite-order_square_tiling
apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,4,∞}. It has infinitely many order-4 apeirogonal
Order-4-5 pentagonal honeycomb
Order-4-5_pentagonal_honeycomb
Tiling method in hyperbolic geometry
In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}. All vertices are ideal, located
Infinite-order_square_tiling
Rational circle tangent to the real line
horocycles are circumscribed by apeirogons they tile the hyperbolic plane with an order-3 apeirogonal tiling. There is a link between the area of Ford circles
Ford_circle
Coxeter); The uniform tilings (infinite polyhedra) 11 Euclidean convex uniform tilings; 28 Euclidean nonconvex or apeirogonal uniform tilings; Infinite number
List_of_uniform_polyhedra
Trapezohedron with eight faces
engraving Stars. The tetragonal trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above
Tetragonal_trapezohedron
5-dimensional hyperbolic space, square tiling honeycomb, {4,4,3}, in 3-dimensional hyperbolic space, and the order-3 apeirogonal tiling, {∞,3} of 2-dimensional hyperbolic
Cubic_honeycomb_honeycomb
Three-dimensional solid
Hendecagonal prism Dodecagonal prism ... Apeirogonal prism Polyhedron image ... Spherical tiling image Plane tiling image Vertex config. 2.4.4 3.4.4 4.4.4
Cylinder
Geometrical concept
4-dimensional hyperbolic space, square tiling honeycomb, {4,4,3}, in 3-dimensional hyperbolic space, and the order-3 apeirogonal tiling, {∞,3} of 2-dimensional hyperbolic
Tesseractic honeycomb honeycomb
Tesseractic_honeycomb_honeycomb
Partition of a sphere's surface into polygons
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
Spherical_polyhedron
Curve whose normals converge asymptotically
The order-3 apeirogonal tiling, {∞,3}, fills the hyperbolic plane with apeirogons whose vertices exist along horocyclic paths.
Horocycle
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
Regular geometrical object in hyperbolic space
It is similar to the 2D hyperbolic pentaapeirogonal tiling, r{5,∞} with pentagon and apeirogonal faces. The truncated order-6 dodecahedral honeycomb,
Order-6 dodecahedral honeycomb
Order-6_dodecahedral_honeycomb
Natural number
convex pentagonal tilings, with eight being edge-to-edge. There are 15 regular and semiregular tilings when infinite (improper) apeirogonal forms are counted:
15_(number)
These mutations can occur from spherical tilings to Euclidean tilings to hyperbolic tilings. Hyperbolic tilings can also be divided between compact, paracompact
Uniform tiling symmetry mutations
Uniform_tiling_symmetry_mutations
Solid with 2 parallel n-gonal bases connected by n parallelograms
{4,3,3}×{ } (tesseractic prism) are lower symmetry forms of a 5-cube. Apeirogonal prism Equiprojective polyhedra Rectified prism Prismanes List of shapes
Prism_(geometry)
Polyhedron made of 12 congruent kites
symmetry, order 6. The hexagonal trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above
Hexagonal_trapezohedron
Spatial tiling of convex uniform polyhedra
unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding
Convex_uniform_honeycomb
Solid with 10 faces
antiprism. The square antiprism is first in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n. If a {\displaystyle a} denotes the side length
Square_antiprism
3D shape obtained by revolving a rhombus about one of its axes of symmetry
Pentagonal bipyramid Hexagonal bipyramid ... Apeirogonal bipyramid Polyhedron image ... Spherical tiling image Plane tiling image Face config. V2.4.4 V3.4.4 V4
Bicone
Polyhedron; 2 hexagonal pyramids joined base-to-base
each other, perpendicular to the horizontal plane. It can be drawn as a tiling on a sphere which also represents the fundamental domains of [3,2], *322
Hexagonal_bipyramid
Polyhedron made of congruent kites arranged radially
trapezohedron Hexagonal trapezohedron ... Apeirogonal trapezohedron Polyhedron image ... Spherical tiling image Plane tiling image Face configuration V2.3.3.3
Trapezohedron
Polyhedron with parallel bases connected by triangles
antiprism Heptagonal antiprism ... Apeirogonal antiprism Polyhedron image ... Spherical tiling image Plane tiling image Vertex config. 2.3.3.3 3.3.3.3
Antiprism
12-sided box
Hendecagonal prism Dodecagonal prism ... Apeirogonal prism Polyhedron image ... Spherical tiling image Plane tiling image Vertex config. 2.4.4 3.4.4 4.4.4
Dodecagonal_prism
Prism with a 7-sided base
to the Cavalieri's principle. The heptagonal prism can also be seen as a tiling on a sphere: Sapiña, R. "Area and volume calculator of a heptagonal prism"
Heptagonal_prism
Antiprism with a five-sided base
antiprism Heptagonal antiprism ... Apeirogonal antiprism Polyhedron image ... Spherical tiling image Plane tiling image Vertex config. 2.3.3.3 3.3.3.3
Pentagonal_antiprism
Prism with a 5-sided base
Hendecagonal prism Dodecagonal prism ... Apeirogonal prism Polyhedron image ... Spherical tiling image Plane tiling image Vertex config. 2.4.4 3.4.4 4.4.4
Pentagonal_prism
Polyhedron with 6 congruent rhombus faces
can be assembled to form a rhombic dodecahedron. The same rhombohedra also tile space in the trigonal trapezohedral honeycomb. The trigonal trapezohedra
Trigonal_trapezohedron
Antiprism with 6-sided caps
antiprism Heptagonal antiprism ... Apeirogonal antiprism Polyhedron image ... Spherical tiling image Plane tiling image Vertex config. 2.3.3.3 3.3.3.3
Hexagonal_antiprism
Prism with an 8-sided base
joining two regular octagon caps. The octagonal prism can also be seen as a tiling on a sphere: In optics, octagonal prisms are used to generate flicker-free
Octagonal_prism
Non-planar polygon with infinitely many sides
zig-zag skew apeirogons exist as Petrie polygons of the three regular tilings of the plane: {4,4}, {6,3}, and {3,6}. These regular zig-zag skew apeirogons
Infinite_skew_polygon
Classification system for symmetry groups in geometry
isomorphic to [2p]. In the limit, going down to one dimension, the full apeirogonal group is obtained when the angle goes to zero, so [∞], abstractly the
Coxeter_notation
Generalization of a polytope in real space
honeycombs also have van Oss apeirogons. For example, the real square tiling and triangular tiling have apeirogons {∞} van Oss apeirogons. If it exists, the van
Complex_polytope
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