geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an apeirogonal hosohedron or infinite hosohedronConway (2008), p. 263 is a tiling of the plane consisting of two vertices at infinity. It may be considered an improper

regular tiling Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his (Latin language, Latin: ''The Har ...

of the Euclidean plane, with

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

Related tilings and polyhedra

The apeirogonal hosohedron is the arithmetic limit of the family of

hosohedra In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune hav ...

, as ''p'' tends to

infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...

, thereby turning the hosohedron into a Euclidean tiling. All the vertices have then receded to infinity and the digonal faces are no longer defined by closed circuits of finite edges. Similarly to the

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fac ...

and the

uniform tiling In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to t ...

s, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and

omnitruncated In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each Flag (geometry), flag of the original polytope and a Facet (geometry), facet for each face of any dimension of the original pol ...

forms are also duplicated, therefore reducing the number of unique forms to four: the

apeirogonal tiling In geometry, an apeirogonal tiling is a tessellation of the Euclidean plane, hyperbolic plane, or some other two-dimensional space by apeirogons. Tilings of this type include: * Order-2 apeirogonal tiling, Euclidean tiling of two half-spaces * Order ...

, the apeirogonal hosohedron, the apeirogonal prism, and the apeirogonal antiprism.

Notes

References

* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,

External links

Jim McNeill: Tessellations of the Plane
Apeirogonal tilings Euclidean tilings Isogonal tilings Isohedral tilings Regular tilings Digonal tilings Infinite-order tilings {{geometry-stub