geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedronConway (2008), p. 263 is a tiling of the

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

consisting of two apeirogons. 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 ''Harmonices Mundi'' (Latin langua ...

of the Euclidean plane, with Schläfli symbol Two apeirogons, joined along all their edges, can completely fill the entire plane as an apeirogon is infinite in size and has an

interior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

of 180°, which is half of a full 360°.

Related tilings and polyhedra

The apeirogonal tiling is the arithmetic limit of the family of

dihedra A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...

, as ''p'' tends to infinity, thereby turning the dihedron into a Euclidean tiling. Similarly to the

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

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 space, hyperbolic plane. Uniform tilings ar ...

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

apeirogonal prism In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane.Conway (2008), p.263 Thorold Gosset called it a ''2-dimensional semi-check ...

, and the

apeirogonal antiprism In geometry, an apeirogonal antiprism or infinite antiprismConway (2008), p. 263 is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane. If the sides are equilateral triangles, ...

Notes

References

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

External links

Jim McNeill: Tessellations of the Plane
Apeirogonal tilings Euclidean tilings Isogonal tilings Isohedral tilings Order-2 tilings Regular tilings {{geometry-stub

Related tilings and polyhedra

See also

Notes

References

External links