Apeirogonal hosohedron

Apeirogonal hosohedron
Type	Regular tiling
Vertex configuration	2∞ [[File:|40px]]
Schläfli symbol(s)	{2,∞}
Wythoff symbol(s)	∞ | 2 2
Coxeter diagram(s)
Symmetry	[∞,2], (*∞22)
Rotation symmetry	[∞,2]+, (∞22)
Dual	Order-2 apeirogonal tiling
Properties	Vertex-transitive, edge-transitive, face-transitive

In geometry, an apeirogonal hosohedron or infinite hosohedronis a tiling of the plane consisting of two vertices at infinity. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol {2,∞}.

Related tilings and polyhedra

The apeirogonal hosohedron is the arithmetic limit of the family of hosohedra {2,p}, as p tends to infinity, thereby turning the hosohedron into a Euclidean tiling.

Similarly to the uniform polyhedra and the uniform tilings, 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, and the apeirogonal antiprism.

(∞ 2 2)	Parent	Truncated	Rectified	Bitruncated	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Wythoff	2 | ∞ 2	2 2 | ∞	2 | ∞ 2	2 ∞ | 2	∞ | 2 2	∞ 2 | 2	∞ 2 2 |	| ∞ 2 2
Schläfli	t0{∞,2}	t0,1{∞,2}	t1{∞,2}	t1,2{∞,2}	t2{∞,2}	t0,2{∞,2}	t0,1,2{∞,2}	s{∞,2}
Coxeter
Image Vertex figure	{∞,2}	∞.∞	∞.∞	4.4.∞	{2,∞}	4.4.∞	60px 4.4.∞	3.3.3.∞

References

• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5

External links

• Jim McNeill: Tessellations of the Plane