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Apeirogonal antiprism

## Summary

Uniform apeirogonal antiprism

Type Semiregular tiling
Vertex configuration
3.3.3.∞
Schläfli symbol sr{2,∞} or ${\displaystyle s{\begin{Bmatrix}\infty \\2\end{Bmatrix}}}$
Wythoff symbol | 2 2 ∞
Coxeter diagram
Symmetry [∞,2+], (∞22)
Rotation symmetry [∞,2]+, (∞22)
Bowers acronym Azap
Dual Apeirogonal deltohedron
Properties Vertex-transitive

In geometry, an apeirogonal antiprism or infinite antiprism[1] 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, it is a uniform tiling. In general, it can have two sets of alternating congruent isosceles triangles, surrounded by two half-planes.

The apeirogonal antiprism is the arithmetic limit of the family of antiprisms sr{2, p} or p.3.3.3, as p tends to infinity, thereby turning the antiprism 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.

Order-2 regular or uniform apeirogonal tilings
(∞ 2 2) Wythoff
symbol
Schläfli
symbol
Coxeter
diagram
Vertex
config.
Tiling image Tiling name
Parent 2 | ∞ 2 {∞,2}       ∞.∞   Apeirogonal
dihedron
Truncated 2 2 | ∞ t{∞,2}       2.∞.∞
Rectified 2 | ∞ 2 r{∞,2}       2.∞.2.∞
Birectified
(dual)
∞ | 2 2 {2,∞}       2   Apeirogonal
hosohedron
Bitruncated 2 ∞ | 2 t{2,∞}       4.4.∞   Apeirogonal
prism
Cantellated ∞ 2 | 2 rr{∞,2}
Omnitruncated
(Cantitruncated)
∞ 2 2 | tr{∞,2}       4.4.∞
Snub | ∞ 2 2 sr{∞,2}       3.3.3.∞   Apeirogonal antiprism Notes edit
1. ^ Conway (2008), p. 263

## References

• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5
• Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.
• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900