# Apeirogonal antiprism

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.

Apeirogonal antiprism

TypeSemiregular tiling
Vertex configuration
3.3.3.
Schläfli symbolsr{2,} or ${\displaystyle s{\begin{Bmatrix}\infty \\2\end{Bmatrix}}}$
Wythoff symbol| 2 2
Coxeter diagram
Symmetry[,2+], (22)
Rotation symmetry[,2]+, (22)
Bowers acronymAzap
DualApeirogonal deltohedron
PropertiesVertex-transitive

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 apeirogonal tilings
(∞ 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 {∞,2} t{∞,2} r{∞,2} t{2,∞} {2,∞} rr{∞,2} tr{∞,2} sr{∞,2}
Coxeter
Image
Vertex figure

{∞,2}

∞.∞

∞.∞

4.4.∞

{2,∞}

4.4.∞

4.4.∞

3.3.3.∞

## Notes

1. Conway (2008), p. 263

## References

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