apeirogonal antiprism
{{Short description|Antiprism with an infinite-sided polygon base}}
{{Uniform tiles db|Uniform tiling stat table|Uainfin}}
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, it is a uniform tiling. In general, it can have two sets of alternating congruent isosceles triangles, surrounded by two half-planes.
Related tilings and polyhedra
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.
File:Infinite prism.svg|The apeirogonal antiprism can be constructed by applying an alternation operation to an apeirogonal prism.
File:Apeirogonal trapezohedron.svg|The dual tiling of an apeirogonal antiprism is an apeirogonal deltohedron.
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}}
Notes
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References
- The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{ISBN|978-1-56881-220-5}}
- {{cite book | author=Grünbaum, Branko | author-link=Branko Grünbaum | author2=Shephard, G. C. | author2-link=G.C. Shephard | title=Tilings and Patterns | publisher=W. H. Freeman and Company | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }}
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
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