apeirogonal prism
{{short description|Prism with an infinite-sided polygon base}}
{{Uniform tiles db|Uniform tiling stat table|Uinfin}}
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, like a single row of a checkerboard.{{cn|date=October 2020|reason=Please give page number}}
If the sides are squares, it is a uniform tiling. If colored with two sets of alternating squares it is still uniform.{{cn|date=October 2020|reason=Coloring not in Conway}}
File:Infinite prism alternating.svg|Uniform variant with alternate colored square faces.
File:Infinite_bipyramid.svg|Its dual tiling is an apeirogonal bipyramid.
Related tilings and polyhedra
The apeirogonal tiling is the arithmetic limit of the family of prisms t{2, p} or p.4.4, as p tends to infinity, thereby turning the prism into a Euclidean tiling.
An alternation operation can create an apeirogonal antiprism composed of three triangles and one apeirogon at each vertex.
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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
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
- {{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 }}
- {{cite book | title = The Symmetries of Things | year = 2008 | first1 = John H. | last1 = Conway | author2 = Heidi Burgiel | author3 = Chaim Goodman-Strauss | publisher = Taylor & Francis | isbn = 978-1-56881-220-5 }}
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