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Truncated tetraapeirogonal tiling
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In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.
Truncated tetraapeirogonal tiling | |
---|---|
![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex configuration | 4.8.∞ |
Schläfli symbol | tr{∞,4} or |
Wythoff symbol | 2 ∞ 4 | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [∞,4], (*∞42) |
Dual | Order 4-infinite kisrhombille |
Properties | Vertex-transitive |
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Related polyhedra and tilings
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Symmetry
Summarize
Perspective
The dual of this tiling represents the fundamental domains of [∞,4], (*∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,∞,1+,4,1+] (∞2∞2) is the commutator subgroup of [∞,4].
A larger subgroup is constructed as [∞,4*], index 8, as [∞,4+], (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞4), and another [∞*,4], index ∞ as [∞+,4], (∞*2) with gyration points removed as (*2∞). And their direct subgroups [∞,4*]+, [∞*,4]+, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2∞).
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See also
Wikimedia Commons has media related to Uniform tiling 4-8-i.
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
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