In
geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
. It has
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
of t.
Uniform colorings
A half symmetry coloring is tr, has two types of apeirogons, shown red and yellow here. If the apeirogonal curvature is too large, it doesn't converge to a single ideal point, like the right image, red apeirogons below.
Coxeter diagram are shown with dotted lines for these divergent,
ultraparallel mirrors.
Symmetry
From
ˆž,∞symmetry, there are 15 small index subgroup 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 symmetry can be doubled as
∞42 symmetry by adding a mirror bisecting the fundamental domain. The
subgroup index In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the
number of left cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''.
The index is denoted , G:H, or :H/math> or (G ...
-8 group,
+,∞,1+,∞,1+(∞∞∞∞) is the
commutator subgroup of
ˆž,∞
Related polyhedra and tiling
See also
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Uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on its v ...
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List of regular polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ' ...
References
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John H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
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External links
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Hyperbolic and Spherical Tiling Gallery
{{Tessellation
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Order-4 tilings
Truncated tilings
Uniform tilings