In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one
square, one
octagon
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon.
A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...
, and one
apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to the ...
on each
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...
. 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 tr.
Related polyhedra and tilings
Symmetry
The dual of this tiling represents the fundamental domains of
��,4 (*∞42) symmetry. There are 15 small index subgroups constructed from
��,4by 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 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+,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*sup>+,
��*,4sup>+, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2
∞).
See also
*
Tilings of regular polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his ''Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619).
Notation of Eucli ...
*
List of uniform planar tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dua ...
References
*
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)
*
External links
*
*
Hyperbolic and Spherical Tiling Gallery
{{Tessellation
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Semiregular tilings
Truncated tilings