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The dual of this tiling represents the fundamental domains of ˆž,3 *∞32 symmetry. There are 3 small index subgroup constructed from ˆž,3by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A special index 4 reflective subgroup, is ∞,∞,3) (*∞∞3), and its direct subgroup ∞,∞,3)sup>+, (∞∞3), and semidirect subgroup ∞,∞,3+) (3*∞).Norman W. Johnson and Asia Ivic Weiss, ''Quadratic Integers and Coxeter Groups'', Can. J. Math. Vol. 51 (6), 1999 pp. 1307–133
/ref> Given ˆž,3with generating mirrors , then its index 4 subgroup has generators . An index 6 subgroup constructed as ˆž,3* becomes ∞,∞,∞) (*∞∞∞).
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 triapeirogonal tiling is a uniform tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the f ...
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'' ...
with a 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.
Symmetry
The dual of this tiling represents the fundamental domains of ˆž,3 *∞32 symmetry. There are 3 small index subgroup constructed from ˆž,3by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A special index 4 reflective subgroup, is ∞,∞,3) (*∞∞3), and its direct subgroup ∞,∞,3)sup>+, (∞∞3), and semidirect subgroup ∞,∞,3+) (3*∞).Norman W. Johnson and Asia Ivic Weiss, ''Quadratic Integers and Coxeter Groups'', Can. J. Math. Vol. 51 (6), 1999 pp. 1307–133/ref> Given ˆž,3with generating mirrors , then its index 4 subgroup has generators . An index 6 subgroup constructed as ˆž,3* becomes ∞,∞,∞) (*∞∞∞).
Related polyhedra and tiling
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For ''p'' < 6, the members of the sequence are omnitruncated polyhedra (zonohedron
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
s), shown below as spherical tilings. For ''p'' > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling
In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of
Uniform colorings
There is only o ...
.
See also
*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 ...
*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 ...
*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 ...
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)
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External links
* * {{Tessellation Apeirogonal tilings Hyperbolic tilings Isogonal tilings Truncated tilings Uniform tilings