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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 triapeirogonal tiling (or trigonal-horocyclic tiling) is a uniform tiling of the hyperbolic plane 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 mor ...
of r.
Uniform colorings
The half-symmetry form, , has two colors of triangles: :Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniformquasiregular polyhedra
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular polygon, regular faces, which alternate around each vertex (geometry), vertex. They are vertex-transitive and edge-transitive, hence a step closer ...
with vertex configurations (3.n.3.n), and ,3Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
symmetry.
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 ...
*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 it ...
References
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) *External links
* * * http://bendwavy.org/klitzing/incmats/o3xinfino.htm * o3x∞o {{Tessellation Apeirogonal tilings Hyperbolic tilings Isogonal tilings Isotoxal tilings Uniform tilings