In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
s and one
octagon
In geometry, an octagon () 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, which alternates two types of edges. A truncated octagon, t is a ...
on each
vertex. It has
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
of t.
Uniform colors
Symmetry
The dual of this tiling represents the fundamental domains of *443 symmetry. It only has one subgroup 443, replacing mirrors with gyration points.
This symmetry can be doubled to
832 symmetry
In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one Square (geometry), square, one hexagon, and one hexadecagon (16-sides) on each vertex (geometry), vertex. It has Schläfli symbol of ''tr ...
by adding a bisecting mirror to the fundamental domain.
Related tilings
From a
Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.
Construction process ...
there are ten hyperbolic
uniform tilings
A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
that can be based from the regular octagonal tiling.
It can also be generated from the (4 3 3) hyperbolic tilings:
This hyperbolic tiling is topologically related as a part of sequence of uniform
truncated polyhedra with
vertex configuration
In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...
s (n.6.6), 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
*
Triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
*
Order-3 octagonal tiling
*
Order-8 triangular tiling
In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of ', having eight regular triangles around each vertex.
Uniform colorings
The half symmetry +,8,3= 4,3,3)can be sho ...
*
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 (Latin: ''The Harmony of the World'', 1619).
Notation of Euclidean tilings
Eucl ...
*
List of uniform 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 du ...
References
*
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
*
External links
*
*
Hyperbolic and Spherical Tiling Gallery*
ttp://www.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Order-8 tilings
Semiregular tilings
Triangular tilings
Truncated tilings
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