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 triheptagonal tiling is a semiregular
tiling
Tiling may refer to:
*The physical act of laying tiles
*Tessellations
Computing
*The compiler optimization of loop tiling
*Tiled rendering, the process of subdividing an image by regular grid
*Tiling window manager
People
*Heinrich Sylvester The ...
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'' ...
. There are one
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
, one
hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
, and one
tetradecagon
In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.
Regular tetradecagon
A '' regular tetradecagon'' has Schläfli symbol and can be constructed as a quasiregular truncated heptagon, t, which alternates two t ...
(14-sides) 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
Uniform colorings
There is only one
uniform coloring
In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following differ ...
of a truncated triheptagonal tiling. (Naming the colors by indices around a vertex: 123.)
Symmetry
Each
triangle in this dual tiling, order 3-7 kisrhombille, represent a fundamental domain of the
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
...
for the symmetry group
,3
Related polyhedra and tilings
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.
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 eight hyperbolic
uniform tilings
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
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*
ttp://www.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Semiregular tilings
Truncated tilings
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