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Hyperbolic and Spherical Tiling Gallery
* ttp://www.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch Hyperbolic tilings Isogonal tilings Semiregular tilings {{geometry-stub
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 ...
, the rhombitriheptagonal tiling is a semiregular tiling 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' ...
. At 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 positio ...
of the tiling there is one triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
and one heptagon
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.
The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than '' hepta-'', a Greek-derived nu ...
, alternating between two 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 a ...
s. The tiling has Schläfli symbol rr. It can be seen as constructed as a rectified triheptagonal tiling, r, as well as an expanded heptagonal tiling or expanded order-7 triangular tiling
In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of .
Hurwitz surfaces
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is ...
.
Dual tiling
The dual tiling is called a ''deltoidal triheptagonal tiling'', and consists of congruentkites
A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the face ...
. It is formed by overlaying an order-3 heptagonal tiling and an order-7 triangular tiling
In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of .
Hurwitz surfaces
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is ...
.
:
Related polyhedra and tilings
From aWythoff 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 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.
Symmetry mutations
This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of thehyperbolic 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' ...
. These vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
figures have (*n32) reflectional symmetry.
See also
* Rhombitrihexagonal tiling * Order-3 heptagonal tiling * Tilings of regular polygons *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 dual ...
* Kagome lattice
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2. ...
References
* John H. Conway, 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 {{geometry-stub