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In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the
hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares. The tiling has
Hyperbolic and Spherical Tiling Gallery
* ttp://www.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch Hyperbolic tilings Isogonal tilings Semiregular tilings {{geometry-stub
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 ...
rr. It can be seen as constructed as a rectified triheptagonal tiling
In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r.
Compar ...
, r, as well as an expanded heptagonal tiling
In geometry, a heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of , having three regular heptagons around each vertex.
Images
Related polyhedra and tilings
This tiling is topologically r ...
or expanded order-7 triangular tiling.
Dual tiling
The dual tiling is called a ''deltoidal triheptagonal tiling'', and consists of congruent kites. It is formed by overlaying an order-3 heptagonal tiling and an order-7 triangular tiling. :
Related polyhedra and tilings
From a Wythoff construction there are eight hyperbolicuniform 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.
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 the hyperbolic plane. These vertex-transitive figures have (*n32) reflectionalsymmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
.
See also
* Rhombitrihexagonal tiling * Order-3 heptagonal tiling *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 tilings
* Kagome lattice
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 {{geometry-stub