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The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7)
Hyperbolic and Spherical Tiling Gallery
* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch {{Tessellation Hyperbolic tilings Isogonal tilings Isohedral tilings Regular tilings
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 ...
, 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 related as a part of sequence of regular polyhedra with Schläfli symbol . From a Wythoff construction 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.Hurwitz surfaces
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere ( spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defin ...
. This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
s with maximal symmetry group), giving them a tiling by heptagons whose symmetry group equals their automorphism group as Riemann surfaces. The smallest Hurwitz surface is the Klein quartic (genus 3, automorphism group of order 168), and the induced tiling has 24 heptagons, meeting at 56 vertices.
The dual order-7 triangular tiling has the same symmetry group, and thus yields triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle ...
s of Hurwitz surfaces.
See also
*Hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathema ...
*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 ...
*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 ...
* List of regular polytopes
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.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch {{Tessellation Hyperbolic tilings Isogonal tilings Isohedral tilings Regular tilings