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 order-6 square tiling is a
regular
Regular may refer to:
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings
Other uses
* Regular character, ...
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' ...
. 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 .
Symmetry
This tiling represents a hyperbolic
kaleidoscope
A kaleidoscope () is an optical instrument with two or more reflecting surfaces (or mirrors) tilted to each other at an angle, so that one or more (parts of) objects on one end of these mirrors are shown as a symmetrical pattern when viewed fro ...
of 4 mirrors meeting as edges of a square, with six squares around every vertex. This symmetry by
orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Horton Conway, John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curv ...
is called (*3333) with 4 order-3 mirror intersections. In
Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, ...
can be represented as
,4* removing two of three mirrors (passing through the square center) in the
,4symmetry. The *3333 symmetry can be doubled to
663 symmetry by adding a mirror bisecting the fundamental domain.
This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has 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 ...
t
1. A second 6-color symmetry can be constructed from a hexagonal symmetry domain.
Example artwork
Around 1956,
M.C. Escher
Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics.
Despite wide popular int ...
explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician
H.S.M. Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in England and educated ...
inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's wood engravings Circle Limit I–IV demonstrate this concept. The last one ''Circle Limit IV (Heaven and Hell)'', (1960) tiles repeating
angel
An angel is a spiritual (without a physical body), heavenly, or supernatural being, usually humanoid with bird-like wings, often depicted as a messenger or intermediary between God (the transcendent) and humanity (the profane) in variou ...
s and
devil
A devil is the mythical personification of evil as it is conceived in various cultures and religious traditions. It is seen as the objectification of a hostile and destructive force. Jeffrey Burton Russell states that the different conce ...
s by (*3333) symmetry on a hyperbolic plane in a
Poincaré disk projection.
The artwork seen below has an approximate hyperbolic mirror overlay added to show the square symmetry domains of the order-6 square tiling. If you look closely, you can see one of four angels and devils around each square are drawn as back sides. Without this variation, the art would have a 4-fold
gyration point
In geometry, a gyration is a rotation in a discrete subgroup of symmetries of the Euclidean plane such that the subgroup does not also contain a reflection symmetry whose axis passes through the center of rotational symmetry. In the orbifold co ...
at the center of each square, giving (4*3),
,4+symmetry.
[Conway, The Symmetry of Things (2008), p.224, Figure 17.4]
''Circle Limit IV''
:
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4
n).
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with
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 ...
, and
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in England and educated ...
, progressing to infinity.
See also
*
Square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille.
Structure and properties
The square tili ...
*
Uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as Face (geometry), faces and is vertex-transitive (Tran ...
*
List of regular polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
Overview
This table shows a summary of regular polytope counts by rank.
There are no Euclidean regular star tessellations in any number of dimensions.
...
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.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don HatchGenusView 0.4 previewView of hyperbolic tiling, and matching 3D torus surface.
{{Tessellation
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-6 tilings
Regular tilings
Square tilings