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 ...
, the order-6 square tiling is a
regular 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 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 regular symmetrical pattern when v ...
of 4 mirrors meeting as edges of a square, with six squares around every vertex. This symmetry by
orbifold notation is called (*3333) with 4 order-3 mirror intersections. In
Coxeter notation 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
In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of t. It can also be identically constructed as a cantic order-6 square tiling, h2
Uniform colorings
By ...
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 Mathematics and art, mathematically inspired woodcuts, lithography, lithographs, and mezzotints.
Despite wide popular interest, Escher was for ...
explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician
H.S.M. Coxeter 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
In various theistic religious traditions an angel is a supernatural spiritual being who serves God.
Abrahamic religions often depict angels as benevolent celestial intermediaries between God (or Heaven) and humanity. Other roles inclu ...
s and
devil
A devil is the 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 conceptions of ...
s by (*3333) symmetry on a hyperbolic plane in a
Poincaré disk
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* L ...
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 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 , and
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, 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. It has Schläfli symbol of meaning it has 4 squares around every vertex.
Conway called it a quadrille.
The internal angle of th ...
*
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 faces and is vertex-transitive ( transitive on its v ...
*
List of regular polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli ...
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.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