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 c ...
, a symmetry mutation is a mapping of
fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
s between two symmetry groups.
[''Two Dimensional symmetry Mutations'' by Daniel Huson]
/ref> They are compactly expressed in orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advant ...
. These mutations can occur from spherical tiling
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most ...
s to Euclidean tiling
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
s to hyperbolic tiling
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 ...
s. Hyperbolic tilings can also be divided between compact, paracompact and divergent cases.
The uniform tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
Uniform tilings can exist in both the Euclidean plane and Hyperbolic space, hyperbolic plane. Uniform tilings ar ...
s are the simplest application of these mutations, although more complex patterns can be expressed within a fundamental domain.
This article expressed progressive sequences of uniform tilings within symmetry families.
Mutations of orbifolds
Orbifolds with the same structure can be mutated between different symmetry classes, including across curvature domains from spherical, to Euclidean to hyperbolic. This table shows mutation classes.
*''n''22 symmetry
Regular tilings
Prism tilings
Antiprism tilings
*''n''32 symmetry
Regular tilings
Truncated tilings
Quasiregular tilings
Expanded tilings
Omnitruncated tilings
Snub tilings
*''n''42 symmetry
Regular tilings
Quasiregular tilings
Truncated tilings
Expanded tilings
Omnitruncated tilings
Snub tilings
*''n''52 symmetry
Regular tilings
*''n''62 symmetry
Regular tilings
*''n''82 symmetry
Regular tilings
References
Sources
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, {{ISBN, 978-1-56881-220-5}
From hyperbolic 2-space to Euclidean 3-space: Tilings and patterns via topology
Stephen Hyde
Polyhedra
Euclidean tilings
Hyperbolic tilings