A plane symmetry is a symmetry of a pattern in the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

: that is, a transformation of the plane that carries any direction lines to lines and preserves many different distances. If one has a pattern in the plane, the set of plane symmetries that preserve the pattern forms a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

. The groups that arise in this way are plane symmetry groups and are of considerable mathematical interest. There are several kinds of plane symmetry groups: *

Reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent co ...

s. These are plane symmetry groups that are generated by reflections, possibly limited to reflections in lines through the origin. *

Rotation group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

s. These groups consist of rotations around a point. * Translation groups. * Symmetries of

geometrical figure A shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type. In geometry, ''shape'' excludes information ...

s. Some of these are reflection groups, e.g., the group of symmetries of the

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

or the

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

. The symmetry group of the

flag of Hong Kong The Regional Flag of the Hong Kong Special Administrative Region of the People's Republic of China depicts a white stylised five-petal Hong Kong orchid tree (''Bauhinia × blakeana'') flower in the centre of a Chinese red field. Its design i ...

or any similar figure without an

axis of symmetry An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...

is a rotation group.

Notes

Euclidean geometry {{Elementary-geometry-stub