In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also possible for a figure/object to have more than one line of symmetry.

The types of symmetries that are possible for a geometric object depend on the set of geometric transforms available, and on what object properties should remain unchanged after a transformation. Because the composition of two transforms is also a transform and every transform has, by definition, an inverse transform that undoes it, the set of transforms under which an object is symmetric form a mathematical group, the symmetry group of the object.

Euclidean symmetries in general The most common group of transforms applied to objects are termed the Euclidean group of "isometries", which are distance-preserving transformations in space commonly referred to as two-dimensional or three-dimensional (i.e., in plane geometry or solid geometry Euclidean spaces). These isometries consist of reflections, rotations, translations, and combinations of these basic operations. Under an isometric transformation, a geometric object is said to be symmetric if, after transformation, the object is indistinguishable from the object before the transformation. A geometric object is typically symmetric only under a subset or "subgroup" of all isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups, and by the types of object invariance that are possible in geometry.

By the Cartan–Dieudonné theorem, an orthogonal transformation in n-dimensional space can be represented by the composition of at most n reflections. + Basic isometries by dimension1D2D3D4DReflectionsPointAffinePointAffinePointAffinePointAffine1ReflectionReflectionReflectionReflection2 Translation Rotation Translation Rotation Translation Rotation Translation3 Transflection Rotoreflection Transflection Rotoreflection Transflection4 Rotary translation Double rotation Rotary translation5 Rotary transflection

Reflectional symmetry

Reflectional symmetry, linear symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection.

In one dimension, there is a point of symmetry about which reflection takes place; in two dimensions, there is an axis of symmetry (a.k.a., line of symmetry), and in three dimensions there is a plane of symmetry. An object or figure for which every point has a one-to-one mapping onto another, equidistant from and on opposite sides of a common plane is called mirror symmetric (for more, see mirror image).

The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical as mirror images of each other. For example. a square has four axes of symmetry, because there are four different ways to fold it and have the edges match each other. Another example would be that of a circle, which has infinitely many axes of symmetry passing through its center for the same reason.

If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called vertical symmetry. Thus one can describe this phenomenon unambiguously by saying that "T has a vertical symmetry axis", or that "T has left-right symmetry".

The triangles with reflection symmetry are isosceles, the quadrilaterals with this symmetry are kites and isosceles trapezoids.

For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions for more), one of the three types of order two (involutions), hence algebraically isomorphic to C2. The fundamental domain is a half-plane or half-space.

Point reflection and other involutive isometries

Reflection symmetry can be generalized to other isometries of -dimensional space which are involutions, such as

in a certain system of Cartesian coordinates. This reflects the space along an -dimensional affine subspace. If  = , then such a transformation is known as a point reflection, or an inversion through a point. On the plane ( = 2), a point reflection is the same as a half-turn (180°) rotation; see below. Antipodal symmetry is an alternative name for a point reflection symmetry through the origin.

Such a "reflection" preserves orientation if and only if is an even number.William H. Barker, Roger Howe Continuous Symmetry: From Euclid to Klein (Google eBook) American Mathematical Soc This implies that for  = 3 (as well as for other odd ), a point reflection changes the orientation of the space, like a mirror-image symmetry. That explains why in physics, the term P-symmetry (P stands for parity) is used for both point reflection and mirror symmetry. Since a point reflection in three dimensions changes a left-handed coordinate system into a right-handed coordinate system, symmetry under a point reflection is also called a left-right symmetry.

Rotational symmetry

Rotational symmetry is symmetry with respect to some or all rotations in -dimensional Euclidean space. Rotations are direct isometries, which are isometries that preserve orientation.Vladimir G. Ivancevic, Tijana T. Ivancevic (2005) Natural Biodynamics World Scientific Therefore, a symmetry group of rotational symmetry is a subgroup of the special Euclidean group E+().

Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations (because translations are compositions of rotations about distinct points), and the symmetry group is the whole E+(). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.

For symmetry with respect to rotations about a point, one can take that point as origin. These rotations form the special orthogonal group SO(), which can be represented by the group of orthogonal matrices with determinant 1. For  = 3, this is the rotation group SO(3).

Phrased slightly differently, the rotation group of an object is the symmetry group within E+(), the group of rigid motions; that is, the intersection of the full symmetry group and the group of rigid motions. For chiral objects, it is the same as the full symmetry group.

Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. For more, see rotational invariance.

Translational symmetry

Translational symmetry leaves an object invariant under a discrete or continuous group of translations .Stenger, Victor J. (2000) and Mahou Shiro (2007). Timeless Reality. Prometheus Books. Especially chapter 12. Nontechnical. The illustration on the right shows four congruent footprints generated by translations along the arrow. If the line of footprints were to extend to infinity in both directions, then they would have a discrete translational symmetry; any translation that mapped one footprint onto another would leave the whole line unchanged.

Glide reflection symmetry

In 2D, a glide reflection symmetry (also called a glide plane symmetry in 3D, and a transflection in general) means that a reflection in a line or plane combined with a translation along the line or in the plane, results in the same object (such as in the case of footprints).. The composition of two glide reflections results in a translation symmetry with twice the translation vector. The symmetry group comprising glide reflections and associated translations is the frieze group p11g, and is isomorphic with the infinite cyclic group Z.

Rotoreflection symmetry

In 3D, a rotary reflection, rotoreflection or improper rotation is a rotation about an axis combined with reflection in a plane perpendicular to that axis.Robert O. Gould, Steffen Borchardt-Ott (2011)Crystallography: An Introduction Springer Science & Business Media The symmetry groups associated with rotoreflections include: if the rotation angle has no common divisor with 360°, the symmetry group is not discrete. if the rotoreflection has a 2n-fold rotation angle (angle of 180°/n), the symmetry group is S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; the abstract group is C2n). A special case is n = 1, an inversion, because it does not depend on the axis and the plane. It is characterized by just the point of inversion. The group Cnh (angle of 360°/n); for odd n, this is generated by a single symmetry, and the abstract group is C2n, for even n. This is not a basic symmetry but a combination.

For more, see point groups in three dimensions.

Helical symmetry

In 3D geometry and higher, a screw axis (or rotary translation) is a combination of a rotation and a translation along the rotation axis.Bottema, O, and B. Roth, Theoretical Kinematics, Dover Publications (September 1990)

Helical symmetry is the kind of symmetry seen in everyday objects such as springs, Slinky toys, drill bits, and augers. The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at a constant angular speed, while simultaneously translating at a constant linear speed along its axis of rotation. At any point in time, these two motions combine to give a coiling angle that helps define the properties of the traced helix.George R. McGhee (2006) The Geometry of Evolution: Adaptive Landscapes and Theoretical Morphospaces Cambridge University Press p.64 When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°. Conversely, if the object rotates slowly and translates quickly, the coiling angle will approach 90°.

Three main classes of helical symmetry can be distinguished, based on the interplay of the angle of coiling and translation symmetries along the axis: