Symmetry (from Greek συμμετρεῖν symmetría "measure together") generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise.
Although the meanings are distinguishable in some contexts, both meanings of "symmetry" are related and discussed in parallel.
The precise notions of symmetry have various measures and operational definitions. For example, symmetry may be observed
This article describes these notions of symmetry from four perspectives. The first is that of symmetry in geometry, which is the most familiar type of symmetry for many people. The second perspective is the more general meaning of symmetry in mathematics as a whole. The third perspective describes symmetry as it relates to science and technology. In this context, symmetries underlie some of the most profound results found in modern physics, including aspects of space and time. Finally, a fourth perspective discusses symmetry in the humanities, covering its rich and varied use in history, architecture, art, and religion.
The opposite of symmetry is asymmetry.
The most familiar type of symmetry for many people is geometrical symmetry. Formally, this means symmetry under a sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space. These isometries consist of reflections, rotations, translations and combinations of these basic operations.
Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection.
In 1D, there is a point of symmetry. In 2D there is an axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric.
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: the two halves are each other's mirror image. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry, for the same reason.
If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called vertical symmetry. One can better use an unambiguous formulation; e.g., "T has a vertical symmetry axis" or "T has left-right symmetry".
The triangles with this symmetry are isosceles, the quadrilaterals with this symmetry are the kites and the isosceles trapezoids.
For each line or plane of reflection, the symmetry group is isomorphic with Cs, one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is a half-plane or half-space.
Bilateria (bilateral animals, including humans) are more or less symmetric with respect to the sagittal plane.
Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries; i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of E+(m).
Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole E+(m). 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 we can take that point as origin. These rotations form the special orthogonal group SO(m), the group of m × m orthogonal matrices with determinant 1. For m = 3 this is the rotation group SO(3).
In another meaning of the word, the rotation group of an object is the symmetry group within E+(m), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. 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. See also rotational invariance.
A glide reflection symmetry (in 3D: a glide plane symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. It implies translational symmetry with twice the translation vector. The symmetry group is isomorphic with Z.
Although the meanings are distinguishable in some contexts, both meanings of "symmetry" are related and discussed in parallel.
The precise notions of symmetry have various measures and operational definitions. For example, symmetry may be observed
- with respect to the passage of time;
- as a spatial relationship;
- through geometric transformations such as scaling, reflection, and rotation;
- through other kinds of functional transformations; and
- as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.
This article describes these notions of symmetry from four perspectives. The first is that of symmetry in geometry, which is the most familiar type of symmetry for many people. The second perspective is the more general meaning of symmetry in mathematics as a whole. The third perspective describes symmetry as it relates to science and technology. In this context, symmetries underlie some of the most profound results found in modern physics, including aspects of space and time. Finally, a fourth perspective discusses symmetry in the humanities, covering its rich and varied use in history, architecture, art, and religion.
The opposite of symmetry is asymmetry.
The most familiar type of symmetry for many people is geometrical symmetry. Formally, this means symmetry under a sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space. These isometries consist of reflections, rotations, translations and combinations of these basic operations.
Reflection symmetry
In 1D, there is a point of symmetry. In 2D there is an axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric.
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: the two halves are each other's mirror image. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry, for the same reason.
If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called vertical symmetry. One can better use an unambiguous formulation; e.g., "T has a vertical symmetry axis" or "T has left-right symmetry".
The triangles with this symmetry are isosceles, the quadrilaterals with this symmetry are the kites and the isosceles trapezoids.
For each line or plane of reflection, the symmetry group is isomorphic with Cs, one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is a half-plane or half-space.
Bilateria (bilateral animals, including humans) are more or less symmetric with respect to the sagittal plane.
Rotational symmetry
Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole E+(m). 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 we can take that point as origin. These rotations form the special orthogonal group SO(m), the group of m × m orthogonal matrices with determinant 1. For m = 3 this is the rotation group SO(3).
In another meaning of the word, the rotation group of an object is the symmetry group within E+(m), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. 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. See also rotational invariance.
Translational symmetry
Translational symmetry leaves an object invariant under a discrete or continuous group of translations 
Glide reflection symmetry
A glide reflection symmetry (in 3D: a glide plane symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. It implies translational symmetry with twice the translation vector. The symmetry group is isomorphic with Z.
Rotoreflection symmetry
In 3D, rotoreflection or improper rotation in the strict sense is rotation about an axis, combined with reflection in a plane perpendicular to that axis. As symmetry groups with regard to a roto-reflection we can distinguish:
- the angle has no common divisor with 360°, the symmetry group is not discrete
- 2n-fold rotoreflection (angle of 180°/n) with symmetry group S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); a special case is n = 1, inversion, because it does not depend on the axis and the plane, it is characterized by just the point of inversion.
- 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. See also point groups in three dimensions.
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