Ingeometry, an object hassymmetry if there is anoperation ortransformation (such astranslation,scaling,rotation orreflection) that maps the figure/object onto itself (i.e., the object has aninvariance under the transform).^[1] Thus, a symmetry can be thought of as an immunity to change.^[2] 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 besymmetric under rotation or to haverotational symmetry. If the isometry is the reflection of aplane figure about a line, then the figure is said to havereflectional symmetry orline symmetry;^[3] it is also possible for a figure/object to have more than one line of symmetry.^[4]

The types of symmetries that are possible for a geometric object depend on the set ofgeometric 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 mathematicalgroup, thesymmetry group of the object.^[5]

The most common group of transforms applied to objects are termed theEuclidean group of "isometries", which are distance-preserving transformations in space commonly referred to as two-dimensional or three-dimensional (i.e., inplane geometry orsolid geometryEuclidean spaces). These isometries consist ofreflections,rotations,translations, and combinations of these basic operations.^[6] 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.^[7] 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 theCartan–Dieudonné theorem, anorthogonal transformation inn-dimensional space can be represented by the composition of at mostn reflections.

Reflectional symmetry, linear symmetry, mirror symmetry, mirror-image symmetry, orbilateral symmetry is symmetry with respect to reflection.^[8]

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.^[3][9] 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, seemirror image).

The axis of symmetry of a two-dimensional figure is a line such that, if aperpendicular 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. asquare 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 acircle, which has infinitely many axes of symmetry passing through its center for the same reason.^[10]

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".

Thetriangles with reflection symmetry areisosceles, thequadrilaterals with this symmetry arekites and isoscelestrapezoids.^[11]

For each line or plane of reflection, thesymmetry group isisomorphic with C_s (seepoint groups in three dimensions for more), one of the three types of order two (involutions), hence algebraically isomorphic to C₂. Thefundamental domain is ahalf-plane orhalf-space.^[12]

Reflection symmetry can be generalized to otherisometries ofm-dimensional space which areinvolutions, such as

(x₁, ...,x_m) ↦ (−x₁, ..., −x_k, x_k+1, ...,x_m)

in a certain system ofCartesian coordinates. This reflects the space along an(m−k)-dimensionalaffine subspace.^[13] Ifk = m, then such a transformation is known as apoint reflection, or aninversion through a point. On theplane (m = 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.^[14]

Such a "reflection" preservesorientation if and only ifk is aneven number.^[15] This implies that form = 3 (as well as for other odd m), a point reflection changes the orientation of the space, like a mirror-image symmetry. That explains why in physics, the termP-symmetry (P stands forparity) is used for both point reflection and mirror symmetry. Since a point reflection in three dimensions changes aleft-handed coordinate system into aright-handed coordinate system, symmetry under a point reflection is also called a left-right symmetry.^[16]

Rotational symmetry is symmetry with respect to some or all rotations inm-dimensional Euclidean space. Rotations aredirect isometries, which are isometries that preserveorientation.^[17] Therefore, a symmetry group of rotational symmetry is a subgroup of the special Euclidean groupE⁺(m).

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),^[18] 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, one can take that point as origin. These rotations form thespecial orthogonal group SO(m), which can be represented by the group ofm × morthogonal matrices withdeterminant 1. Form = 3, this is therotation group SO(3).^[19]

Phrased slightly differently, the rotation group of an object is the symmetry group within E⁺(m), the group of rigid motions;^[20] 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 ofNoether's theorem, rotational symmetry of a physical system is equivalent to theangular momentumconservation law.^[21] For more, seerotational invariance.

Translational symmetry leaves an object invariant under a discrete or continuous group oftranslations${\displaystyle T_a(p)=p+a}$.^[22] 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.

In 2D, aglide reflection symmetry (also called aglide plane symmetry in 3D, and atransflection 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).^[2][23] 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 thefrieze groupp11g, and is isomorphic with the infinite cyclic groupZ.

In 3D, arotary reflection,rotoreflection orimproper rotation is a rotation about an axis combined with reflection in a plane perpendicular to that axis.^[24] 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 isS_2n of order 2n (not to be confused withsymmetric groups, for which the same notation is used; the abstract group isC_2n). A special case isn = 1, aninversion, because it does not depend on the axis and the plane. It is characterized by just the point of inversion.
• The groupC_nh (angle of 360°/n); for oddn, this is generated by a single symmetry, and the abstract group isC_2n, for evenn. This is not a basic symmetry but a combination.

For more, seepoint groups in three dimensions.

In 3D geometry and higher, a screw axis (or rotary translation) is a combination of a rotation and a translation along the rotation axis.^[25]

Helical symmetry is the kind of symmetry seen in everyday objects such assprings,Slinky toys,drill bits, andaugers. The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at a constantangular 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 acoiling angle that helps define the properties of the traced helix.^[26] 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:

• Infinite helical symmetry: If there are no distinguishing features along the length of ahelix or helix-like object, the object will have infinite symmetry much like that of a circle, but with the additional requirement of translation along the long axis of the object—to return it to its original appearance.^[27] A helix-like object is one that has at every point the regular angle of coiling of a helix, but which can also have across section of indefinitely high complexity, provided only that precisely the same cross section exists (usually after a rotation) at every point along the length of the object. Simple examples include evenly coiled springs, slinkies, drill bits, and augers. Stated more precisely, an object has infinite helical symmetries if for any small rotation of the object around its central axis, there exists a point nearby (the translation distance) on that axis at which the object will appear exactly as it did before. It is this infinite helical symmetry that gives rise to the curious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices to move materials along their length, provided that they are combined with a force such as gravity or friction that allows the materials to resist simply rotating along with the drill or auger.
• n-fold helical symmetry: If the requirement that every cross section of the helical object be identical is relaxed, then additional lesser helical symmetries would become possible. For example, the cross section of the helical object may change, but may still repeat itself in a regular fashion along the axis of the helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle. If the angle of rotation at which the symmetry occurs divides evenly into a full circle (360°), then the result is the helical equivalent of a regular polygon. This case is calledn-fold helical symmetry, wheren = 360° (such as the case of adouble helix). This concept can be further generalized to include cases where${\displaystyle m\theta }$ is a multiple of360° – that is, the cycle does eventually repeat, but only after more than one full rotation of the helical object.
• Non-repeating helical symmetry: This is the case in which the angle of rotation θ required to observe the symmetry isirrational. The angle of rotation never repeats exactly, no matter how many times the helix is rotated. Such symmetries are created by using a non-repeatingpoint group in two dimensions.DNA, with approximately 10.5base pairs per turn, is an example of this type of non-repeating helical symmetry.^[28]

In 4D, a double rotation symmetry can be generated as the composite of two orthogonal rotations.^[29] It is similar to 3D screw axis which is the composite of a rotation and an orthogonal translation.

A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:

• The group ofsimilarity transformations;^[30] i.e.,affine transformations represented by amatrix A that is a scalar times anorthogonal matrix. Thushomothety is added,self-similarity is considered a symmetry.
• The group of affine transformations represented by a matrix A with determinant 1 or −1; i.e., the transformations which preserve area.^[31]

  This adds, e.g., obliquereflection symmetry.

• The group of all bijectiveaffine transformations.
• The group ofMöbius transformations which preservecross-ratios.

  This adds, e.g.,inversive reflections such ascircle reflection on the plane.

InFelix Klein'sErlangen program, each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent.^[32] For example, the Euclidean group definesEuclidean geometry, whereas the group of Möbius transformations definesprojective geometry.

Scale symmetry means that if an object is expanded or reduced in size, the new object has the same properties as the original.^[33] This self-similarity is seen in many natural structures such as cumulus clouds, lightning, ferns and coastlines, over a wide range of scales. It is generally not found in gravitationally bound structures, for example the shape of the legs of anelephant and amouse (so-calledallometric scaling). Similarly, if a soft wax candle were enlarged to the size of a tall tree, it would immediately collapse under its own weight.

A more subtle form of scale symmetry is demonstrated byfractals. As conceived byBenoît Mandelbrot, fractals are a mathematical concept in which the structure of a complex form looks similar at any degree ofmagnification,^[34] well seen in theMandelbrot set. Acoast is an example of a naturally occurring fractal, since it retains similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand. The branching of trees, which enables small twigs to stand in for full trees indioramas, is another example.

Because fractals can generate the appearance ofpatterns in nature, they have a beauty and familiarity not typically seen with mathematically generated functions. Fractals have also found a place incomputer-generated movie effects, where their ability to create complex curves with fractal symmetries results in more realisticvirtual worlds.

With every geometry,Felix Klein associated an underlyinggroup of symmetries. The hierarchy of geometries is thus mathematically represented as a hierarchy of thesegroups, and hierarchy of theirinvariants. For example, lengths, angles and areas are preserved with respect to theEuclidean group of symmetries, while only theincidence structure and thecross-ratio are preserved under the most generalprojective transformations. A concept ofparallelism, which is preserved inaffine geometry, is not meaningful inprojective geometry. Then, by abstracting the underlyinggroups of symmetries from the geometries, the relationships between them can be re-established at the group level. Since the group of affine geometry is asubgroup of the group of projective geometry, any notion invariant in projective geometry isa priori meaningful in affine geometry; but not the other way round. If you add required symmetries, you have a more powerful theory but fewer concepts and theorems (which will be deeper and more general).

William Thurston introduced a similar version of symmetries in geometry. Amodel geometry is asimply connectedsmooth manifoldX together with a transitive action of aLie groupG onX with compact stabilizers. TheLie group can be thought of as the group of symmetries of the geometry.

A model geometry is calledmaximal ifG is maximal among groups acting smoothly and transitively onX with compact stabilizers, i.e. if it is the maximal group of symmetries. Sometimes this condition is included in the definition of a model geometry.

Ageometric structure on a manifoldM is a diffeomorphism fromM toX/Γ for some model geometryX, where Γ is a discrete subgroup ofG acting freely onX. If a given manifold admits a geometric structure, then it admits one whose model is maximal.

A3-dimensional model geometryX is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled onX. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes calledThurston geometries. (There are also uncountably many model geometries without compact quotients.)

• Fractal
• Symmetric relation

• Calotta: A World of Symmetry
• Dutch: Symmetry Around a Point in the Plane

• Symmetry

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