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In
geometry Geometry (from the grc, γεωμετρία; ''geo-'' "earth", ''-metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and ...
, circular symmetry is a type of
continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to anothe ...
for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the
circle group In mathematics, the circle group, denoted by \mathbb T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers :\mathbb T = \~. The circle group ...
in the
complex plane Image:Complex conjugate picture.svg, Geometric representation of ''z'' and its conjugate ''z̅'' in the complex plane. The distance along the light blue line from the origin to the point ''z'' is the ''modulus'' or ''absolute value'' of ''z''. The ...
, or the
special orthogonal 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. The ...
SO(2), and
unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an a ...
U(1). Reflective circular symmetry is isomorphic with the
orthogonal 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. The ...
O(2).


Two dimensions

A 2-dimensional object with circular symmetry would consist of
concentric circle In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point), ...
s and
annular Annulus (or anulus) or annular may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus communis'', around the optic nerve * Annular ligament (disamb ...
domains. Rotational circular symmetry has all
cyclic symmetry In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element&n ...
, Z''n'' as subgroup symmetries. Reflective circular symmetry has all
dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geome ...
, Dih''n'' as subgroup symmetries.


Three dimensions

In 3-dimensions, a
surface Water droplet lying on a damask. Surface tension">damask.html" style="text-decoration: none;"class="mw-redirect" title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating ...
or
solid of revolution In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the ''axis of revolution'') that lies on the same plane. Assuming that the curve does not cross t ...
has circular symmetry around an axis, also called cylindrical symmetry or axial symmetry. An example is a right circular
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines connecti ...
. Circular symmetry in 3 dimensions has all
pyramidal symmetry In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') that does not change the object. They are ...
, C''n''v as subgroups. A double-cone,
bicone100px, right A bicone or dicone (''bi- ''comes from Latin,'' di-'' from Greek) is the three-dimensional surface of revolution of a rhombus around one of its axes of symmetry. Equivalently, a bicone is the surface created by joining two congruent righ ...
,
cylinder A cylinder (from Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can havi ...
,
toroid In mathematics, a toroid is a surface of revolution with a hole in the middle, like a doughnut, forming a solid body. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated a ...
and
spheroid A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry. If the ellipse is rotate ...
have circular symmetry, and in addition have a
bilateral symmetry Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take the face of a human being which has a plan ...
perpendular to the axis of system (or half cylindrical symmetry). These reflective circular symmetries have all discrete prismatic symmetries, D''n''h as subgroups.


Four dimensions

In four dimensions, an object can have circular symmetry, on two orthogonal axis planes, or duocylindrical symmetry. For example the
duocylinderframe, ridge (see below), as a flat torus">Ridge (geometry)">ridge (see below), as a flat torus. The ridge is rotating on XW plane. The duocylinder, or double cylinder, is a geometric object embedded in 4-dimensional Euclidean space, defined ...
and
Clifford torus In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon C ...
have circular symmetry in two orthogonal axes. A
spherinder In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball (or solid 2-sphere), radius ''r''1 and a line segment of length 2''r''2: :D = \ Like the du ...
has spherical symmetry in one 3-space, and circular symmetry in the orthogonal direction.


Spherical symmetry

An analogous 3-dimensional equivalent term is spherical symmetry. Rotational spherical symmetry is isomorphic with the
rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a trans ...
, and can be parametrized by the Davenport chained rotations pitch, yaw, and roll. Rotational spherical symmetry has all the discrete chiral 3D Point groups in three dimensions, point groups as subgroups. Reflectional spherical symmetry is isomorphic with the
orthogonal 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. The ...
O(3) and has the 3-dimensional discrete point groups as subgroups. A scalar field has spherical symmetry if it depends on the distance to the origin only, such as the potential of a central force. A vector field has spherical symmetry if it is in radially inward or outward direction with a magnitude and orientation (inward/outward) depending on the distance to the origin only, such as a central force.


See also

* Isotropy * Rotational symmetry * Particle in a spherically symmetric potential * Gauss's theorem


References

* * *{{springer, title=Orthogonal group, id=p/o070300 Symmetry Rotation