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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 ano ...
for a planar object that can be rotated by any arbitrary
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
and map onto itself. Rotational circular symmetry is isomorphic with the
circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, 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 = \. ...
in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by 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. ...
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 ...
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. ...
O(2).


Two dimensions

A 2-dimensional object with circular symmetry would consist of concentric circles and
annular Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * '' Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus co ...
domains. Rotational circular symmetry has all
cyclic symmetry In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
, 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, g ...
, Dih''n'' as subgroup symmetries.


Three dimensions

In 3-dimensions, a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
or
solid of revolution In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the '' axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is ...
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 con ...
. Circular symmetry in 3 dimensions has all pyramidal symmetry, C''n''v as subgroups. A double-cone,
bicone In geometry, a bicone or dicone (from la, bi-, and Greek: ''di-'', both meaning "two") 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 ...
,
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an ...
,
toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
and
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, 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 ...
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 ...
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
duocylinder The duocylinder, also called the double cylinder or the bidisc, is a geometric object embedded in 4- dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii ''r''1 and ''r''2: :D = \left\ It is analogo ...
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 Kingd ...
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) of radius ''r''1 and a line segment of length 2''r''2: :D = \ Like th ...
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 ...
, and can be parametrized by the
Davenport chained rotations In physics and engineering, Davenport chained rotations are three chained intrinsic rotations about body-fixed specific axes. Euler rotations and Tait–Bryan rotations are particular cases of the Davenport general rotation decomposition. The ang ...
pitch, yaw, and roll. Rotational spherical symmetry has all the discrete chiral 3D
point group In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...
s 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. ...
O(3) and has the 3-dimensional discrete point groups as subgroups. A
scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
has spherical symmetry if it depends on the distance to the origin only, such as the
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
of a
central force In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...
. 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 Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
*
Rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...
*
Particle in a spherically symmetric potential In the quantum mechanics description of a particle in spherical coordinates, a spherically symmetric potential, is a potential that depends only on the distance between the particle and a defined centre point. One example of a spherical potential ...
*
Gauss's theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...


References

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