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In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
Rotational circular symmetry is isomorphic with the
In 3-dimensions, a surface or
An analogous 3-dimensional equivalent term is spherical symmetry.
Rotational spherical symmetry is isomorphic with the rotation group SO(3), and can be parametrized by 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. T ...
SO(2), and unitary group U(1). Reflective circular symmetry is isomorphic with the orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
O(2).
Two dimensions
A 2-dimensional object with circular symmetry would consist ofconcentric 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 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 bin ...
, Z''n'' as subgroup symmetries. Reflective circular symmetry has all dihedral symmetry, Dih''n'' as subgroup symmetries.
Three dimensions
In 3-dimensions, a surface 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 the ...
has circular symmetry around an axis, also called cylindrical symmetry or axial symmetry. An example is a right circular cone. 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, bicone, cylinder, toroid and spheroid 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 pla ...
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 andClifford 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 Kingdo ...
have circular symmetry in two orthogonal axes. A spherinder 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), 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 angle ...
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 p ...
s as subgroups. Reflectional spherical symmetry is isomorphic with the orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
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 (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
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
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 (mathematics), surface to the ''divergence'' o ...
References
* * *{{springer, title=Orthogonal group, id=p/o070300 Symmetry Rotation