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 In 2-dimensions an archery target has circular symmetry. |  A surface of revolution has circular symmetry around an axis in 3-dimensions. |
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 circle group in the complex plane, or the special orthogonal group SO(2), and unitary group U(1). Reflective circular symmetry is isomorphic with the orthogonal group O(2).
Two dimensions
A 2-dimensional object with circular symmetry would consist of concentric circles and annular domains.
Rotational circular symmetry has all cyclic symmetry, Zn as subgroup symmetries. Reflective circular symmetry has all dihedral symmetry, Dihn as subgroup symmetries.
Three dimensions
In 3-dimensions, a surface or solid of revolution 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, Cnv as subgroups.
A double-cone, bicone, cylinder, toroid and spheroid have circular symmetry, and in addition have a bilateral symmetry perpendicular to the axis of system (or half cylindrical symmetry). These reflective circular symmetries have all discrete prismatic symmetries, Dnh as subgroups.
Four dimensions
 (simple) |  1:5 |  5:1 |
| Cylindrical | Duocylindrical | |
|---|---|---|
In four dimensions, an object can have circular symmetry, on two orthogonal axis planes, or duocylindrical symmetry. For example, the duocylinder and Clifford torus 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 pitch, yaw, and roll. Rotational spherical symmetry has all the discrete chiral 3D point groups as subgroups. Reflectional spherical symmetry is isomorphic with the orthogonal group 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)[citation needed] 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
- Weisstein, Eric W. "Solid of Revolution". MathWorld.
- Weisstein, Eric W. "Surface of Revolution". MathWorld.
- "Orthogonal group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
This article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations Please help improve this article by introducing more precise citations February 2017 Learn how and when to remove this message In 2 dimensions an archery target has circular symmetry A surface of revolution has circular symmetry around an axis in 3 dimensions 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 circle group in the complex plane or the special orthogonal group SO 2 and unitary group U 1 Reflective circular symmetry is isomorphic with the orthogonal group O 2 Two dimensionsA 2 dimensional object with circular symmetry would consist of concentric circles and annular domains Rotational circular symmetry has all cyclic symmetry Zn as subgroup symmetries Reflective circular symmetry has all dihedral symmetry Dihn as subgroup symmetries Three dimensionsA double cone is a surface of revolution generated by a line In 3 dimensions a surface or solid of revolution 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 Cnv as subgroups A double cone bicone cylinder toroid and spheroid have circular symmetry and in addition have a bilateral symmetry perpendicular to the axis of system or half cylindrical symmetry These reflective circular symmetries have all discrete prismatic symmetries Dnh as subgroups Four dimensionsClifford torus stereographic projections simple 1 5 5 1Cylindrical Duocylindrical In four dimensions an object can have circular symmetry on two orthogonal axis planes or duocylindrical symmetry For example the duocylinder and Clifford torus have circular symmetry in two orthogonal axes A spherinder has spherical symmetry in one 3 space and circular symmetry in the orthogonal direction Spherical symmetryAn unmarked sphere has reflectional 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 pitch yaw and roll Rotational spherical symmetry has all the discrete chiral 3D point groups as subgroups Reflectional spherical symmetry is isomorphic with the orthogonal group 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 citation needed depending on the distance to the origin only such as a central force See alsoIsotropy Rotational symmetry Particle in a spherically symmetric potential Gauss s theoremReferencesWeisstein Eric W Solid of Revolution MathWorld Weisstein Eric W Surface of Revolution MathWorld Orthogonal group Encyclopedia of Mathematics EMS Press 2001 1994
