geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, circular symmetry is a type of

continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant u ...

for a planar object that can be rotated by any arbitrary

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

, or the special orthogonal group 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 of concentric circles and annular domains. Rotational circular symmetry has all cyclic symmetry, 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 has circular symmetry around an axis, also called cylindrical symmetry or axial symmetry. An example is a right circular

cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

. 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 , 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 two con ...

, cylinder, toroid and

spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...

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, the face of a human being has a plane of symme ...

perpendicular 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 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) 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 ...

, and can be parametrized by the Davenport chained rotations pitch, yaw, and roll. Rotational spherical symmetry has all the discrete chiral 3D

point group In geometry, a point group is a group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...

s as subgroups. Reflectional spherical symmetry is isomorphic with the

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 each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...

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. 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.

References

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

Two dimensions

Three dimensions

Four dimensions

Spherical symmetry

See also

References