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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, C''n''v as subgroups.
A double-cone, bicone,
An analogous 3-dimensional equivalent term is spherical symmetry.
Rotational spherical symmetry is isomorphic with the
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 the complex plane, 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 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 domains. Rotational circular symmetry has allcyclic 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, 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. Circular symmetry in 3 dimensions has all pyramidal symmetry, C''n''v as subgroups.
A double-cone, bicone, 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 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 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 and Clifford torus have circular symmetry in two orthogonal axes. Aspherinder
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 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. ...
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
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
* 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