Rotational symmetry

Rotational symmetry is invariance under rotation about an axis or around a point. It occurs when a nontrivial rotation maps an object onto itself.

Rotation: A transformation that turns points around a fixed point or axis by a specified angle. Rotational symmetry tests whether an object stays unchanged under this transformation.
Cyclic group: A group generated by repeated application of one element. A finite set of planar rotations forms a cyclic group.
Regular polygon: A polygon with all sides equal and all interior angles equal. A regular n-sided polygon is preserved by rotations in steps of 360°/n.
Reflection symmetry: Invariance under reflection across a line or plane. Reflection reverses orientation, while a rotation preserves it.
Symmetry: A property of an object that remains unchanged under a specified transformation. Rotational symmetry is one specific kind of invariance.
Point group: A group of spatial transformations that leaves at least one point fixed. The rotational symmetries of a three-dimensional object belong to its point group.
Circle: The set of points in a plane at a fixed distance from one center. Every rotation about its center preserves a circle.
Translational symmetry: Invariance under shifting an object by a fixed distance in a specified direction. Translation moves the symmetry pattern rather than turning it about a center.
Group theory: The mathematical study of sets equipped with operations satisfying identity, inverse, and associativity laws. Rotations preserving an object form a group under composition.
Order of rotational symmetry: The number of distinct rotations that map an object onto itself in one full turn. This number counts the object's rotational symmetry operations in the plane.

Linked from 13 pages