In mathematics, rotation is a notion that originated in geometry. Any rotation is a movement of **a specific space** that retains at least one point. It can, for example, explain the motion of a rigid body around a fixed point. The magnitude of **a clockwise rotation** is negative, whereas the magnitude of a counterclockwise rotation is positive. Rotation can be defined for any dimension, but it is usually defined for three dimensions (3D), which we will discuss here.

A rotation in 3D space can be described by an angle whose degree is measured by a compass or calculator. This angle represents the number of degrees by which the three axes of a coordinate system are rotated. For example, a 90-degree rotation would take the x axis over to the y axis and y back to x. A 180-degree rotation would take all three axes over again in the same direction. There are two ways to specify a rotation explicitly using angles: counterclockwise rotation is specified by a negative angle, while clockwise rotation is specified by a positive angle.

Rotation is also used in physics to describe the transformation of physical quantities such as vectors or tensors under **some operation**. In physics, rotation is often used to describe the transformation of physical quantities that depend on position.

A rotation is the movement of a geometric form around **a central axis**. The amount of rotation is expressed in degrees. When the degrees are positive, the rotation is counterclockwise; when the degrees are negative, the rotation is clockwise. A rotation can be thought of as the opposite of a reflection: If you reflect **a line segment** across its midpoint, it will rotate out from the middle point.

In mathematics, especially in plane geometry, to rotate something is to change its orientation without moving its place. For example, if we want to rotate a triangle around one of its sides, we can do so by adding or subtracting **multiple angles** from 360 degrees. There are two ways to rotate a triangle: either by an integer number of degrees or by a fractional number of degrees. An angle that can be divided into 180 degrees is an integer angle; otherwise it is a fractional angle. For example, 30 degrees is not an integer angle because it can be divided into 180 degrees but 60 degrees is an integer angle because it cannot be divided into 180 degrees.

The term "rotate" also has other meanings in mathematics and physics. In algebra, rotating elements within a matrix means multiplying each element by a specified number (usually a power of some integer number $n \geq 1$). This operation rotates the vector represented by the column vector or row vector into the $n$-th position.

Rotation in mathematics refers to the transformation that revolves a figure around a fixed point known as the center of rotation. Furthermore, the shape and size of an item and its rotation are the same. The figurines, however, may be rotated in different orientations. Furthermore, rotation can occur in either a clockwise or counter-clockwise direction. A rotation of an object about its own axis is called self-rotation.

Figure rotation is used in many fields of science and technology including but not limited to physics, chemistry, engineering, and astronomy. Figure rotation is also useful in mathematics for transforming graphical elements (such as angles or curves) into other figures (such as circles or lines).

In physics, a figure-eight orbit is an elliptical orbit that a body sets itself in motion on. In other words, it is an orbit that a body takes itself. A figure-eight orbit is useful in that it uses less energy than **a circular orbit** of the same radius, so a spacecraft using this technique could reach a destination point by traveling along **a lower-energy course**.

In chemistry, a rotation is a cyclic change in the orientation of a molecule with respect to its environment. This can be achieved by rotating a molecule within **a crystal structure**, or by rotating the crystal around an axis perpendicular to the plane of **the crystal lattice**.