Rotation
A rotation turns a figure by a specified angle about a fixed center point, producing a congruent image.
Formula
(x,y) \to (x\cos t - y\sin t, \ x\sin t + y\cos t)
Definition
A rotation turns a shape around a fixed center point by a specified angle and direction (clockwise or counterclockwise), keeping the shape's size and angles the same. A rotation by angle $\theta$ counterclockwise about the origin maps $(x, y)$ to $(x\cos\theta - y\sin\theta, \ x\sin\theta + y\cos\theta)$; to rotate about an arbitrary center $(h, k)$, translate so $(h,k)$ is at the origin, rotate, then translate back, and rotations are orientation-preserving isometries. Formally, rotation by $\theta$ about the origin is the linear map $R(\theta) = \begin{bmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{bmatrix}$, an element of the special orthogonal group $SO(2)$, where composing rotations gives $R(\theta_1)R(\theta_2) = R(\theta_1+\theta_2)$ ($SO(2)$ is abelian); in 3-D, rotations form the non-abelian group $SO(3)$, where order of rotations about different axes matters.
Example
Rotating a square $90$ degrees counterclockwise about its center leaves it looking the same, thanks to its $4$-fold rotational symmetry, while rotating a scalene triangle $90$ degrees about the origin moves each vertex to a new location. Rotating point $(3, 0)$ by $90$ degrees counterclockwise gives $(0, 3)$; rotating that by another $90$ degrees gives $(-3, 0)$, then $(0, -3)$, and a fourth $90$-degree rotation returns to the start. In 3-D, rotating $90$ degrees about the x-axis then $90$ degrees about the y-axis gives a different result than the reverse order, the non-commutativity of $SO(3)$ fundamental in robotics (gimbal lock) and quantum mechanics (spin operators).
Key Insight
Rotating $180$ degrees ends at the same place whether you go clockwise or counterclockwise, but $90$ degrees clockwise differs from $90$ degrees counterclockwise. A rotation by $\theta$ and one by $\theta+360^\circ$ are identical, so rotational symmetry is described by the smallest rotation that maps a shape to itself; for a regular n-gon, the minimum rotation is $360/n$ degrees. The non-commutativity of 3-D rotations has profound physical consequences, underlying the non-commutativity of quantum angular momentum operators $[L_x, L_y] = i\hbar L_z$, connecting geometry directly to the quantum mechanical structure of atoms.