SO(3) is the group of 3D rotations: 3×3 orthogonal matrices with determinant +1. Every rotation can be described by Euler angles, an axis-angle pair (Rodrigues' formula), or a unit quaternion. Quaternions double-cover SO(3) — both q and −q represent the same rotation, revealing the topological connection π₁(SO(3)) = ℤ/2.