Charts on SO(3) closed set under composition. This visualization can be extended to a general rotation in 3-dimensional space. The identity rotation is a point, and a small angle of rotation about some axis can be represented as a point on a sphere with a small radius. As the angle of rotation grows, the sphere grows, until the angle of rotation reaches 180 degrees, at which point the sphere begins to shrink, becoming a point as the angle approaches 360 degrees (or zero degrees from the negative direction). This set of expanding and contracting spheres represents a hypersphere in four-dimensional space (a 3-sphere).
Degrees of freedom (physics and chemistry) for 3 degrees of freedom. In addition, the molecule has two rotational degrees of motion and one vibrational mode. The rotations occur around the two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical rotation. This yields, for a diatomic molecule, a decomposition of: For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition: which means that an -atom molecule has vibrational degrees of freedom for . In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited