  Grassmann angles improve upon similar concepts of angle between subspaces
that measure volume contraction in orthogonal projections, working for real or
complex subspaces, and being more efficient when dimensions are different.
Their relations with contractions, inner and exterior products of multivectors
are used to obtain formulas for computing these or similar angles in terms of
arbitrary bases, and various identities for the angles with certain families of
subspaces. These include generalizations of the Pythagorean trigonometric
identity $\cos^2\theta+\sin^2\theta=1$ for high dimensional and complex
subspaces, which are connected to generalized Pythagorean theorems for volumes,
quantum probabilities and Clifford geometric product.
