  We provide explicit, simple, geometric formulas for free involutions rho of
Euclidean spheres that are not conjugate to the antipodal involution. Therefore
the quotient S^n/rho is a manifold that is homotopically equivalent but not
diffeomorphic to RP^n. We use these formulas for constructing explicit
non-trivial elements in pi_1 Diff(S^5) and pi_1 Diff(S^13) and to provide
explicit formulas for non-cancellation phenomena in group actions.
