  We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a
simple closed geodesic and a minimal subsphere Sigma^5, which can be
equivariantly identified with the Brieskorn sphere W^5_3. As applications we in
particular determine the full isometry group of Sigma^7, classify all closed
subgroups that act freely, determine the homotopy type of the corresponding
orbit spaces, identify the Hirsch-Milnor involution in dimension 5 with the
Calabi involution of W^5_3, and obtain explicit formulas for diffeomorphisms
between the Brieskorn spheres W^5_3 and W^13_3 with standard Euclidean spheres.
