A multidimensional continued fraction expansion is given which finds provably good Diophantine approximations for all θ∈Rd. For anyQ>1 it finds some approximation (p,q) ∈ Zd+1with 1 ⩽q⩽Qsatisfying ‖qθ−p‖ ⩽ √d+ 1Q−1/d. This expansion consists of a sequence of reduced lattice bases for a parametrized series of lattice bases Bt(θ) (of different lattices) in GL(d+ 1, R), where the positive real parametertvaries. This parametrized familyBt(θ) forms a geodesic in GL(d+ 1, R), and also projects to a geodesicgθin the Riemannian symmetric spacePd+1of all positive definite symmetric matrices. The multidimensional continued fraction expansion is a ‘cutting sequence’ expansion forgθusing a Minkowski fundamental domain of GL(d+ 1, Z)\Pd+1. This method generalizes to give continued fraction expansions finding good Diophantine approximations to an arbitrary set of linear forms.