  Lawson-Osserman constructed three types of non-parametric minimal cones of
high codimensions based on Hopf maps between spheres, which correspond to
Lipschitz but non-differentiable solutions to the minimal surface equations,
thereby making sharp contrast to the regularity theorem for minimal graphs of
codimension 1. In this paper, we develop the constructions in a more general
scheme. Once a mapping f between unit spheres is composited of a harmonic
Riemannian submersion and a homothetic (i.e., up to a constant factor,
isometric) minimal immersion, certain twisted graph of f can yield a
non-parametric minimal cone. Because the choices of the second component
usually form a huge moduli space, our constructions produce a constellation of
uncountably many examples. For each such cone, there exists an entire minimal
graph whose tangent cone at infinity is just the given one. Moreover, new
phenomena on the existence, non-uniqueness and non-minimizing of solutions to
the related Dirichlet problem are discovered.
