  Minkowski space, conformal group, compactification, conformal infinity,
conformal inversion, light cone at infinity, SU(2,2), SO(4,2), Hodge star
operator, Clifford algebra, spinors, twistors, antilinear operators, exterior
algebra, bivectors, isotropic subspaces, null geodesics, Lie spheres, Dupin
cyclides, gravitation} \abstract{Maxwell's equations are invariant not only
under the Lorentz group but also under the conformal group. Irving E. Segal has
shown that while the Galilei group is a deformation of the Poincar\'e group,
and the Poincar\'e group is a deformation of the conformal group, the conformal
group ends the road, it is {\em rigid\}. There are thus compelling mathematical
and physical reasons for promoting the conformal group to the role of the
fundamental symmetry of space--time, more important than the Poincar\'e group
that formed the group-theoretical basis of special and general theories of
relativity. While the action of the conformal group on Minkowski space is
singular, it naturally extends to a nonsingular action on the compactified
Minkowski space, often referred to in the literature as "Minkowski space plus
light-cone at infinity". Unfortunately in some textbooks the true structure of
the compactified Minkowski space is sometimes misrepresented, including false
proofs and statements that are simply wrong. In this paper we present in, a
simple way, two different constructions of the compactified Minkowski space,
both stemming from the original idea of Roger Penrose, but putting stress on
the mathematically subtle points and relating the constructions to the Clifford
algebra tools. In particular the little-known antilinear Hodge star operator is
introduced in order to connect real and complex structures of the algebra. A
possible relation to Waldyr Rodrigues' idea of gravity as a plastic deformation
of Minkowski's vacuum is also indicated.
