  We present a periodic infinite chain of finite generalisations of the
exceptional structures, including e8, the exceptional Jordan algebra (and
pair), and the octonions. We demonstrate that the exceptional Jordan algebra is
part of an infinite family of finite-dimensional matrix algebras (corresponding
to a particular class of cubic Vinberg's T-algebras). Correspondingly, we prove
that e8 is part of an infinite family of algebras (dubbed "Magic Star"
algebras) that resemble lattice vertex algebras.
