  The problem of classifying off-shell representations of the $N$-extended
one-dimensional super Poincar\'{e} algebra is closely related to the study of a
class of decorated $N$-regular, $N$-edge colored bipartite graphs known as {\em
Adinkras}. In this paper we {\em canonically} realize these graphs as
Grothendieck ``dessins d'enfants,'' or Belyi curves uniformized by certain
normal torsion-free subgroups of the $(N,N,2)$-triangle group. We exhibit an
explicit algebraic model over $\mathbb{Q}(\zeta_{2N})$, as a complete
intersection of quadrics in projective space, and use Galois descent to prove
that the curves are, in fact, definable over $\mathbb{Q}$ itself. The stage is
thereby set for the geometric interpretation of the remaining Adinkra
decorations in Part II.
