  We propose and develop a new method to classify orbits of the spin group
${\rm Spin}(2d)$ in the space of its semi-spinors. The idea is to consider
spinors as being built as a linear combination of their pure constituents,
imposing the constraint that no pair of pure spinor constituents sums up to a
pure spinor. We show that this leads to a simple combinatorial problem that has
a finite number of solutions in dimensions up to and including fourteen. We
call each distinct solution a combinatorial type of an impure spinor. We
represent each combinatorial type graphically by a simplex, with vertices
corresponding to the pure constituents of a spinor, and edges being labelled by
the dimension of the totally null space that is the intersection of the
annihilator subspaces of the pure spinors living at the vertices. We call the
number of vertices in a simplex the impurity of an impure spinor. In dimensions
eight and ten the maximal impurity is two. Dimension twelve is the first
dimension where one gets an impurity three spinor, represented by a triangle.
In dimension fourteen the generic orbit has impurity four, while the maximal
impurity is five. We show that each of our combinatorial types uniquely
corresponds to one of the known spinor orbits, thus reproducing the
classification of spinors in dimensions up to and including fourteen from
simple combinatorics. Our methods continue to work in dimensions sixteen and
higher, but the number of the possible distinct combinatorial types grows
rather rapidly with the dimension.
