  Using an octonionic formalism, we introduce a new mechanism for reducing 10
spacetime dimensions to 4 without compactification. Applying this mechanism to
the free, 10-dimensional, massless (momentum space) Dirac equation results in a
particle spectrum consisting of exactly 3 generations. Each generation contains
1 massive spin-1/2 particle with 2 spin states, 1 massless spin-1/2 particle
with only 1 helicity state, and their antiparticles --- precisely one
generation of leptons. There is also a single massless spin-1/2
particle/antiparticle pair with the opposite helicity and no generation
structure.
  We conclude with a discussion of some further consequences of this approach,
including those which could arise when using the formalism on a curved
spacetime background, as well as the implications for the nature of spacetime
itself.
