  We briefly review previous work on the invariant theory of 3 x 3 x 3 arrays.
We then recall how to generate arrays of arbitrary size m_1 x ... x m_k with
hyperdeterminant 0. Our main result is an explicit formula for the 3 x 3 x 3
hyperdeterminant as a polynomial in the fundamental invariants of degrees 6, 9
and 12 for the action of the Lie group SL(3,C) x SL(3,C) x SL(3,C). We apply
our calculations to Nurmiev's classification of normal forms for 3 x 3 x 3
arrays.
