  It is well known that Pascal's triangle exhibits fractal behavior when
reduced modulo a prime. We show that the triangle of Fibonomial coefficients
has a similar nature modulo two. Specifically, for any $m \ge 0$, the
subtriangle consisting of the first $3 \cdot 2^m$ rows is duplicated on the
left and right sides of the next $3 \cdot 2^m$ rows, with an inverted triangle
of zeros in between. We give three proofs of this fact. The first uses a
combinatorial interpretation of the Fibonomials due to Sagan and Savage. The
second employs an analogue of Lucas' congruence for the parity of binomial
coefficients. The final one is inductive. We also use induction to show that
the Fibonomial triangle has a similar structure modulo three. We end with some
open questions.
