  The Casimir effect for massless scalar fields satisfying Dirichlet boundary
conditions on the parallel plates in the presence of one fractal extra
compactified dimension is analyzed. We obtain the Casimir energy density by
means of the regularization of multiple zeta function with one arbitrary
exponent. We find a limit on the scale dimension like $\delta>1/2$ to keep the
negative sign of the renormalized Casimir energy which is the difference
between the regularized energy for two parallel plates and the one with no
plates. We derive and calculate the Casimir force relating to the influence
from the fractal additional compactified dimension between the parallel plates.
The larger scale dimension leads to the greater revision on the original
Casimir force. The two kinds of curves of Casimir force in the case of
integer-numbered extra compactified dimension or fractal one are not
superposition, which means that the Casimir force show whether the
dimensionality of additional compactified space is integer or fraction.
