  We consider the fragmentation process with mass loss and discuss self-similar
properties of the arising structure both in time and space, focusing on
dimensional analysis. This exhibits a spectrum of mass exponents $\theta$,
whose exact numerical values are given for which $x^{-\theta}$ or $t^{\theta
z}$ has the dimension of particle size distribution function $\psi(x,t)$, where
$z$ is the kinetic exponent. We obtained conditions for which the scaling and
fragmentation process altogether break down and give explicit scaling solution
for special case. Finally, we identify a new class of fractals ranging from
random to non-random and show that the fractal dimension increases with
increasing order and a transition to strictly self-similar pattern occurs when
randomness completely ceases.
