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We consider the fragmentation process with mass loss and discuss self-similarproperties of the arising structure both in time and space, focusing ondimensional analysis. This exhibits a spectrum of mass exponents $\theta$,whose exact numerical values are given for which $x^{-\theta}$ or $t^{\thetaz}$ has the dimension of particle size distribution function $\psi(x,t)$, where$z$ is the kinetic exponent. We obtained conditions for which the scaling andfragmentation process altogether break down and give explicit scaling solutionfor special case. Finally, we identify a new class of fractals ranging fromrandom to non-random and show that the fractal dimension increases withincreasing order and a transition to strictly self-similar pattern occurs whenrandomness completely ceases.Comment: 5 pages, latex, No figur

Transition from random to ordered fractals in fragmentation of particles in an open system