Intrinsically anomalous self-similarity of randomly folded matter

We found that randomly folded thin sheets exhibit unconventional scale invariance, which we termed as an intrinsically anomalous self-similarity, because the self-similarity of the folded configurations and of the set of folded sheets are characterized by different fractal dimensions. Besides, we found that self-avoidance does not affect the scaling properties of folded patterns, because the self-intersections of sheets with finite bending rigidity are restricted by the finite size of crumpling creases, rather than by the condition of self-avoidance. Accordingly, the local fractal dimension of folding structures is found to be universal Dl=2.64±0.05 and close to expected for a randomly folded phantom sheet with finite bending rigidity. At the same time, selfavoidance is found to play an important role in the scaling properties of the set of randomly folded sheets of different sizes, characterized by the material-dependent global fractal dimension DDl. So intrinsically anomalous self-similarity is expected to be an essential feature of randomly folded thin matter.