Fractal topology of hand-crumpled paper

Abstract

We study the statistical topology of folding configurations of hand folded paper balls. Specifically, we are studying the distribution of two sides of the sheet along the ball surface and the distribution of sheet fragments when the ball is cut in half. We found that patterns obtained by mapping of ball surface into unfolded flat sheet exhibit the fractal properties characterized by two fractal dimensions which are independent on the sheet size and the ball diameter. The mosaic patterns obtained by sheet reconstruction from fragments of two parts (painted in two different colors) of the ball cut in half also possess a fractal scale invariance characterized by the box fractal dimension DBF=1.68 ± 0.04 , which is independent on the sheet size. Furthermore, we noted that DBF, at least numerically, coincide with the universal fractal dimension of the intersection of hand folded paper ball with a plane. Some other fractal properties of folding configurations are recognized.

Fractal features of a crumpling network in randomly folded thin matter and mechanics of sheet crushing

“…Examples range from crumpled nanosheets of graphene [1][2][3], graphene oxide [4][5][6], and proteins [7][8][9], to hand folded paper [10][11][12][13][14][15] and geological formations [16]. Even when the crumpling processes appear quite haphazard, the geometry of randomly folded materials are well defined in a statistical sense and rather well reproducible in experiments [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], because the topology and self-avoiding interactions are the two most important factors when dealing with the crushing of thin matter [17][18][19]. Accordingly, the studies of crumpling phenomena remain an active area of research, both theoretically and experimentally (see, for example Refs.…”

“…(5) and (6). On the other hand, in the case of elastoplastic sheets, such as paper, it was observed that once the folding force is withdrawn, the ball diameter logarithmically increases in time due to elastic strain relaxation [11][12][13][14][15]. In this way, it was found that after complete relaxation during approximately 10 days the final ball diameter is almost independent of the force applied during the sheet crushing [11].…”

Edwards's statistical mechanics of crumpling networks in crushed self-avoiding sheets with finite bending rigidity

“…P, and a are defined by Eqs. (7), (10), (24), and (25), respectively. Figure 2 shows the graphs of P(l) versus ///max for different pimm.…”

“…[22], In this context, it is pertinent to point out that the fractal dimensions of ball configuration and the fractal dimension of a set of balls folded by the same forces are generally different due to elastic strain relaxation after the confinement force is withdrawn [23]. It was also recognized that both fractal dimensions are independent of the sheet elastic properties [20][21][22][23][24][25][26][27][28][29][30], but may change due to plastic deformations of the sheet material [26,28,31]. Consequently, although the crumpling processes appear quite haphazard, the crumpling behavior is well defined in a statistical sense and rather well reproducible in experiments [13][14][15][16][17][18][19][20]32], Moreover, almost all thin materials display nearly the same scale invariant crumpling behavior [22], This has allowed the authors of Ref.…”

“…Another noteworthy feature of crumpling networks in randomly crushed thin sheets is their statistical scale invariance within a wide range of length scales [21,22], This gives rise to the fractal geometry of both a crumpled sheet configuration [22][23][24][25][26] and a set of balls folded from sheets of different sizes under the same confinement force [1,13,14,20,[27][28][29][30][31]. The relationships between the fractal dimensions of a crumpling network and sheet configuration were established in Ref.…”