Fractal topology of hand-crumpled paper Alexander S. Balankin, Didier Samayoa Ochoa, Israel Andrés Miguel, Julián Patiño Ortiz, and Miguel Ángel Martínez Cruz Grupo “Mecánica Fractal,” Instituto Politécnico Nacional, México D.F. 07738, México Received 22 March 2010; published 17 June 2010 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 D BF = 1.68  0.04, which is independent on the sheet size. Furthermore, we noted that D BF , 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. DOI: 10.1103/PhysRevE.81.061126 PACS numbers: 46.65.+g, 89.75.Da, 05.45.Df Crumpled configurations of thin materials are very com- mon in nature, ranging from the microscopic level-folded proteins 1 and nanoparticle membranes 2—to the macro- scopic level - folded paper 3,4 and fault-related geological formations 5. In mathematics, Riemann has used a crumpled ball of paper with bookworms to explain the hid- den dimensions in non-Euclidean geometry 6. Accordingly, the mechanical and topological properties of folding configu- rations have attached much interest from both fundamental and applied points of view 1–11. It was found that a set of balls folded from thin sheets of different sizes L under the same force F = const obeys a fractal scaling law M  R D , where M = hL 2 is the sheet mass,  is the material density, R is the ball diameter, and D is the global fractal dimension of the set 4,12. For elastic mem- branes with thickness h  R  L the global fractal dimension D is expected to be universal 13–15. Specifically, theoreti- cal considerations and numerical simulations suggest that a set of balls folded from the phantom membranes is charac- terized by D =8 / 3. For balls folded from self-avoiding elastic membranes numerical simulations performed in 13,14 lead to somewhat different universal values D =2.3 and D = 2.5, respectively 16. In the case of balls folded from elastoplas- tic materials, such as a paper, the ball diameter increases due to strain relaxation after the folding force is withdrawn 17. Accordingly, the global fractal dimension of the set of balls folded from elastoplastic sheets of different sizes is found to be the material dependent 4,15,17. Here it should be pointed out the difference between experiments with hand folded papers 4,15,17 and the numerical simulations of elastoplastic sheets folding performed in 14. In contrast to experiments with hand folded paper, in numerical simula- tions 14 the stress relaxation leads to the decrease of the ball diameter under a fixed folding force. Because of re- stricted relaxation, the compactification of elastoplastic sheets under crumpling is less effective than in the elastic case 14. Accordingly, the difference arises from the lack of similarity of the elasto-plastic ridge patterns 14. On the other hand, in experiments with hand folded papers it was found that the internal structure of paper balls after strain relaxation obeys the fractal scale invariance m  r D l , where m is the mass of sheet within the box of size r  R and D l is the fractal dimension of the folding configuration, which is ex- pected to be the material independent 4,15. Specifically, in experiments with different kinds of paper it was found that D l = 2.64  0.05 4. In contrast to this, the local fractal dimension of folding configurations of predominantly plastic sheets, such as aluminum foils 18, is found to be a func- tion of the compaction ratio k = R / L 19. This is easy to understand taking into account that D l =3, when k = k min  1.25h / L 1/3 , whereas D l  3, if k  k min . Furthermore, the authors of 20 have observed a spontaneous symmetry breaking in folding configurations of randomly crumpled aluminum foils as k decreases. The fundamental differences in the folding behavior of elastic, elastoplastic, and predomi- nantly plastic sheets were pointed out in 15. The bending deformations of paper are energetically more favorable than stretching 11. Topologically, in the limit h → 0, it is not possible to confine an unstretchable two- dimensional sheet into a small three-dimensional volume by only smooth deformations 21. Such confinement necessar- ily requires singular crumpling along sharp lines and vertices 11. In real sheets these singularities smoothen resulting in a balance between stretching and bending energies 10. The energetically preferred configurations of crumpled thin sheets consist of mostly flat regions facets bounded by an almost straight folds crumpling creases that meet in sharp vertices developable cones10–17. Folding singularities leave the impression of crumpling network in the unfolded paper sheet 17,22,23. It was found that the crumpling net- work patterns in unfolded sheets of different kinds of paper are characterized by the same box fractal dimension D BN = 1.83  0.03 23. The roughness of hand folded paper ball surfaces was studied in 17. It was found that surfaces of crumpled balls display self-affine invariance with the univer- sal local roughness exponent  = 0.72  0.04 17, whereas the global roughness of set of balls folded from sheets of different sizes is characterized by the material dependent glo- bal roughness exponent  =2 / D 17. In this work we study the statistical topology of randomly folded paper balls. Specifically, we are interesting in the dis- tribution of two sides of the sheet along the ball surface and in the distribution of sheet fragments when the ball is cut in half. A hand folded paper ball is a very ill-defined system, PHYSICAL REVIEW E 81, 061126 2010 1539-3755/2010/816/0611266 ©2010 The American Physical Society 061126-1