# Crumpling

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In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density. The geometry of crumpled structures is the subject of some interest to the mathematical community within the discipline of topology.[1] Crumpled paper balls have been studied and found to exhibit surprisingly complex structures with compressive strength resulting from frictional interactions at locally flat facets between folds.[2] The unusually high compressive strength of crumpled structures relative to their density is of interest in the disciplines of materials science and mechanical engineering.

## Significance

The packing of a sheet by crumpling is a complex phenomenon that depends on material parameters and the packing protocol. Thus the crumpling behaviour of foil, paper and poly-membranes differs significantly and can be interpreted on the basis of material foldability.[3] The high compressive strength exhibited by dense crumple formed cellulose paper is of interest towards impact dissipation applications and has been proposed as an approach to utilising waste paper.[4]

From a practical standpoint, crumpled balls of paper are commonly used as toys for domestic cats.

## References

1. ^ Cerda, Enrique; Chaieb, Sahraoui; Melo, Francisco; Mahadevan, L (1999). "Conical dislocations in crumpling". Nature. 401 (6748): 46–49. Bibcode:1999Natur.401...46C. doi:10.1038/43395. S2CID 4331085.
2. ^ Cambou, Anne Dominique; Narayanan, Menon (2011). "Three-dimensional structure of a sheet crumpled into a ball". Proceedings of the National Academy of Sciences. 108 (36): 14741–14745. arXiv:1203.5826. Bibcode:2011PNAS..10814741C. doi:10.1073/pnas.1019192108. PMC 3169141. PMID 21873249.
3. ^ Habibi, M; Bonn, D (2017). "Effect of the material properties on the crumpling of a thin sheet". Soft Matter. 3 (22): 4029–4034. Bibcode:2017SMat...13.4029H. doi:10.1039/C6SM02817A. PMID 28512658.
4. ^ Hanaor, D.A.H.; Johnson, E.A. Flores; Wang, S.; Quach, S.; Dela-Torre, K.N.; Gan, Y.; Shen, L. (2017). "Mechanical properties in crumple-formed paper derived materials subjected to compression". Heliyon. 3 (6): e00329. Bibcode:2017Heliy...300329H. doi:10.1016/j.heliyon.2017.e00329. PMC 5477149. PMID 28653042.