# Crumpling

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Deformation of sheet into complex 3D structure

"Crumpled" and "Crumple" redirect here. For the deformation feature, see [Crumple zone](/source/Crumple_zone). For the album by Code Kunst, see [Crumple (album)](/source/Crumple_(album)).

Ball of crumpled paper

In [geometry](/source/Geometry) and [topology](/source/Topology), **crumpling** is the process whereby a sheet of [paper](/source/Paper) or other two-dimensional [manifold](/source/Manifold) undergo disordered [deformation](/source/Deformation_(engineering)) to yield a [three-dimensional](/source/Three-dimensional_space) structure comprising a random network of [ridges](/source/Face_(geometry)#Ridge_or_.28n-2.29-face) and [facets](/source/Facet) with variable density. The geometry of crumpled structures is of interest to mathematicians studying topology.[1] Crumpled paper balls have been studied and found to exhibit surprisingly complex structures with [compressive strength](/source/Compressive_strength) resulting from [frictional](/source/Friction) interactions at [locally](/source/Local_property) 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](/source/Materials_science) and [mechanical engineering](/source/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](/source/Cellulose) paper is of interest towards impact dissipation applications and has been proposed as an approach to [utilising waste paper](/source/Paper_recycling).[4]

From a practical standpoint, crumpled balls of paper are commonly used as [toys](/source/Cat_play_and_toys) for [domestic cats](/source/Cat).

## References

1. **[^](#cite_ref-1)** Cerda, Enrique; Chaieb, Sahraoui; Melo, Francisco; Mahadevan, L (1999). "Conical dislocations in crumpling". *Nature*. **401** (6748): 46–49. [Bibcode](/source/Bibcode_(identifier)):[1999Natur.401...46C](https://ui.adsabs.harvard.edu/abs/1999Natur.401...46C). [doi](/source/Doi_(identifier)):[10.1038/43395](https://doi.org/10.1038%2F43395). [S2CID](/source/S2CID_(identifier)) [4331085](https://api.semanticscholar.org/CorpusID:4331085).

1. **[^](#cite_ref-2)** Cambou, Anne Dominique; Narayanan, Menon (2011). ["Three-dimensional structure of a sheet crumpled into a ball"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3169141). *Proceedings of the National Academy of Sciences*. **108** (36): 14741–14745. [arXiv](/source/ArXiv_(identifier)):[1203.5826](https://arxiv.org/abs/1203.5826). [Bibcode](/source/Bibcode_(identifier)):[2011PNAS..10814741C](https://ui.adsabs.harvard.edu/abs/2011PNAS..10814741C). [doi](/source/Doi_(identifier)):[10.1073/pnas.1019192108](https://doi.org/10.1073%2Fpnas.1019192108). [PMC](/source/PMC_(identifier)) [3169141](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3169141). [PMID](/source/PMID_(identifier)) [21873249](https://pubmed.ncbi.nlm.nih.gov/21873249).

1. **[^](#cite_ref-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](/source/Bibcode_(identifier)):[2017SMat...13.4029H](https://ui.adsabs.harvard.edu/abs/2017SMat...13.4029H). [doi](/source/Doi_(identifier)):[10.1039/C6SM02817A](https://doi.org/10.1039%2FC6SM02817A). [PMID](/source/PMID_(identifier)) [28512658](https://pubmed.ncbi.nlm.nih.gov/28512658).

1. **[^](#cite_ref-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"](http://www.heliyon.com/article/e00329/). *Heliyon*. **3** (6) e00329. [Bibcode](/source/Bibcode_(identifier)):[2017Heliy...300329H](https://ui.adsabs.harvard.edu/abs/2017Heliy...300329H). [doi](/source/Doi_(identifier)):[10.1016/j.heliyon.2017.e00329](https://doi.org/10.1016%2Fj.heliyon.2017.e00329). [PMC](/source/PMC_(identifier)) [5477149](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5477149). [PMID](/source/PMID_(identifier)) [28653042](https://pubmed.ncbi.nlm.nih.gov/28653042).

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Adapted from the Wikipedia article [Crumpling](https://en.wikipedia.org/wiki/Crumpling) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Crumpling?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.