geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

and

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, crumpling is the process whereby a sheet of

paper Paper is a thin sheet material produced by mechanically or chemically processing cellulose fibres derived from wood, Textile, rags, poaceae, grasses, Feces#Other uses, herbivore dung, or other vegetable sources in water. Once the water is dra ...

or other two-dimensional

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

undergo disordered deformation to yield a

three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...

structure comprising a random network of

ridge A ridge is a long, narrow, elevated geomorphologic landform, structural feature, or a combination of both separated from the surrounding terrain by steep sides. The sides of a ridge slope away from a narrow top, the crest or ridgecrest, wi ...

s and

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...

s with variable density. The geometry of crumpled structures is of interest to mathematicians studying topology. Crumpled paper balls have been studied and found to exhibit surprisingly complex structures with

compressive strength In mechanics, compressive strength (or compression strength) is the capacity of a material or Structural system, structure to withstand Structural load, loads tending to reduce size (Compression (physics), compression). It is opposed to ''tensil ...

resulting from

friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...

al interactions at locally flat facets between folds. The unusually high compressive strength of crumpled structures relative to their density is of interest in the disciplines of

materials science Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries. The intellectual origins of materials sci ...

and

mechanical engineering Mechanical engineering is the study of physical machines and mechanism (engineering), mechanisms that may involve force and movement. It is an engineering branch that combines engineering physics and engineering mathematics, mathematics principl ...

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. The high compressive strength exhibited by dense crumple formed

cellulose Cellulose is an organic compound with the chemical formula, formula , a polysaccharide consisting of a linear chain of several hundred to many thousands of glycosidic bond, β(1→4) linked glucose, D-glucose units. Cellulose is an important s ...

paper is of interest towards impact dissipation applications and has been proposed as an approach to utilising waste paper. From a practical standpoint, crumpled balls of paper are commonly used as toys for domestic cats.

References

Topology Manifolds Deformation (mechanics) Structural analysis Materials science Mechanical engineering {{geometry-stub