geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

and

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, 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, rags, grasses or other vegetable sources in water, draining the water through fine mesh leaving the fibre evenly distrib ...

or other two-dimensional manifold undergoes disordered

deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...

to yield a three-dimensional structure comprising a random network of ridges 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 cut ...

s with variable density. The geometry of crumpled structures is the subject of some interest the mathematical community within the discipline of topology. Crumpled paper balls have been studied and found to exhibit surprisingly complex structures with compressive strength resulting from

friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...

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 and

mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, an ...

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 formula , a polysaccharide consisting of a linear chain of several hundred to many thousands of β(1→4) linked D-glucose units. Cellulose is an important structural component of the primary cell w ...

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 A toy or plaything is an object that is used primarily to provide entertainment. Simple examples include toy blocks, board games, and dolls. Toys are often designed for use by children, although many are designed specifically for adults and pe ...

for domestic cats.

References

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