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 c ...

and

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, 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 distributed ...

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 ...

undergoes disordered deformation to yield a three-dimensional structure comprising a random network of

ridge A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...

s and facets 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 In mechanics, compressive strength or compression strength is the capacity of a material or structure to withstand loads tending to reduce size (as opposed to tensile strength which withstands loads tending to elongate). In other words, compre ...

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 t ...

al interactions at

locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). Pr ...

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, and ...

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 wall ...

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 The cat (''Felis catus'') is a domestic species of small carnivorous mammal. It is the only domesticated species in the family Felidae and is commonly referred to as the domestic cat or house cat to distinguish it from the wild members of t ...

References

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