Structural rigidity
Encyclopedia
In discrete geometry
Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...

and mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, structural rigidity is a combinatorial theory
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

for predicting the flexibility of ensembles formed by rigid bodies
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

A mechanical linkage is an assembly of bodies connected together to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for...

s or hinge
Hinge
A hinge is a type of bearing that connects two solid objects, typically allowing only a limited angle of rotation between them. Two objects connected by an ideal hinge rotate relative to each other about a fixed axis of rotation. Hinges may be made of flexible material or of moving components...

s.

## Definitions

Rigidity
Stiffness
Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.-Calculations:...

is the property of a structure that it does not bend or flex under an applied force. The opposite of rigidity is flexibility. In structural rigidity theory, structures are formed by collections of objects that are themselves rigid bodies, often assumed to take simple geometric forms such as straight rods (line segments), with pairs of objects connected together by flexible hinges. A structure is rigid if it cannot flex; that is, if there is no continuous motion of the structure that preserves the shape of its rigid components and the pattern of their connections at the hinges.

There are two essentially different kinds of rigidity. Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a positive amount. Infinitesimal rigidity means that the structure will not flex by even an amount that is too small to be detected even in theory. (Technically, that means certain differential equations have no nonzero solutions.) The importance of finite rigidity is obvious, but infinitesimal rigidity is also crucial because infinitesimal flexibility in theory corresponds to real-world minuscule flexing, and consequent deterioration of the structure.

A rigid graph is an embedding
Graph embedding
In topological graph theory, an embedding of a graph G on a surface Σ is a representation of G on Σ in which points of Σ are associated to vertices and simple arcs are associated to edges in such a way that:...

of a graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

in a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

which is structurally rigid. That is, a graph is rigid if the structure formed by replacing the edges by rigid rods and the vertices by flexible hinges is rigid. It is also possible to consider rigidity problems for graphs in which some edges represent compression elements (able to stretch to a longer length, but not to shrink to a shorter length) while other edges represent tension elements (able to shrink but not stretch). A rigid graph with edges of these types forms a mathematical model of a tensegrity
Tensegrity
Tensegrity, tensional integrity or floating compression, is a structural principle based on the use of isolated components in compression inside a net of continuous tension, in such a way that the compressed members do not touch each other and the prestressed tensioned members delineate the...

structure.

## Mathematics of rigidity

The fundamental problem is how to predict the rigidity of a structure by theoretical analysis, without having to build it. Key results in this area include the following:
• In any dimension, the rigidity of rod-and-hinge linkages is described by a matroid
Matroid
In combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces....

. The bases of the two-dimensional rigidity matroid (the minimally rigid graphs in the plane) are the Laman graph
Laman graph
In graph theory, the Laman graphs are a family of sparse graphs describing the minimally rigid systems of rods and joints in the plane. Formally, a Laman graph is a graph on n vertices such that, for all k, every k-vertex subgraph has at most 2k −3 edges, and such that the whole graph...

s.
• Cauchy's theorem
Cauchy's theorem (geometry)
Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states thatconvex polytopes in three dimensions with congruent corresponding faces must be congruent to each other...

states that a three-dimensional convex polyhedron constructed with rigid plates for its faces, connected by hinges along its edges, forms a rigid structure.
• Flexible polyhedra
Flexible polyhedron
In geometry, a flexible polyhedron is a polyhedral surface that allows continuous non-rigid deformations such that all faces remain rigid. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex .The first examples of flexible polyhedra, now called Bricard's...

, non-convex polyhedra that are not rigid, were constructed by Raoul Bricard
Raoul Bricard
Raoul Bricard is a French engineer and a mathematician. He is best known for his work in work in geometry, especially descriptive geometry and scissors congruence, and kinematics, especially mechanical linkages.- Biography :...

, Robert Connelly
Robert Connelly
Robert Connelly is a mathematician specializing in discrete geometry and rigidity theory. He is best known for discovering embedded flexible polyhedra. One such polyhedron is in the National Museum of American History....

, and others. The bellows conjecture, now proven, states that any continuous motion of a flexible polyhedron must preserve its volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

.

However, in many other simple situations it is not yet always known how to analyze the rigidity of a structure mathematically despite the existence of considerable mathematical theory.

## History

One of the founders of the mathematical theory of structural rigidity was the great physicist James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

. The late twentieth century saw an efflorescence of the mathematical theory of rigidity, which continues in the twenty-first century.