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 geome ...

and

mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...

, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by

rigid bodies In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...

connected by flexible

linkage Linkage may refer to: * ''Linkage'' (album), by J-pop singer Mami Kawada, released in 2010 *Linkage (graph theory), the maximum min-degree of any of its subgraphs *Linkage (horse), an American Thoroughbred racehorse * Linkage (hierarchical cluster ...

s or

hinge A hinge is a mechanical 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: all other ...

Definitions

Rigidity 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 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 In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...

of a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

in a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

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. A graph that is not rigid is called ''flexible''. More formally, a graph embedding is flexible if the vertices can be moved continuously, preserving the distances between adjacent vertices, with the result that the distances between some nonadjacent vertices are altered. The latter condition rules out Euclidean congruences such as simple translation and rotation. 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, tensional integrity or floating compression is a structural principle based on a system of isolated components under compression inside a network of continuous tension, and arranged in such a way that the compressed members (usually ...

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 In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being ...

. The bases of the two-dimensional

rigidity matroid In the mathematics of structural rigidity, a rigidity matroid is a matroid that describes the number of Degrees of freedom (mechanics), degrees of freedom of an undirected graph with rigid edges of fixed lengths, embedded into Euclidean space. In ...

(the minimally rigid graphs in the plane) are the

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 ...

s. * Cauchy's theorem states that a three-dimensional

convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

constructed with rigid plates for its faces, connected by hinges along its edges, forms a rigid structure. *

Flexible polyhedra In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such ...

, non-convex polyhedra that are not rigid, were constructed by

Raoul Bricard Raoul Bricard (23 March 1870 – 26 November 1943) was a French engineer and a mathematician. He is best known for his work in geometry, especially descriptive geometry and scissors congruence, and kinematics, especially mechanical linkages. ...

Robert Connelly Robert Connelly (born July 15, 1942) is a mathematician specializing in discrete geometry and rigidity theory. Connelly received his Ph.D. from University of Michigan in 1969. He is currently a professor at Cornell University. Connelly is best ...

, and others. The

bellows conjecture In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy's theorem (geometry), Cauchy rigidity theorem shows t ...

, now proven, states that every continuous motion of a flexible polyhedron preserves its

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

. *In the

grid bracing In the mathematics of structural rigidity, grid bracing is a problem of adding cross bracing to a square grid to make it into a rigid structure. It can be solved optimally by translating it into a problem in graph theory on the connectivity of bipa ...

problem, where the framework to be made rigid is a

square grid In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of th ...

with added diagonals as

cross bracing In construction, cross bracing is a system utilized to reinforce building structures in which diagonal supports intersect. Cross bracing is usually seen with two diagonal supports placed in an X-shaped manner. Under lateral force (such as wind or ...

, the rigidity of the structure can be analyzed by translating it into a problem on the connectivity of an underlying

bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...

. 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 (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...

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

" theory of the equilibrium and deflections of frameworks subjected to the action of forces is acting on the hardnes of quality... in cases in which the framework ... is strengthened by additional connecting pieces ... in cases of three dimensions, by the regular method of equations of forces, every point would have three equations to determine its equilibrium, so as to give 3''s'' equations between ''e'' unknown quantities, if ''s'' be the number of points and ''e'' the number of connexions ic There are, however, six equations of equilibrium of the system which must be fulfilled necessarily by the forces, on account of the equality of action and reaction in each piece. Hence if ''e'' = 3''s'' − 6, the effect of any eternal force will be definite in producing tensions or pressures in the different pieces; but if ''e'' > 3''s'' − 6, these forces will be indeterminate...."

Notes

References

*. *. *. *. *. * {{DEFAULTSORT:Structural Rigidity Mathematics of rigidity Mechanics

Definitions

Mathematics of rigidity

History

See also

Notes

References