Structural rigidity

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.

Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a positive amount.

A rigid graph with edges of these types forms a mathematical model of a tensegrity structure.

"[A] 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 3s equations between e unknown quantities, if s be the number of points and e the number of connexions[sic].

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.

Graphs are drawn as rods connected by rotating hinges. The cycle graph C 4 drawn as a square can be tilted over by the blue force into a parallelogram, so it is a flexible graph. K 3 , drawn as a triangle, cannot be altered by any force that is applied to it, so it is a rigid graph.
The Moser spindle , a rigid graph and an example of a Laman graph .