Crystallography & Symmetry
patterns derived from their groups
Concept

Rigidity — where it appears

The property of a structure that no motion preserves its constraints except the rigid motions of space as a whole. For a framework of bars it is decided by the rank of a matrix; for a polyhedron of rigid plates it is Cauchy's theorem, and convexity is the hypothesis doing the work.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

One framework has a count of zero, one mechanism and one self-stress. Every net this collection has a placement for, as a periodic bar-and-joint framework in a fixed cell: its point group, the joints and bars of one cell, the scalar Maxwell count 2n − e − 2, and the mechanisms and self-stresses found exactly from the rank of the rigidity matrix. The scalar count is always the difference of the last two, which is Maxwell's identity — and the bathroom net is the row that shows what the identity costs: nought equals one minus one, and a framework that reads isostatic moves.

The mechanisms a count cannot see

Maxwell's count subtracts constraints from freedoms, and a mechanism and a state of self-stress cancel in the subtraction — so a framework with one of each reports the same number as a rigid one. The bathroom net reports nought and moves. Doing the same subtraction with representations instead of numbers separates them, because a mechanism and a self-stress cancel only when they belong to the same representation.

applied · Rigidity Five solids, twice each. Each Platonic solid as a framework of rods hinged at the corners, and again with its faces made rigid by adding their diagonals. The rank of the rigidity matrix reaches 3V − 6 exactly when the framework cannot move; the shortfall counts the ways it can. Three of the five are rigid as rods and all five are rigid as plates, which is Cauchy's theorem in the form a rank computation can see.

The polyhedra that can flex

A cube of rods folds and a cube of cardboard does not, and the difference is a rank. Cauchy proved in 1813 that a convex polyhedron with rigid faces is rigid; the rank of a rigidity matrix sees it directly, and it also sees where the hypothesis is doing the work. Drop convexity and an octahedron flexes — followed here for forty steps with every edge length held to five parts in a thousand million million.

applied · Rigidity One determinant, three dimensions. The Cayley–Menger determinant of a set of squared distances, at three sizes. Its value is the squared content of the simplex those distances describe, times a factor that alternates in sign with the dimension. At three points it is Heron's formula rewritten; at four it gives a tetrahedron's volume from its six edge lengths with no coordinates anywhere. The alternating sign is not a convention — a value of the wrong sign means the distances belong to no set of points at all.

What six lengths decide and nine do not

A tetrahedron's volume is a determinant in its six edge lengths, with no coordinates anywhere. Add a fifth vertex and the lengths stop deciding: two shapes with identical edges and identical faces have volumes in the ratio 2.6. What survives is that the possibilities are finite — which is the whole reason a flexing polyhedron cannot change its volume.

applied · Rigidity Nine frameworks, counted and then decided. Maxwell's count subtracts bars from twice the joints; the pebble game inserts the bars one at a time and discards any that cannot be paid for. The two agree on most of these frameworks and not on all, and where they differ the count is the one that is wrong — it assumes every bar is an independent constraint, and a bar added to a part that is already rigid is not. The redundant column is how many bars the game refused.

A game that decides what counting only bounds

Maxwell's count subtracts bars from twice the joints and is a bound, not an answer, because it assumes every bar constrains something new. In the plane there is an exact repair: Laman's condition, run as a game in which each joint holds two pebbles and a bar is admitted only if four can be gathered at its ends. Two rigid bodies sharing a joint are what the count gets backwards.

applied · Rigidity

Named alongside it

The objects these essays reach for when they reach for this one.

Degrees of freedomMechanismConvexityMaxwell countPolyhedronCharacterCrystal netIrreducible representationPermutationPoint groupRepresentationSelf stress

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