Maxwell's counting rule for frameworks

Builds on Truss Rigidity of frameworks Rank of a matrix

Statement

Let $d \ge 1$ and let $(G, p)$ be a framework in $\R^d$ with $j$ joints and $b$ bars, whose joints affinely span $\R^d$ . Let $R$ be its rigidity matrix, the $b \times dj$ matrix with one row for each bar. The row of the bar $uv$ has $p(u) - p(v)$ in the columns of $u$ , $p(v) - p(u)$ in the columns of $v$ , and zeros elsewhere. Let $s = \dim \ker R^{T}$ be the number of independent states of self-stress. Let $m = \dim \ker R - d(d+1)/2$ be the number of independent infinitesimal mechanisms, not counting the rigid motions of $\R^d$ . Then $s - m = b - d j + \frac{d(d+1)}{2}.$ In particular, if the framework is infinitesimally rigid, then $b \ge d j - d(d+1)/2$ ; in the plane this reads $b \ge 2j - 3$ and in space $b \ge 3j - 6$ .

Remarks

The proof is the rank–nullity theorem applied twice. If $r$ is the rank of $R$ , then $\dim \ker R = dj - r$ and $\dim \ker R^{T} = b - r$ . The infinitesimal rigid motions of $\R^d$ form a space of dimension $d(d+1)/2$ in $\ker R$ because the joints span $\R^d$ affinely. Subtracting the two equalities gives the formula.

Example: a triangle in the plane has $j = 3$ , $b = 3$ and $b - 2j + 3 = 0$ ; it has no mechanism and no self-stress, so $s = m = 0$ . A square of four bars has $b - 2j + 3 = -1$ , and indeed it has one mechanism, the shear into a rhombus.

The count is necessary but not sufficient for rigidity. A framework can have enough bars and still have a mechanism if some part has more bars than it needs, which then gives a state of self-stress. In the plane, the exact combinatorial condition for generic frameworks is given by Laman graphs.

Going deeper

For a student

The rigidity matrix $R$ maps joint velocities to bar length rates, and its transpose $R^{T}$ maps bar forces to the joint forces they produce. So $\ker R$ holds the infinitesimal motions and $\ker R^{T}$ holds the self-stresses, and the theorem links the statically indeterminate degree of a truss with its kinematic freedom. A framework with $s = m = 0$ is statically and kinematically determinate, which is the isostatic case used in engineering design.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Rank of a matrix concept
Rigidity of frameworks concept
Truss concept
Maxwell's counting rule for frameworks statement

Tag	`AT0J8EWQGH`
URI	`m2://AT0J8EWQGH`
Kind	statement, theorem
Writing	W0 stub
MSC	52C25, 70C20, 74K10
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About	Truss, Rigidity of frameworks
Uses	Truss, Rigidity of frameworks, Rank of a matrix
Changed	3 November 2026