---
tag: RHDAR5YY8S
kind: concept
name: Structural rigidity
msc: [52C25, 05C62, 70B15]
sources:
  - url: https://en.wikipedia.org/wiki/Structural_rigidity
same_as:
  wikidata: Q7625050
  wikipedia: Structural rigidity
---

## Summary

Rigidity theory studies when a framework of rigid bars joined at flexible joints cannot be deformed except by moving it as a whole.
A framework is a [[graph|00C0BR7D7H]] together with positions of its vertices in [[Euclidean space|0XPR1E85W0]], with each edge a bar of fixed length.
The subject links geometry, combinatorics and linear algebra, and it is used in engineering, chemistry and robotics.
Classical results include Cauchy's rigidity theorem for convex polyhedra and Laman's characterisation of generically rigid graphs in the plane.

## Explanations

### L1 human

Build a shape out of sticks and hinges and ask whether it wobbles.
A square of four sticks wobbles: it can be pushed into a rhombus.
A triangle of three sticks does not, since three side lengths fix a triangle.
Adding one diagonal stick to the square splits it into two triangles and makes it rigid.
Rigidity theory turns this into precise mathematics and counts how many sticks a rigid structure needs.