---
tag: V5YS9BZX0N
kind: concept
name: Rigidity of frameworks
msc: [52C25, 51K99]
sources:
  - url: https://en.wikipedia.org/wiki/Structural_rigidity
close_to:
  - system: wikipedia
    id: Structural rigidity
    note: This node is a duplicate of node RHDAR5YY8S, which holds the same_as link. The two nodes wait for a merge.
  - system: wikidata
    id: Q7625050
    note: This node is a duplicate of node RHDAR5YY8S, which holds the same_as link. The two nodes wait for a merge.
---

## Summary

A framework is a [[graph|00C0BR7D7H]] whose vertices are placed at points of Euclidean space, with the edges read as rigid bars joined at the vertices.
The framework is rigid when it cannot be deformed a little without changing some bar length, except by moving it as a whole.
Rigidity theory decides this from the graph and the positions, and in the plane [[Laman graphs|C81JPZ51NA]] answer the question for generic positions.
It belongs to [[distance geometry|HPW5F0GQ4B]] and has uses in engineering, robotics and molecular biology.

## Explanations

### L1 human

Build a shape from sticks joined by loose hinges at the ends.
A triangle of sticks holds its shape: three side lengths fix the triangle.
A square of four sticks does not: you can push it over into a rhombus while every stick keeps its length.
Adding one diagonal stick to the square makes it rigid again.
Rigidity theory asks which stick figures behave like the triangle and which like the square.