Graphs, Digraphs, and Rigidity of Grids (UMAP)

This module studies the rigidity of frameworks made of rigid rods in the plane that are connected by two-dimensional ball joints; in particular, grids of squares are considered. It discusses strategies for bracing grids with rods, and criteria for rigidity of the grid is formulated in terms of linear algebra and in terms of the brace graph of the grid. The efficiency and fault tolerance of bracings is briefly considered. Bracing a grid with cables rather than rods leads to a tensegrity structure and to criteria for rigidness in terms of a system of inequalities and in terms of the cable-brace directed graph of the grid. An appendix gives suggestions for the instructor.

Table of Contents:

RIGIDITY OF FRAMEWORKS
Frameworks
Walls and Grids
Bracing a Grid
The Degree of Freedom of a Grid

GRAPHS AND GRIDS
The Brace Graph
Connected Brace Graphs
An Algorithm for Detecting Connectivity
Trees and Efficiency
Fault-Tolerant Bracings
Amazing Transformations

DIRECTED GRAPHS AND TENSEGRITY
Bracing Grids with Cables
Cables and Linear Inequalities
Buckminster Fuller
The Cable-Brace Graph
An Algorithm for Directed Graphs

SOLUTIONS TO THE EXERCISES

APPENDIX: NOTES FOR THE INSTRUCTOR

REFERENCES

ABOUT THE AUTHOR