How to Brace a One-Story Building

This paper describes the economical placing of braces in the walls and ceiling of a rectangular one-story building. First, efficient schemes for bracing the ceiling are shown to correspond to trees in a bipartite graph. Then a combinatorial analysis shows how certain sets of braces in vertical walls cause some ceiling braces to be redundant. We also give a complete description of bracing schemes for plane grids of squares.

The fundamental tools are elementary vector space theory and combinatorial geometry (matroid theory). However, our style is deliberately discursive, with many worked examples, because we hope to attract readers possessing mathematical sophistication and mechanical intuition at a wide range of levels.