Flexibility of Bricards planar linkages and other structures via resultants and computer algebra

TL;DR

This paper uses polynomial equations, resultants, and computer algebra to analyze the flexibility of Bricard's quadrilaterals and discovers new flexible arrangements, demonstrating the power of symbolic computation in mathematical discovery.

Contribution

It introduces a symbolic algorithm that analyzes the flexibility of structures via resultants, filling gaps in Bricard's classical work and uncovering new flexible configurations.

Findings

Successfully solved Bricard's quadrilaterals arrangement

Discovered new flexible arrangements previously unknown

Validated the software's effectiveness for mathematical discovery

Abstract

Flexibility of structures is extremely important for chemistry and robotics. Following our earlier work, we study flexibility using polynomial equations, resultants, and a symbolic algorithm of our creation that analyzes the resultant. We show that the software solves a classic arrangement of quadrilaterals in the plane due to Bricard. We fill in several gaps in Bricard's work and discover new flexible arrangements that he was apparently unaware of. This provides strong evidence for the maturity of the software, and is a wonderful example of mathematical discovery via computer assisted experiment.