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Geometric Constraint System Stories: Evariste Galois, CAD and Crystallography

Professor Stephen Power, Department of Mathematics and Statistics

Tuesday 15 December 2009, 1005-1030
Lecture Theatre 1, Management School Building

Geometric constraint systems are large systems of equations that arise from a configuration of geometrical objects and which feature in many structural models in applied science. I will outline three stories which have at their heart the analysis of the flexibility and rigidity of bar-joint structures. A bar (= strong bond) is inextensible while a joint (= weak angular bond) can swivel. For a 3D example imagine a telegraph pylon with all the joints loosened (and whether it will fall down).

James Clerk-Maxwell noted simple counting rules that hold when typical bar-joint systems in 2D or 3D are rigid structures. Nevertheless it is still an open problem to determine what exactly determines rigidity and in what manner bar-joint structures can flex or deform.

Evariste Galois, "revolutionary and geometer", died in a dual in 1832 before at the age of 21. The night before, the story goes, he wrote down his profound theory on the solvability of polynomial equations. In my second story I will explain how "Galois' Obstacle", as I call it, is relevant to the geometric constraint systems within core CAD software.

Max Born and Theodore von Karman developed the classical theory of crystallography. This theory incorporates infinite periodic bar-joint structures as mathematical models in the analysis of the low energy spectrum. Can the mathematics of such models shed light on the physics of crystals or, for that matter, more amorphous materials?

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