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An excerpt from the 2024 Department Newsletter research article...

Informally, a mechanical linkage is a system of rigid links (rods or bars) connected by ideal joints and moving in the plane or in space. This definition suffices for engineering purposes, and one can find it in some form in many engineering books. However, from the mathematical viewpoint, this is not a satisfactory definition. Mathematically speaking, an abstract linkage is a finite connected metric graph L = (G, l), a graph G together with a length function l which assigns to every edge e ∈ E(G) of G a positive real number, its length l(e). Given such a graph and a target metric space (X, d) (for the purpose of this article, X will be Euclidean space of some dimension), one de fines the realization space R(L, X), as the space of maps from the vertex set of G to the target space X, f : V(G) -> X, subject to the condition d( f(v), f(w) ) = l ([v, w]).

Here v, w ∈ V(G)are vertices of G and [v, w] ∈ E(G) are the edges connecting these vertices. In terms of the informal definition of a mechanical linkage, each realization f ∈ R(L, X) defines a system of rods f([v, w]) connected at joints f(v).

For the full article, check out our 2024 Department Newsletter! This article starts on page 5.