A construction of laminates from rank-1 polygons

We construct non-trivial laminates distributed continuously along a smooth closed 'evolute' curve by approximating an auxiliary ('involute') curve by a polygon with rank-1 sides. We give an explicit example in which the evolu te has no rank-1 connections. Using these techniques, given a finite set of matrices B-1,..., B-r and any other matrix A, we show how to construct a l aminate with positive mass at each Bi, any proportion less than 1 of its su pport on the B-i, and centre of mass at A.