Limit cycles for successive projections onto hyperplanes in ℝⁿ

In this paper we consider successive orthogonal projections onto m hyperplanes in ℝⁿ, where m ≥ 2 and n ≥ 2. A limit cycle is defined to be a sequence of points formed by projecting onto each of the hyperplanes once in a prescribed order, with the last projection giving the starting point. Several examples, including triangles, quadrilaterals, regular polygons, and arbitrary collections of lines in ℝ², are solved for the limit cycle. Limit cycles are found in various ways, including by a limiting process and by solving an mn × mn linear system of equations. The latter approach will produce all the limit cycles for an arbitrary ordered set of m hyperplanes in ℝⁿ.