This paper describes an approach to the computational homotopy of CW-spaces. Computational advantages are obtained by considering spaces that a tessellated in the sense that the space is the union of the closure of its \(n\)-cells and all closures of the \(n\)-cell have face posets isomorphic to that of some fixed polytope. The case where this is a permutahedral one is particularly useful. The authors describe a “zig-zag” homotopy retraction approach to reducing the number of cells of low-dimensional lattice spaces. It is important that this approach is algorithmic. Applications to feature recognition in low-dimensional images is given as an illustration of the technique. The ideas have been implemented in the HAP (homological algebra package http://www.gap-system.org/Pacjkages/hap.html) for the GAP computational system. Sessions with the package are given in the paper.