From Path-Segment Tiles to Loops and Labyrinths

Robert Bosch, Sarah Fries, Mäneka Puligandla and Karen Ressler
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013)
Pages 119–126 Regular Papers


We explain how to use integer programs to assemble a collection of path-segment tiles—squares decorated with path segments—on a grid so that the path segments join together to form a single closed loop or a single open path through the centers of the squares of the grid. We describe how to optimize (maximize or minimize) the number of bends (90-degree turns) in the loop or path. We show how to force loops to be symmetric and to encourage paths to be as close to symmetric as possible. We conclude by displaying laser-cut and 3D-printed artwork designed with our integer programs.