Hopeless Love and Other Lattice Walks

Tom Verhoeff and Koos Verhoeff
Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture
Pages 197–204 Regular Papers


The Hopeless Love theme arose from observations about chess bishops and their walks on the chessboard. In chess, there are two types of bishops: one confined to the white squares and the other to the black squares. If two bishops of opposite type fall in love, then they can walk around each other, coming very close together, but they can never meet. Their love is hopeless. In this article, we explore the generalization of bishop walks to the 3D chessboard and define when their paths are considered a hopeless love. This has led to several mathematical sculptures. A fictitious 3D chess piece that walks in directions of the main diagonals of the cube we call a rhomber. Again, hopeless love is in the picture, and has resulted in a mathematical sculpture.