Construction of Toroidal Polyhedra corresponding to perfect Chains of isosceles Tetrahedra

Abstract In 1957, Steinhaus conjectured that a chain of regular tetrahedra, meeting face-to-face and forming a closed loop, does not exist; this was later proven by Świerczkowski. We show that modifying the statement by requiring the tetrahedra of a chain to be isosceles results in the first examples of closed chains with all tetrahedra being congruent. As a result, we provide a census of toroidal polyhedra arising from closed chains consisting of up to 20 isosceles tetrahedra. Moreover, we establish the existence of an infinite family of toroidal polyhedra emerging from chains of isosceles tetrahedra. Additionally, we generalise the notion of chains of isosceles tetrahedra and therefore introduce clusters of isosceles tetrahedra along with a combinatorial tool called tetrahedral symbol to describe them. Finally, we exploit our methods to construct clusters of isosceles tetrahedra that yield polyhedra of higher genera.