Construction of Toroidal Polyhedra corresponding to perfect Chains of wild Tetrahedra

In 1957, Steinhaus proved that a chain of regular tetrahedra, meeting face-to-face and forming a closed loop does not exist. Over the years, various modifications of this statement have been considered and analysed. Weakening the statement by only requiring the tetrahedra of a chain to be wild, i.e. having all faces congruent, results in various examples of such chains. In this paper, we elaborate on the construction of these chains of wild tetrahedra. We therefore introduce the notions of chains and clusters of wild tetrahedra and relate these structures to simplicial surfaces. We establish that clusters and chains of wild tetrahedra can be described by polyhedra in Euclidean 3-space. As a result, we present methods to construct toroidal polyhedra arising from chains and provide a census of such toroidal polyhedra consisting of up to 20 wild tetrahedra. Here, we classify toroidal polyhedra with respect to self-intersections and reflection symmetries. We further prove the existence of an infinite family of toroidal polyhedra emerging from chains of wild tetrahedra and present clusters of wild tetrahedra that yield polyhedra of higher genera.