The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubi...

Self-assembly of octahedral Fe(II) ions and linear perylene bisimide (PBI) dyes with 2,2'-bipyridine groups covalently attached at the imide positions quantitatively yields an Fe4(PBI)6 tetrahedron by the directional bonding approach. With an edge length of 3.9 nm and estimated internal volume >950 Å(3), tetrahedron T is one of the largest M4L6 tetrahedra ever reported. Importantly, many of the...

In this paper we derive from arguments of string scattering a set of eight tetrahedron equations, with different index orderings. It is argued that this system of equations is the proper system that represents integrable structures in three dimensions generalising the Yang-Baxter equation. Under additional restrictions this system reduces to the usual tetrahedron equation in the vertex form. Mo...

The action of the symmetric group S4 on the Tetrahedron algebra, introduced by Hartwig and Terwilliger [HT05], is studied. This action gives a grading of the algebra which is related to its decomposition in [HT05] into a direct sum of three subalgebras isomorphic to the Onsager algebra. The ideals of both the Tetrahedron algebra and the Onsager algebra are determined.