A generalised tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalised triangle groups and whose edge groups are finite cyclic. We prove an improved Spelling Theorem for generalised triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalised tetrahedron groups ac...

We report progress in constructing Boltzmann weights for integrable 3dimensional lattice spin models. We show that a large class of vertex solutions to the modified tetrahedron equation can be conveniently parameterized in terms of N -th roots of theta-functions on the Jacobian of a compact algebraic curve. Fay’s identity guarantees the Fermat relations and the classical equations of motion for...

We present a method to adapt a tetrahedron mesh together with a surface mesh with respect to a size criterion. Both surface and tetrahedron mesh adaptation are carried out simultaneously and no CAD is required to adapt the surface mesh. The adaptation procedure consists in splitting or removing interior and surface edges in order to enforce a given size criterion. The enrichment process is base...

One of the challenges of Volume Modelling is the definition of theoretical frameworks to support object manipulation and representation. Opposite to what is seen in the field of surface modelling, few satisfactory tools have been presented that ensure topological robustness to volumetric models. In this paper we offer one such mathematical framework for volumetric model definition based on tetr...

SUMMARY We present a new shape measure for tetrahedral elements that is optimal in that it gives the distance of a tetrahedron from the set of inverted elements. This measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume. Using this shape measure, we formulate two optimization objective funct...

A tetrahedral strip is a tetrahedron-tetrahedron truss where any tetrahedron has two neighbors except those in the extremes which have only one. Unless any of the tetrahedra degenerate, such a truss is rigid. In this case, if the distance between the strip endpoints is imposed, any rod length in the truss is constrained by all the others to attain discrete values. In this paper, it is shown how...

The reaction of 17,17-dichloro-androst-16(E)-chloromethylene with secondary amine base afforded substitution products exocyclic D-ring ketones, in contrast to the alkaline which cleaved steroidal des-D formyl alkyne. alkyne was employed prepare annulated isoxazolo steroid via 1,3-dipolar nitrile oxide cycloaddition reaction.

In this paper we discuss the structure of A 4 M 4 alkali(= A)–tetralide (= M) (= group 14) clusters. Without polarization these polyions consist of a central tetralide tetrahedron with each face capped by an alkali ion. We show that ionic polarization can lead to quite different cluster structures by breaking up of the tetralide tetrahedron into pairs M 2 , and it can even destroy covalent bond...