The present paper regards the volume function of a doubly truncated hyperbolic tetrahedron. Starting from the previous results of J. Murakami, U. Yano and A. Ushijima, we have developed a unified approach to expressing the volume in different geometric cases by dilogarithm functions and to treat properly the many analytic strata of the latter. Finally, several numeric examples are given.

A polytope P of 3-space, which meets a given lattice L only in its vertices, is called L-elementary. An L-elementary tetrahedron has volume ≥ (1/6). det(L), in case equality holds it is called L-primitive. A result of Knudsen, Mumford and Waterman, tells us that any convex polytope P admits a linear simplicial subdivision into tetrahedra which are primitive with respect to the bigger lattice (1...

In 3 dimensions, the Ising model is in the same universality class as φ-theory, whose massive 3-loop tetrahedral diagram, C, was of an unknown analytical nature. In contrast, all single-scale 4-dimensional tetrahedra were reduced, in hep-th/9803091, to special values of exponentially convergent polylogarithms. Combining dispersion relations with the integer-relation finder PSLQ, we find that C/...

As a secondary structure of DNA, DNA tetrahedra exhibit intriguing charge transport phenomena and provide promising platform for wide applications like biosensors, as shown in recent electrochemical experiments. Here, we study multi-terminal tetrahedron, finding that its properties strongly depend upon the interplay among contact position, on-site energy disorder, base-pair mismatch. Our result...

For two-dimensional lattice equations one definition of integrability is that the model can be naturally and consistently extended to three dimensions, i.e., that it is “consistent around a cube” (CAC). As a consequence of CAC one can construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted a search based on this principle and certain additional assumptions. One of those ...

It is well known that three-dimensional Euclidean space cannot be tiled by regular tetrahedra. But how well can we do? In this work, we give several constructions that may answer the various senses of this question. In so doing, we provide some solutions to packing, tiling, and covering problems of tetrahedra. Our results suggest that the regular tetrahedron may not be able to pack as densely a...

This paper presents a new methodology of using tetrahedron configurations to determine the forward kinematics of 6-dof parallel manipulators. The approach is introduced in the form of the Tetrahedron Proposition and Theorem using the special characteristics of tetrahedral geometry which are then applied to obtaining the kinematic solutions of two 6-dof parallel manipulators such as the Hunt-Pri...