Elementary formulas for a hyperbolic tetrahedron

We derive some elementary formulas expressing the relation between the dihedral angles and edge lengths of a tetrahedron in hyperbolic space.     

Volume of a doubly truncated hyperbolic tetrahedron

The present paper regards the volume function of a doubly truncated hyperbolic tetrahedron. Starting from the earlier results of J. Murakami, U. Yano and A. Ushijima, we have developed a unified approach to express 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.     

Gauss legendre quadrature formulas over a tetrahedron

In this article we consider the Gauss Legendre Quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)|0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three‐dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2‐cube: {(ξ, η, ζ)| ? 1 ≤ ζ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points. The effectiveness of the formulas is demonstrated by applying them to the integration of three nonpolynomial, three polynomial functions and to the evaluation of integrals for element stiffness matrices in linear three‐dimensional elasticity. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Functional tetrahedron equation

We describe a method for constructing classical integrable models in a (2+1)-dimensional discrete spacetime based on the functional tetrahedron equation, an equation that makes the symmetries of a model obvious in a local form. We construct a very general “block-matrix model,” find its algebraic-geometric solutions, and study its various particular cases. We also present a remarkably simple quantization scheme for one of those cases. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 3, pp. 370–384, December, 1998.     

Regular simple geodesic loops on a tetrahedron

A regular simple geodesic loop on a tetrahedron is a simple geodesic loop which does not pass through any vertex of the tetrahedron. It is evident that such loops meet each face of the tetrahedron. Among these loops, the minimal loops are those which meet each face exactly once. Necessary and sufficient conditions for the existence of minimal loops are obtained. These conditions fall naturally into two categories, conditions in the first category being called coherence conditions and conditions in the second category being called separation conditions. It is shown that for the existence of three distinct minimal loops through any point on the face of a tetrahedron it is necessary and sufficient that the tetrahedron be isosceles, which, in turn, amounts to the tetrahedron satisfying three coherence conditions. All other regular simple geodesic loops on an isosceles tetrahedron are then classified. Finally, coherence conditions for the existence of similar loops on an arbitrary tetrahedron are found.     

Imbedding a “soap bubble” into a tetrahedron

Translated from Matematicheskie Zametki, Vol. 56, No. 2, pp. 90–93, August, 1994.     

The Gergonne and Nagel centers of a tetrahedron

The Gergonne center of a triangle is the intersection of the cevians through the points where the incircle touches the sides. This does not admit a direct generalization to tetrahedra since the cevians of a tetrahedron through the points where the insphere touches the faces are not necessarily concurrent. This article introduces an alternative definition of the Gergonne center that coincides with the previous definition for the triangle and that admits a generalization to tetrahedra. The same is done for the Nagel center. Received 4 February 2000; revised 1 May 2000.     

Geometry of the tetrahedron space

Let X^° be the space of all labeled tetrahedra in P³. In [E. Babson, P.E. Gunnells, R. Scott, A smooth space of tetrahedra, Adv. Math. 165(2) (2002) 285-312] we constructed a smooth symmetric compactification of X^°. In this article we show that the complement is a divisor with normal crossings, and we compute the cohomology ring .     

Pythagorean theorems on rectangular tetrahedron

Two generalized forms of the Pythagorean Theorem for rectangular tetrahedron are proved using only elementary methods: Pythagorean Theorem and Heron's formula for the area of a triangle.     

Relationship between tetrahedron shape measures

Tetrahedron shape measures are used for evaluating the quality of tetrahedra in finite element meshes. Three shape measures, theminimum solid angle _min theradius ratio , and themean ratio , are discussed in this paper. A new formula for the computation of a solid angle of tetrahedron is derived. For different shape measures andv (with values 1), we establish a relationship between andv of the form wherec ₀,c ₁,e ₀, ande ₁ are positive constants. This means that if one measure approaches zero for a poorly-shaped tetrahedron, so does the other. Combined with the property that each measure attains a maximum value only for the regular tetrahedron, this means that the shape measures are equivalent.This work was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

The combinatorics of some tetrahedron manifolds

We study a family of closed connected orientable 3-manifolds (which are examples of tetrahedron manifolds) obtained by pairwise identifications of the boundary faces of a standard tetrahedron. These manifolds generalize those considered in previous papers due to Grasselli, Piccarreta, Molnár and Sieradski. Then we completely describe our tetrahedron manifolds in terms of Seifert fibered spaces, and determine their Seifert invariants. Moreover, we obtain different representations of our manifolds as 2-fold coverings, and give examples of non-equivalent knots with the same tetrahedron manifold as 2-fold branched covering space.     

Equilibrium for a combinatorial Ricci flow with generalized weights on a tetrahedron

Chow and Lou [2] showed in 2003 that under certain conditions the combinatorial analogue of the Hamilton Ricci flow on surfaces converges to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [2] was that the weights are nonnegative. We have recently shown that the same statement on convergence can be proved under weaker conditions: some weights can be negative and should satisfy certain inequalities. In this note we show that there are some restrictions for weakening the conditions. Namely, we show that in some situations the combinatorial Ricci flow has no equilibrium or has several points of equilibrium and, in particular, the convergence theorem is no longer valid.     

The tetrahedron equation and algebraic geometry

The tetrahedron equation arises as a generalization of the famous Yang-Baxter equation to the2+1-dimensional quantum field theory and three-dimensional statistical mechanics. Not much is known about its solutions. In the present paper, a systematic method of constructing nontrivial solutions to the tetrahedron equation with spin-like variables on the links is described. The essence of this method is the use of the so-called tetrahedral Zamolodchikov algebras. Bibliography:12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 137–149. Translated by I. G. Korepanov.     

The lattice points of ann-dimensional tetrahedron

We show that the number of orderedm-tuples of points on the integer lattice, inside or on then-dimensional tetrahedron bounded by the hyperplanesX ₁=0,X ₂=0, ...,X _n=0 andw ₁ X ₁+w ₂ X _n+...+w _n X_n=X, with the property that, for eachj, no more thank such points have non-zerojth ordinate, is asymptotically

Volume of a tetrahedron in terms of dihedral angles and circumradius

The volume of a tetrahedron is represented in terms of the six dihedral angles between the faces and the radius of the sphere circumscribing the tetrahedron.     

No acute tetrahedron is an 8-reptile

An $r$-gentiling is a dissection of a shape into $r\ge 2$ parts which are all similar to the original shape. An $r$-reptiling is an $r$-gentiling of which all parts are mutually congruent. The complete characterization of all reptile tetrahedra has been a long-standing open problem. This note concerns acute tetrahedra in particular. We find that no acute tetrahedron is an $r$-gentile or $r$-reptile for any $r<10$. The proof is based on showing that no acute spherical diangle can be dissected into less than ten acute spherical triangles.     

A Lobatto interpolation grid in the tetrahedron

A sequence of increasingly refined interpolation grids insidethe tetrahedron is proposed with the goal of achieving uniformconvergence and ensuring high interpolation accuracy. The numberof interpolation nodes, N, corresponds to the number of termsin the complete mth-order polynomial expansion with respectto the three tetrahedral barycentric coordinates. The proposedgrid is constructed by deploying Lobatto interpolation nodesover the faces of the tetrahedron, and then computing interiornodes using a simple formula that involves the zeros of theLobatto polynomials. Numerical computations show that the Lebesgueconstant and interpolation accuracy of the proposed grid comparefavourably with those of alternative grids constructed by solvingoptimization problems. The condition number of the mass matrixis significantly lower than that of the uniform grid and comparableto that of optimal grids proposed by previous authors.