Plimpton 322 : A Universal Cuneiform Table for Old Babylonian Mathematicians, Builders, Surveyors and Teachers

This article deals with the damaged and incomplete Old Babylonian tablet Plimpton 322 which contains 4 columns and 15 rows of a cuneiform mathematical text. It has been shown that the presumed original table with its 7 columns and 39 rows represented: a table of square roots of numbers from 0 to 2 for mathematicians; an earliest rudiments of a trigonometric table for builders and surveyors where angles are not measured as an arc in a unit circle but as a side of a unit right-angled triangle; a list of the 39 exercises on reciprocal pairs, unit and integer-side right triangles (rectangles), factorization and square numbers for teachers. The article provides new arguments in favor of old disputes (squares of diagonals or widths; mistakes in previous analysis of errors in P322). Contradictory ideas about P322 are discussed: Is it the table of triangle sides or factorization terms? Was it compiled by a parallel or independent factorization of the sides or of their squares? Are sides of an initial unit triangle enlarged or reduced by such a factorization? Does it contain two or four arithmetical errors? Time and dimensional requirements for calculation and writing of the complete tablet have been also estimated.

The Determination of Heating Shapes and Locations for Triangle Heating

Thermal forming is a method to form a curved plate by inducing local shrinkage and angular distortion through heating and cooling. In this approach, two different methods are available: line heating and triangle heating. Among them, this paper discusses triangle heating and presents algorithms for determining heating shapes and locations. The heating shape is determined by using the in-plane strain distributions, which are calculated by nonlinear kinematics analysis between the designed and initial shapes, field survey results, and mechanics based on the neutral axis. To predict the angular distortion and shrinkage in various heating conditions, a functional relation of residual deformations is formulated. For the formulation, multivariate analysis and multiple regression techniques are used with data obtained from experiments of unit triangle heating and numerical analysis. Using the determined heating shapes and the functional relation for the residual deformations, a correct triangle heating position is determined by an algorithm, which can predict qualitatively correct angular distortion and shrinkage in the interior and quantitatively correct distortion values on the edge. Finally, analytic verification of the proposed method has been done by applying the method to a convex type plate used in the field. The proposed work can be used for automation of curved plate fabrication in the shipyards.

Integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space

This paper concerns with analytical integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space. The volume integration of trivariate polynomials over linear polyhedra is computed as sum of surface integrals in R3 on application of the well known Gauss's divergence theorem and by using triangulation of the linear polyhedral boundary. The surface integrals in R3 over an arbitrary triangle are connected to surface integrals of bivariate polynomials in R2. The surface integrals in R2 over a simple polygon or over an arbitrary triangle are computed by two different approaches. The first algorithm is obtained by transforming the surface integrals in R2 into a sum of line integrals in a one-parameter space, while the second algorithm is obtained by transforming the surface integrals in R2 over an arbitrary triangle into a parametric double integral over a unit triangle. It is shown that the volume integration of trivariate polynomials over linear polyhedra can be obtained as a sum of surface integrals of bivariate polynomials in R2. The computation of surface integrals is proposed in the beginning of this paper and these are contained in Lemmas 1–6. These algorithms (Lemmas 1–6) and the theorem on volume integration are then followed by an example for which the detailed computational scheme has been explained. The symbolic integration formulas presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, for example, the volume, centre of mass, moments of inertia etc., required in engineering design processes.

Compression Deformation Structures of Single Crystal TiAl

ABSTRACTThe compression deformation behavior of single crystalline TiAl was examined by transmission electron microscopy (T.E.M.). The relatively pure Ti–56 a/o Al crystal was containerless processed in ultrapure hydrogen. The crystal growth direction is 18° off (011) and 60° off [111] in the [011]– [111]-[010] unit triangle. At low stresses, a/2 [110] type dislocations were observed. a/2 [110] dislocations appeared at slightly higher stresses. Additional plastic deformation initiates twinning. Twinning plays an important role at higher stresses. Diffraction results indicate that most of the twins have the (111) mirror plane.A small amount of (111) twins were also observed. Superdislocations of the a<011> type were not observed to contribute to the plastic deformation in this crystal. The results indicate that plastic deformation by twinning follows the low density of ordinary dislocations.

The Orientation and Position Accuracy of Electron Diffraction

The misorientations existing between subgrains are of critical importance in understanding the nucleation and growth of recrystallised grains. In this paper it is demonstrated that, because of spherical aberration and focus error in the objective of a conventional 100 kV electron microscope, the accuracy of beam direction determination is a function of the size of the area selected, when this area is small. Various factors are discussed, and it is shown that for beam direction determination to within 2° accuracy over the whole unit triangle using a modern 100 kV microscope, that the minimum area must be greater than 1.5 μm diameter. The implications of this on previous conventional SAD deformation texture studies, where the subgrain size is typically ∼ 2,500 Å, are discussed.