Geometric Terminology Research Articles

A new terminology for vein wall-foliation crosscutting relationships

Confronted with discordant quartz veins with complex vein wall-foliation crosscutting relationships in the High-Ardenne slate belt (Belgium), we developed a new terminology to describe the geometrical relationship between host fabric elements and crosscutting elements. This new terminology introduces a directional sense in the description of the vein wall-foliation geometry. Applying this new terminology on a series of quartz veins with vein wall-foliation crosscutting relationships of variable degrees of complexity, allowed to categorize the veins studied according to two types of vein wall-foliation geometry. While the first vein wall-foliation geometry suggests that the veins acted as competent objects during vein-shape modification, the second vein wall-foliation geometry suggests that the vein-shape modification resulted from deforming fluid-filled cavities with very mobile vein walls. Based on these examples, we tentatively demonstrate that the new geometrical terminology may bring useful, additional elements to the kinematic analysis of irregular veins with complex vein wall-foliation crosscutting relationships.

Geometrothermodynamics for black holes and de Sitter space

A general method to extract thermodynamic quantities from solutions of the Einstein equation is developed. In 1994, Wald established that the entropy of a black hole could be identified as a Noether charge associated with a Killing vector of a global space-time (pseudo-Riemann) manifold. We reconstruct Wald’s method using geometrical language, e.g., via differential forms defined on the local space-time (Minkowski) manifold. Concurrently, the abstract thermodynamics are also reconstructed using geometrical terminology, which is parallel to general relativity. The correspondence between the thermodynamics and general relativity can be seen clearly by comparing the two expressions. This comparison requires a modification of Wald’s method. The new method is applied to Schwarzschild, Kerr, and Kerr–Newman black holes and de Sitter space. The results are consistent with previous results obtained using various independent methods. This strongly supports the validity of the area theorem for black holes.

Occult Spheres, Planes, and Dimensions: Geometric Terminology and Analogy in Modern Esoteric Discourse

This is an article on the history of religion and mathematics, which explores the origins and uses of geometric terminology in late-nineteenth- and early-twentieth-century esoteric discourses, and inquires as to whether or not such adaptions are instances of the discursive strategy identified by Olav Hammer as terminological scientism. For the situation regarding spheric terminology, I argue that the context is not scientistic, being rather an example of the decontextualised modern appropriation of antique terminologies. In the case of planar terminology, I conclude that while its origin with Emanuel Swedenborg as an adaption of anatomical vocabulary does indeed fit Hammer's definition of scientism, modern usage stripped the term of Swedenborg's scientistic rationale — making the scientistic designation something of a vestigial atavism. In dealing with dimensional terminology, my conclusion is that from Johann Zöllner's initial formulation to the broader Spiritualistic and Theosophical applications, it exemplifies terminological scientism through and through.

Napoleon’s quasigroups

Abstract Napoleon’s quasigroups are idempotent medial quasigroups satisfying the identity (ab·b)(b·ba) = b. In works by V. Volenec geometric terminology has been introduced in medial quasigroups, enabling proofs of many theorems of plane geometry to be carried out by formal calculations in a quasigroup. This class of quasigroups is particularly suited for proving Napoleon’s theorem and other similar theorems about equilateral triangles and centroids.

Beyond teaching language: Towards terminological primacy in learners’ geometric conceptualisation

This paper reports on a specific aspect of a broader geometry conceptualisation study that sought to explore and explicate learners’ knowledge of basic geometric terminology in selected Nigerian and South African high schools. It is framed by the notion that students’ acquisition of the correct terminology in school geometry is important for their success in the subject. The original study further aimed to determine the relationship that might exist between a learner’s ability in verbal geometry terminology tasks and his/her ability in visual geometry terminology tasks. A total of 144 learners (72 each from South Africa and Nigeria) were selected for the study, using both the stratified and the fish‐bowl sampling techniques. A questionnaire consisting of a sixty‐item multiple‐choice objective test provided the data for the study. An overall percentage mean score of 44,17% obtained in the test indicated that learners in this study had only a limited knowledge of basic geometric terminology. The Nigerian subsample in the study had a weaker understanding of basic geometric terminology than their South African counterparts. Importantly, there were high positive correlations between participants’ ability in verbal geometry terminology tasks and their ability in visual geometry terminology tasks. These results are consistent with those of several earlier studies, and provide a reasonably firm basis for certain recommendations to be made.

A geometric framework for the kinematics of crystals with defects

Presented is a general theoretical framework capable of describing the finite deformation kinematics of several classes of defects prevalent in metallic crystals. Our treatment relies upon powerful tools from differential geometry, including linear connections and covariant differentiation, torsion, curvature and anholonomic spaces. A length scale dependent, three-term multiplicative decomposition of the deformation gradient is suggested, with terms representing recoverable elasticity, residual lattice deformation due to defect fields, and plastic deformation resulting from defect fluxes. Also proposed is an additional micromorphic variable representing additional degrees-of-freedom associated with rotational lattice defects (i.e. disclinations), point defects, and most generally, Somigliana dislocations. We illustrate how particular implementations of our general framework encompass notable theories from the literature and classify particular versions of the framework via geometric terminology.

Children's descriptions of fractal shapes.

Elementary-school students (N = 359), who had not been introduced to fractal geometry, examined and described the shape of three fractal sets, i.e., batter-fried chicken, popcorn, and curly leaf lettuce. The data were generated during individual, untimed, semistructured interviews. In general, the children used terminology of classical geometry to describe the shape of the popcorn, and they used adjectives, phrases, created words, and classical geometric terminology to describe the shape of the chicken and lettuce. A framework of terminology that children might use in describing fractal shapes was not discernible from the findings, but the results suggest that the children's descriptions may have been influenced by the properties of the different fractal sets.

CHILDREN'S DESCRIPTIONS OF FRACTAL SHAPES

Elementary-school students (N = 359), who had not been introduced to fractal geometry, examined and described the shape of three fractal sets, i.e., batter-fried chicken, popcorn, and curly leaf lettuce. The data were generated during individual, untimed, semistructured interviews. In general, the children used terminology of classical geometry to describe the shape of the popcorn, and they used adjectives, phrases, created words, and classical geometric terminology to describe the shape of the chicken and lettuce. A framework of terminology that children might use in describing fractal shapes was not discernible from the findings but the results suggest that the children's descriptions may have been influenced by the properties of the different fractal sets.

On arcs in path designs of block size 3

For each admissiblev we exhibit a path designP(v, 3, 1) with a spanning set of minimum cardinality and aP(v, 3, 1) with a scattering set of maximum cardinality. Moreover, we study maximal independent sets (or complete arcs in the geometric terminology) having the minimum number of secants, i.e. sets which are both spanning and scattering, and complete arcs with the maximum number of secants.

A SET OF NEW ALL-DIRECTION ROBUST REGULATION INDICES

A set of new all-direction regulation indices (ADRI) is suggested using the accustomedclassical geometrical terminology. With the advantage of the previous indices, they aresuitable for both simple-input simple-output systems and multi-input multi-output systems.Different from the tracking indices type, ADRI summarises the whole regulation specifica-tions of the system for various possible disturbances from all directions in the state space.The characters of the new and the old indices are analyzed. The algorithm of ADRI valuesis given. It is found that both the ADRI and the robustness index of a system simultaneouslyrequire the eigenvectors to be as orthonormal as possible. As a result a functional index isobtained that can improve the robustness and the dynamic regulation performance simulta-neously.

Zonoids with minimal volume-product

Of course, only the "only if" part needs a proof. We give here some preliminaries and notations. First, we make the remark that our approach is based on the analysis of random polytopes made in [7] and we make extensive reference to [7] and [6] in order to avoid repeating calculations that are done there. We use standard geometric terminology as may be found, e.g., in [3] or [4]. The dimension n will be fixed throughout the paper. B is the Euclidean unit ball in ~" , S the unit sphere of ~ " with the relative topology, co is the rotation invariant measure on S, normalized so that

Electron theory of metals and geometry

Modern electron theory of metals abounds in geometric terminology and, by using geometric ideas, provides clear descriptions of many complicated phenomena. Some problems in the theory of normal metals that can be interpreted geometrically are examined in this paper in a language accessible to many physicists. Particular attention is devoted to those phenomena and properties that are connected with qualitative (topological) changes in geometric figures such as Fermi surfaces, plane sections though such surfaces, belts on the Fermi surface, and so on. Graphical illustrations of these ideas are provided.

The geometry of the cartan space associated with general compressible fluid flows

In this paper, an attempt is made to develop a general theory of the motion of a compressible fluid by means of the geometry of the general space propounded by Cartan based on the concept of area. Assuming that the flow is irrotational and isentropic, it is shown in Section I, that the equation of motion of an inviscid compressible fluid is regarded as that of hypersurfaces in the Cartan space. Various hydrodynamical features of the compressible fluid motion including the “shock” phenomena are clarified by the metric properties of the space. The fluid motion in the general case, in which the above assumptions are not valid, is treated in Section 2. The physical quantities such as vorticity, entropy and the Croccian vector are expressed by means of the geometrical terminology of the Cartan space along lines parallel to the Section 1.

Derivation of the Wishart distribution of the second order sample moments by straightforward integration of a multiple integral

1. Introduction. In his textbook of statistics Kendall classifies the methods of deducing exact sampling distributions into four groups: (a) straightforward evaluation of the integral in question by ordinary analytical processes such as a convenient change of variable; (b) the use of geometrical terminology; (c) the use of characteristic functions; and (d) other analytical methods.