Ternary Operations as Primitive Notions for Constructive Plane Geometry IV

In this paper we provide a quantifier-free constructive axiomatization for Euclidean planes in a first-order language with only ternary operation symbols and three constant symbols (to be interpreted as ‘points’). We also determine the algorithmic theories of some ‘naturally occurring’ plane geometries. Mathematics Subject Classification: 03F65, 51M05, 51M15, 03B30.     

Ternary Operations as Primitive Notions for Constructive Plane Geometry V

In this paper we provide a quantifier-free, constructive axiomatization of metric-Euclidean and of rectangular planes (generalizations of Euclidean planes). The languages in which the axiom systems are expressed contain three individual constants and two ternary operations. We also provide an axiom system in algorithmic logic for finite Euclidean planes, and for several minimal metric-Euclidean planes. The axiom systems proposed will be used in a sequel to this paper to provide ‘the simplest possible’ axiom systems for several fragments of plane Euclidean geometry. Mathematics Subject Classification: 51M05, 51M15, 03F65.     

Ternary Operations as Primitive Notions for Constructive Plane Geometry VI

In this paper we provide quantifier-free, constructive axiomatizations for several fragments of plane Euclidean geometry over Euclidean fields, such that each axiom contains at most 4 variables. The languages in which they are expressed contain only at most ternary operations. In some precisely defined sense these axiomatizations are the simplest possible.     

Dragging in a Dynamic Geometry Environment Through the Lens of Variation

What makes Dynamic Geometry Environment (DGE) a powerful mathematical knowledge acquisition microworld is its ability to visually make explicit the implicit dynamism of thinking about mathematical geometrical concepts. One of DGE’s powers is to equip us with the ability to retain the background of a geometrical configuration while we can selectively bring to the fore dynamically those parts of the whole configuration that interest us. That is, we can visually study the variation of an aspect of a DGE figure while keeping other aspects constant, hence anticipating the emergence of invariant patterns. The aim of this paper is to expound the epistemic value of variation of the Dragging tool in DGE in mathematical discovery. Functions of variation (contrast, separation, generalization, fusion) proposed in Marton’s theory of learning and awareness will be used as a framework to develop a discernment structure which can act as a lens to organize and interpret dragging explorations in DGE. Such a lens focuses very strongly on mathematical aspects of dragging in DGE and is used to re-interpret known dragging modalities (e.g., Arzarello et al.) in a potentially more mathematically-relevant way. The exposition will centre about a specific geometrical problem in which two dragging trajectories are mapped out, consequently resulting in a DGE theorem and a visual theorem. In doing so, a new spectral dragging strategy will be introduced that literally allows one to see the drag mode in action. A model for the lens of variation in the form of a discernment nest structure is proposed as a meta-tool to interpret dragging experiences or as a meta-language to relate different dragging analyses which consequently might give rise to pedagogical and epistemological implications.     

Semistar Operations and Standard Closure Operations

Let R be a commutative ring. It is shown that there is an order isomorphism between a popular class of finite type closure operations on the ideals of R and the poset of semistar operations of finite type.     

Riemann-Finsler Geometry with Applications to Information Geometry

Information geometry is a new branch in mathematics, originated from the applications of differential geometry to statistics. In this paper we briefly introduce Riemann-Finsler geometry, by which we establish Information Geometry on a much broader base, so that the potential applications of Information Geometry will be beyond statistics.     

3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$

It is possible to set up a correspondence between 3D space and \({\mathbb{R}^{3,3}}\), interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become represented as rotors in the geometric algebra of \({\mathbb{R}^{3,3}}\). We show explicitly how various primitive projective transformations (translations, rotations, scalings, perspectivities, Lorentz transformations) are represented, in geometrically meaningful parameterizations of the rotors by their bivectors. Odd versors of this representation represent projective correlations, so (oriented) reflections can only be represented in a non-versor manner. Specifically, we show how a new and useful ‘oriented reflection’ can be defined directly on lines. We compare the resulting framework to the unoriented \({\mathbb{R}^{3,3}}\) approach of Klawitter (Adv Appl Clifford Algebra, 24:713–736, 2014), and the \({\mathbb{R}^{4,4}}\) rotor-based approach by Goldman et al. (Adv Appl Clifford Algebra, 25(1):113–149, 2015) in terms of expressiveness and efficiency.     

Symplectic Geometry

Book Review

Symplectic GeometryA. T. Fomenko: Advanced Studies in Contemporary Mathematics, Volume 5, Gordon and Breach Science Publishers, 1988, 387 pp     

Pseudoquaternion Geometry

We describe complex holomorphic transformations of a quaternion vector space taking left quaternion lines to left quaternion lines and real linear transformations of the quaternion plane simultaneously preserving the sets of left and right quaternion lines.     

Turnaround Students in High School Mathematics: Constructing Identities of Competence Through Mathematical Worlds

This study investigates prospective secondary teachers’ cognitive difficulties and mathematical ideas involved in making connections among representations. We implemented a three-week teaching unit to help prospective secondary mathematics teachers develop understanding of big ideas that are critical to formulating connections among representations, in the context of conic curves. Qualitative analysis of data showed that most undergraduate mathematics majors and minors in this study struggled with variation, the Cartesian Connection, and other affiliated ideas such as graph as a locus of points. Furthermore, they were unable to identify basic metric relations encoded in algebraic expressions such as the distance between points, which further compounded their difficulties in making connections among representations. We argue that mathematics teacher education needs more focus on these ideas so that their graduates can successfully teach these big ideas in their future instruction.