Since its introduction in the late 1980s, Dynamic Geometry Software (DGS) has become one of the most innovative tools in mathematics education. It is defined as graphical software, where geometric figures can be constructed with pre-defined relationships, which will retain when the figures are dynamically manipulated. This digital tool provides a new geometry learning environment inherently different from the traditional paper-and-pencil mode. This research investigates the situation where learners interact directly with this dynamic geometry environment. It examines how learners interpret DGS key features; such as drag-mode and parent-and-child relationship, and how such interpretations relate to their higher-order thinking of reasoning in geometric tasks. Three types of reasoning strategies are pursued in this research. These are: inductive reasoning, deductive reasoning and abductive reasoning. How and to what extent the DGS environment plays a role in the learner's reasoning strategies and arguments is the question at the focus of this research. Vygotsky's model of tool used as mediated activity and Verillon & Rabardel's Instrumented Activity Situation (IAS) model are used as a framework for this research. These models help to distinguish the independent roles of the learner, the DGS tool, the designed tasks and Euclidean geometry in the overall setting. They also help to clarify the influences that each of these entities may have on each other. The research is conducted in Thailand with a Thai version of The Geometer's Sketchpad to a group of 14-15 year-old lower secondary students. The research method used is a task-based interview, where pairs of students perform geometric construction and exploration tasks with Geometer's Sketchpad while the researcher challenges their reasoning. This research finds the tension between the deductive reasoning nature in Euclidean geometry, the inductive nature of visual presentation in the dynamic geometry environment, and the influence of students' experience in the paper-and-pencil environment on their interpretation of dynamic geometry. Abductive reasoning is found to be students' main reasoning strategy, with a combination of inductive and deductive reasoning to support their verification of the hypothesis.

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We study the isospectral deformations of the Eguchi-Hanson spaces along a torus isometric action in the noncompact noncommutative geometry. We concentrate on locality, smoothness and summability conditions of the nonunital spectral triples, and relate them to some geometric conditions to be noncommutative spin manifolds.

It has been recognised that, within the context of face recognition, angular separation between centred feature vectors is a useful measure of dissimilarity. In this thesis we explore this observation in more detail and compare and contrast angular separation with the Euclidean, Manhattan and Mahalonobis distance metrics. This is applied to 2D, 2.5D and 3D face images and the investigation is done in conjunction with various feature extraction techniques such as local binary patterns (LBP) and linear discriminant analysis (LDA). We also employ error-correcting output code (ECOC) ensembles of support vector machines (SVMs) to project feature vectors non-linearly into a new and more discriminative feature space. It is shown that, for both face verification and face recognition tasks, angular separation is a more discerning dissimilarity measure than the others. It is also shown that the effect of applying the feature extraction algorithms described above is to considerably sharpen and enhance the ability of all metrics, but in particular angular separation, to distinguish inter-personal from extra-personal face image differences. A novel technique, known as angularisation, is introduced by which a data set that is well separated in the angular sense can be mapped into a new feature space in which other metrics are equally discriminative. This operation can be performed separately or it can be incorporated into an SVM kernel. The benefit of angularisation is that it allows strong classification methods to take advantage of angular separation without explicitly incorporating it into their construction. It is shown that the accuracy of ECOC ensembles can be improved in this way. A further aspect of the research is to compare the effectiveness of the ECOC approach to constructing ensembles of SVM base classifiers with that of binary hierarchical classifiers (BHC). Experiments are performed which lead to the conclusion that, for face recognition problems, ECOC yields greater classification accuracy than the BHC method. This is attributed primarily to the fact that the size of the training set decreases along a path from the root node to a leaf node of the BHC tree and this leads to great difficulties in constructing accurate base classifiers at the lower nodes.

It is widely held that the role of diagrams in mathematical arguments is merely heuristic, or involves a dubious appeal to a postulated faculty of “intuition”. To many these have seemed to exhaust the available alternatives, and worries about the status of intuition have in turn motivated the dismissal of diagrams. Thus, on a standard interpretation, an important goal of 19th Century mathematics was to supersede appeals to intuition as a ground for knowledge, with Euclid’s geometry—in which diagrams are ubiquitous—an important target. On this interpretation, Euclid’s presentation is insufficient to justify belief or confer knowledge in Euclidean geometry. It was only with the work of Hilbert that a fully rigorous presentation of Euclidean geometry became possible, and such a presentation makes no nonredundant use of diagrams. My thesis challenges these claims. Against the “heuristic” view, it argues that diagrams can be of genuine epistemic value, and it specifically explores the epistemology of diagrams in Euclid’s geometry. Against the “intuitive” view, it claims that this epistemology need make no appeal to a faculty of intuition. It describes in detail how reasoning with diagrams in Euclid’s geometry can be sufficient to justify belief and confer knowledge. And it shows how the background dialectic, by assuming that the “heuristic” or “intuitive” views above are exhaustive, ignores the availability of this further alternative. By using a detailed case study of mathematical reasoning, it argues for the importance of the epistemology of diagrams itself as a fruitful area of philosophical research.

The problem we consider here arises quite naturally from Crum's Problem which asks: What is the maximum number of non-overlapping polyhedra such that any pair of them have a common boundary of positive area? In (1) Besicovitch, by constructing a sequence of polyhedra satisfying the required conditions, showed that the answer to Crum's Problem is infinity. In this thesis we ask: What is the answer when the polyhedra of Crum's Problem are restricted to be tetrahedra? As stated in the Abstract, we prove that the answer is either 8 or 9 and the evidence tends to point to the answer in fact being 8. As the difference in answers would suggest, the methods we use to establish these results are completely different from those used by Besicovitch in his paper. In Chapter One we show that an n-con (for definition see the Abstract) can be represented by an n-Towed matrix whose minors satisfy certain conditional we then develop arguments from which we deduce that n is less than 18. Chapter Two shows that the bound may be reduced to n less than 14. The subsequent five chapters are mainly concerned with the conditions under which a 9-con can exist and we eventually show that if an n-con for n > 9 exists then no plane contains six faces of the tetrahedra of the n-con.5.Our analysis of the 9-con continues in Chapter Eight where we show that what one may describe as the most symmetrical case for a 9-con cannot exist and also that the faces of the tetrahedra of the 9-con must be so arranged that they are contained in either nine or ten planes. To demonstrate that the existence or not of a 9-con is critical, we show in Chapter Nine that a 10-con cannot exist and in Chapter Ten that a 8-con does exist. Chapter Eleven discusses the results obtained.

The Grundlagen der Geometrie brought Euclid’s ancient axioms up to the standards of modern logic, anticipating a completely mechanical verification of their theorems. There are five groups of axioms, each focused on a logical feature of Euclidean geometry. The first two groups give us ordered geometry, a highly limited setting where there is no talk of measure or angle. From these, we mechanically verify the Polygonal Jordan Curve Theorem, a result of much generality given the setting, and subtle enough to warrant a full verification. Along the way, we describe and implement a general-purpose algebraic language for proof search, which we use to automate arguments from the first axiom group. We then follow Hilbert through the preliminary definitions and theorems that lead up to his statement of the Polygonal Jordan Curve Theorem. These, once formalised and verified, give us a final piece of automation. Suitably armed, we can then tackle the main theorem.

Οι γεωμετρικές κατασκευές (με την χρήση κανόνα και διαβήτη) και η παρουσίαση των τριών περίφημων προβλημάτων από την ελληνική αρχαιότητα μέχρι την τελική απάντηση που δόθηκε για αυτά. Η εξιστόρηση της κατασκευής κανονικών πολυγώνων και η διερεύνηση της δυνατό- ητας των μαθητών του Λυκείου να εργαστούν πάνω σε προβλήματα γεωμετρικών κατασκευών, με την αναλυτικο-συνθετική μέθοδο. / Geometrical constructions (using only an unmarked straight edge and compasses)of Euclidean plane geometry and the presentation of the three unsolved famous constructions problems from the greek anciety until the final answer for them. The construction’s narration of regular polygons and the research of high school students possibility to work on geometrical construction problems with the analysis and synthesis method.