Identifying Hazardous Shapes in the Plane

This paper explores the problem of identifying the shapes of invisible hazardous entities in R2 by a set S = {s1, s2, . . . , sk} of mobile sensors (autonomous robots). A hazardous entity, H, is a region that affects the operation of robots that either penetrate the area or come in contact with it. In this paper, we propose algorithms for searching a rectangular region for a stationary hazardous entity, where some a priori geometrical knowledge is given (e.g., edge size range), and if such an entity exists, then determine the area that it occupies. We explore entities that are convex in nature such as line segment, circles (discs), and simple convex shapes. The objectives are to minimize the distance travelled by the robots during the search phase, and to minimize the number of robots that are required to identify the region covered by the hazardous entity. The number of robots required to locate H is three or four robots when H is a line segment, two or three robots when H is a circle, and seven robots are sufficient when H is a triangle. Our results extend to n-vertex convex shapes and we show that 2n + 1 robots are sufficient to determine the coverage of H.

On Euclidean diagrams and geometrical knowledge

We argue against the claim that the employment of diagrams in Euclidean geometry gives rise to gaps in the proofs. First, we argue that it is a mistake to evaluate its merits through the lenses of Hilbert’s formal reconstruction. Second, we elucidate the abilities employed in diagram-based inferences in the Elements and show that diagrams are mathematically reputable tools. Finally, we complement our analysis with a review of recent experimental results purporting to show that, not only is the Euclidean diagram-based practice strictly regimented, it is rooted in cognitive abilities that are universally shared.

VARIOUS PROBLEMS FOR TEACHING GEOMETRY TO THIRD GRADE STUDENTS

The knowledge on geometry are of great importance for the understanding of reality. Spatial notion and geometrical concepts, graphical skills and habits are an important part of the study of geometrical knowledge in elementary school as propedeutics of the system course on geometry in the next school levels. In the recent years, education in Bulgaria follows the trends imposed by the European Union related to the acquiring of some basic key competencies. They promote to the improvement of knowledge, skills, abilities and attitudes of students and their more successful social development. From the school year 2016/2017, the education in the Bulgarian schools is in accordance with the new Law on pre-school and school education. Under this law, students are teached under new curriculum and teaching kits for the corresponding class. According to the new curriculum, the general education of the students of I-IV grade, covers basic groups of key competencies. Here, much more attention is paid also to the results of international researches on the students’ performance in mathematics. Primary school students participate in international competitions and Olympiads, which lead to the need of working on more mathematical problems with geometric content of the relevant specific types. This allows to study and use author’s various mathematical problems for teaching geometry. Their purpose is to contribute to the expansion of space notions of the students, to develop their thinking and imagination. This article is dedicated to the application of author’s various mathematical problems and exercises for teaching students from the third grade through which the geometrical knowledge and skills of the students develop and build. The solving of the mathematical problems is realized on a rich visual-practical basis, providing conditions for inclusion of the students in various activities. The proposed various mathematical problems are developed by themes including fully geometric problems and exercises for teaching mathematics to third grade students. Teaching by using the various mathematical problems was held with 149 students from third grade, from five schools - three in Sofia and two in smaller towns, in the school year 2016/2017.

Cassirer and the Structural Turn in Modern Geometry

The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in his Substanzbegriff und Funktionsbegriff (1910). The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The second aim is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice.

On Surface Continuity in the Procession of Product Design

The purpose of this paper was to verify the way in which CAD systems and their tools for visual surfaces analysis interact with morphological knowledge in the determination of continuity in product design procession. Geometrical knowledge is necessary but not enough for working with this attribute of form in everyday objects, where cultural factors are involved. Geometry establishes a progressive range of surface continuity that involves the concepts of position, tangency and curvature. In product design different degrees of continuity that not necessarily follow this idea of increment. What is understood as discontinuous in products in most cases is geometrically continuous.

The Hemicycle of Notre-Dame of Paris: Gothic Design and Geometrical Knowledge in the Twelfth Century

The Hemicycle of Notre-Dame of Paris: Gothic Design and Geometrical Knowledge in the Twelfth Century analyzes how the layout of four plinths in the hemicycle of Notre-Dame of Paris reflects the state of mathematical knowledge at the time of the first construction phases of the cathedral in the early 1160s. During the first half of the twelfth century, building enterprise was paired with intellectual activity in Paris, where architects experimented with a new building style——Gothic——and scholars explained geometry in treatises. Stefaan Van Liefferinge reconstructs the mathematics used by the Gothic builders at Notre-Dame, notes its resemblance to the geometry of the Parisian scholars, and suggests that this similarity points to either the exchange of knowledge or a common origin.