Geometria das dimensões e quadridimensionalidade

• Autor(a) principal: Souza, Samuel de
• Data de Publicação: 2019
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Biblioteca Digital de Teses e Dissertações da PUC_SP
• Texto Completo: https://tede2.pucsp.br/handle/handle/22934
• topic: Dimensão; Geometria das dimensões; Quadridimensionalidade; Dimension; Geometry of dimensions; Four-dimensionality; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA

This research has as its main focus to propose a theoretical-didactical reflection on the Geometry of Dimensions, based on the visualization and in the different dimensional deconstructions and reconstructions of geometric figures. We start from the development of a crescent geometry vision, going from dimension zero, the dot, to dimension one, the Line Segment, to dimension two, of the Face, to dimension three, of dimensional variable, the Solid, to dimension four, of the Hypersolid variable. We develop this theme, by using four Registers of Semiotics Representation, by alternating between the Geometric Figural Register, the Mother Tongue Discursive Register, the Chart Register, and the Algebric Register. We attempted to produce crescent Dimensional Reconstructions, by means of Instrumental Reconstructions or by means of Mereological Heuristic Reconfigurations, the cutting of specific figures at each dimensional element, followed by an infinitesimal increment to these figures, and their Dimensional Reconstruction in a superposition to the new dimension. Further dimensional reconstructions were also made from the previous dimensions. Complementarily, we developed the concept of fractional dimensions, between the whole dimensions, analogous to what has been done in the General Relativity Theory and non-Euclidean Geometries. We carefully added to each constructed dimensional figure four types of visualization in different dimensional reconstructions from the passage of one dimension to the next. Gauss influenced several mathematicians in the formation of non-Euclidean Geometries, like the Lobachevskyan Geometry and the Riemannian Geometry. The emergence of Maxwell’s Electromagnetism, as well as Einstein’s Relativity Theory already appear naturally as four-dimensional. It was the mathematicians Poincaré and Minkowski who have drawn attention to the four-dimensionality of these theories in Physics. Thus, the time-space continuum had arisen, which had shown to be the physical space of four-dimensional behavior. From then on, the development of Physics begins to make use of dimensions larger than three as an inevitable condition. The mathematicians Euclides, Archimedes, Pappus, Descartes, Newton, Leibniz, Gauss, Euler, Möbius, Lobachevsky, Bolyai, Riemann, Klein, Maxwell, Poincaré, Minkowski and Einstein contributed to the development of many kinds of Geometry, and the dimensions were always present in their analyses. In this work we emphasize the dimensions in Geometry as a research focused on teaching. Einstein’s Gravitation shows that the presence of physical matter in space curves the space into a fourth dimension, making so that the celestial bodies and light follow a so-called geodesic “straight trajectory”, but in which its three-dimensional appearance is actually a curve. These ideas allowed for the emergence of Models to the Universe with a dimensionally curved space, which proves to be the deepest nature of the Geometry of physical objects. Influenced by these studies and the works of Duval, the TRRS, the cognitive experience of Learning, diversity of Visualization and differentiation of Reasonings, we researched these elements for the teaching of the Mathematical Education of the Geometry of Dimensions. At the end, from dimension five onwards, we develop recursive equations in the Discursive, Chart and Algebric Register, generalizing and demonstrating its validity to any whole D-esimal dimension