Artículos con la etiqueta ‘Euclides’

Saccheri’s Rectilinear Quadrilaterals

Por • 19 ago, 2013 • Category: Opinion

We study Saccheri’s three hypotheses on a two right-angled isosceles quadrilateral, with a rectilinear summit side. We claim that in the Hilbert`s foundation of geometry the euclidean parallelism is a theorem and as that it can be used, in the hyperbolic geometry

Einstein’s physical geometry at play: inertial motion, the boostability assumption, the Lorentz transformations, and the so-called conventionality of the one-way speed of light

Por • 10 jun, 2013 • Category: Ciencia y tecnología

In this work, Einstein’s view of geometry as physical geometry is taken into account in the analysis of diverse issues related to the notions of inertial motion and inertial reference frame. Einstein’s physical geometry enables a non-conventional view on Euclidean geometry (as the geometry associated to inertial motion and inertial reference frames) and on the uniform time. Also, by taking into account the implications of the view of geometry as a physical geometry, it is presented a critical reassessment of the so-called boostability assumption (implicit according to Einstein in the formulation of the theory) and also of ‘alternative’ derivations of the Lorentz transformations that do not take into account the so-called ‘light postulate’. Finally it is addressed the issue of the eventual conventionality of the one-way speed of light or, what is the same, the conventionality of distant simultaneity (within the same inertial reference frame). It turns out that it is possible to see the (possible) conventionality of distant simultaneity as a case of conventionality of geometry (in Einstein’s reinterpretation of Poincar\’e’s views).

Saccheri’s Quadrilaterals

Por • 18 mar, 2012 • Category: Ciencia y tecnología

We study Saccheri’s three hypotheses on a two right-angled isosceles quadrilateral, under certain assumptions, with respect of the independence of the euclidean parallel postulate. We also trace the historical circumstances under which, the development of arguments for the consistency of non-Euclidean geometries occurred; indicating an important shift in mathematicians’ attitude towards the fifth Euclidean postulate.

A Relationship between Geometry and Algebra

Por • 8 nov, 2011 • Category: Ambiente

The three key documents for study geometry are: 1) «The Elements» of Euclid, 2) the lecture by B. Riemann at G\»ottingen in 1854 entitled «\»Uber die Hypothesen welche der Geometrie zu Grunde liegen» (On the hypotheses which underlie geometry) and 3) the «Erlangen Program», a document written by F. Klein (1872) on his income as professor at the Faculty of Philosophy and the Senate of the Erlangen University. The latter document F. Klein introduces the concept of group as a tool to study geometry. The concept of a group of transformations of space was known at the time. The purpose of this informative paper is to show a relationship between geometry and algebra through an example, the projective plane. Erlangen program until today continues being a guideline of how to study geometry.