Utilización de sistemas informáticos en la representación gráfica de objetos de cuatro dimensiones

Abstract

Three-dimensional objects have traditionally been represented on two-dimensional medium by applying the laws of perspective to achieve a resemblance to reality or by a number of projec- tions on different planes, sections and details, together with dimensions, that allow a faithful reproduction of the object. Modern computer design programs allow three-dimensional draw- ing of the object, and deduce from it the required two-dimensional views. This quantitative jump increases the types of objects that can be reproduced graphically: four-dimensional ob- jects can be represented by a number of two- or three-dimensional views, and the computer screen allows time to be introduced as a fourth variable, that can be linked to the fourth dimen- sional variable by a simple formula, allowing the representation of four-dimensional objects to three dimensional spectators. A promising line of investigation arises from the relationship thus established between the four dimensional Euclidean space and the Einstenian space-time universe. We will study as four dimensional objects the regular hyper-polyhedres that can be constructed by folding in the fourth dimension a number of equal regular polyhedres. The subject was first covered by the swiss mathematician Ludwig Schläfli at the end of the nineteenth century, determiming that there are six regular four dimensional hiper-polyhedra, or politopes, and only three on all other higher dimensions. Starting with the simplest ones, we will determine the views and sections required for an adequate graphic representation.