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Pérez Águila, Ricardo
- Capítulo 1. Introductio-
4D Orthogonal Polytopes
-- Licenciatura en Ingeniería
en Sistemas Computacionales. - Departamento de Ingeniería en
Sistemas Computacionales. - Escuela de Ingeniería, - Universidad de las Américas


Chapter 1 Introduction 1.1 Historical Overview In the ancient Greek, Euclid said that -a point has no dimension at all.
A line has only one dimension: length.
A plane has two dimensions: length and breadth.
A solid has three dimensions: length, breadth, and height.
And there it stops.
Nothing has four dimensions-. In -The Republic- (370 b.C.) is presented the Plato's allegory of the cave. [Gutiérrez, 95] resumes it as follows: in a dark cave there are some prisoners chained since they were children.
They can't see the daylight, the objects nor the people from the exterior. They just see the shadows that are projected onto the bottom's cave.
Outside the cave there are a road and a torch that origins these shadows.
The prisoners consider the shadows as their only reality.
One of the prisoners escapes and discovers the real world.
He returns to the cave and try to convince the others.
They don't believe him.
An important aspect of the Plato's allegory, is that it introduces the notion of a two-dimensional world and the experience of a being that discovers the existence of a three-dimensional world which includes him and his partners [Rucker, 84]. The first approach to the fourth dimension (4D) was made by August Möbius in 1827.
He speculated that rotations could work as reflections if any body (or figure) is passed through a higher dimension (one higher than the body or figure).
For example, a 1 right hand silhouette (a 2D figure) can be turned into a left hand silhouette passing it through the 3D space [Robbin, 92].
Möbius proposed that a 4D space is needed to turn right-handed three-dimensional crystals into left-handed crystals. In England, Arthur Cayley and John J.
Sylvester described an Euclidean geometry of four dimensions where hyperplanes are determined by noncoplanar quadruples of points. They were able to move into a higher dimension because they added a new axiom: -outside any given three-dimensional hyperplane, there are other points- [Banchoff, 9...

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