# Pérez &Aacute;guila, Ricardo - Capítulo 2. The Hypercub- 4D Orthogonal Polytopes

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Pérez &Aacute;guila, Ricardo
- Capítulo 2. The
Hypercub-
4D Orthogonal Polytopes
-- Licenciatura en Ingeniería
en Sistemas Computacionales. - Departamento de Ingeniería en
Sistemas Computacionales. - Escuela de Ingeniería, - Universidad de las Américas
Puebla.

## Introducción

Chapter 2 The Hypercube 2.1 Obtaining a Segment, a Square, a Cube and a Hypercube In [Rucker, 77] is presented the Claude Bragdon's method to define a series of figures which are called the parallelotopes [Coxeter, 63].
First a 0D point is taken and moved one unit to the right.
The path between the first and the second new point produces a 1D segment.
The first dimension, represented by the X-axis, has appeared (Figure 2.1). X O O FIGURE 2.1 Generation and final 1D unit segment C1 (own elaboration). The new segment is then moved one unit upward.
The path between the first and the second new segment produces a 2D square (a parallelogram).
The second dimension, represented by the Y-axis, has appeared (Figure 2.2). Y O X O X FIGURE 2.2 Generation and final 2D unit square C2 (own elaboration). 14 The new square is then moved one unit forward out this paper.
The path between the first and the second new square produces a 3D cube (a parallelepiped).
The third dimension, represented by the Z-axis, has appeared (Figure 2.3).
Because we are working over a 2D surface (this paper and the computer’s screen), a diagonal between X and Y-axis represents the Z-axis, however it should be interpreted as a line perpendicular to this 2D surface. Y Y Z O X O X FIGURE 2.3 Generation and final 3D unit cube C3 (own elaboration). We know that the fourth dimension has a direction perpendicular to the other three dimensions, in this case the W-axis is presented as a perpendicular line to the Z-axis.
Then the cube is moved one unit in direction of the W-axis.
The path (six cubes perpendicular to the first one) between the first and the second new cube produces the 3D boundary of a 4D hypercube (a 4D parallelotope).
The fourth dimension has appeared (Figure 2.4). 15 Y Y Z Z W X O X O FIGURE 2.4 Generation and final 4D unit hypercube C4 (own elaboration). Definition 2.1: Let Cn be the n-dimensional parallelotope, then C0 is a point and Figures 2.1 to 2.4 correspon...