# Pérez &Aacute;guila, Ricardo - Capítulo 7. Results and Future Wor- 4D Orthogonal Polytopes

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Pérez &Aacute;guila, Ricardo
- Capítulo 7. Results
and Future Wor-
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 7 Results and Future Work 7.1 Results: Resume of Studied Analogies 7.1.1 Analogies Between The Parallelotopes In the 1D, 2D, 3D and 4D space In chapter 2 were studied the properties of the segment, the square, the cube and the hypercube (see Table 7.1):  Segment: 1D Parallelotope (C1), boundary composed by 2 vertices.  Square: 2D Parallelotope (C2), boundary composed by 4 vertices and 4 edges.  Cube: 3D Parallelotope (C3), boundary composed by 8 vertices, 12 edges and 6 faces.  Hypercube: 4D Parallelotope (C4), boundary composed by 16 vertices, 32 edges, 24 faces and 8 volumes. TABLE 7.1 Analogies between the nD Parallelotopes (1D, 2D, 3D and 4D; own elaboration) Segment Square Cube Hypercube 89 7.1.2 Analogies Between The Main Rotations In The 2D, 3D and 4D space In chapter 3 were presented the possible main rotations in the 2D, 3D and 4D space:  2D space: Rotation around a point, 1 possible main rotation around the origin.  3D space: Rotation around an axis, 3 possible main rotations around main axis (X, Y, and Z).  4D space: Rotation around a plane, 6 possible main rotations around main planes (XY, YZ, XZ, XW, YW, and ZW) [Hollasch, 91]. 7.1.3 Analogies Between The Unravellings Of The Parallelotopes In the 2D, 3D and 4D space In chapter 4 were studied the properties of the unravellings for the parallelotopes in 2D, 3D and 4D space (see Table 7.2):  Square: A central segment surrounded by other two through a vertex adjacency with each one; a satellite segment adjacent to any other of the segments except the central. Completely immersed in a 1D space (a straight line).  Cube: A central face surrounded by other four through an edge adjacency with each one; a satellite face adjacent to any other of the faces except the central.
Completely immersed in a 2D space (a plane).  Hypercube: A central volume surrounded by other six through a face adjacency with each one; a satellite volume adjacent to any other of the vol...