# Pérez &Aacute;guila, Ricardo - Capítulo 5. 4D Orthogonal Polytope- 4D Orthogonal Polytopes

Pérez &Aacute;guila, Ricardo
- Capítulo 5. 4D
Orthogonal Polytope-
4D Orthogonal Polytopes
-- Licenciatura en Ingeniería
en Sistemas Computacionales. - Departamento de Ingeniería en
Sistemas Computacionales. - Escuela de Ingeniería, - Universidad de las Américas
Puebla.

## Teaser

Chapter 5 4D Orthogonal Polytopes 5.1 Definition [Coxeter, 63] defines an Euclidean polytope n as a finite region of n-dimensional space enclosed by a finite number of (n-1) dimensional hyperplanes.
The finiteness of the region implies that the number Nn-1 of bounding hyperplanes satisfies the inequality Nn-1 n. The part of the polytope that lies on one of these hyperplanes is called a cell.
Each cell of a n is an (n-1)-dimensional polytope, n-1.
The cells of a n-1 are n-2's, and so on; we thus obtain a descending sequence of elements n-3, n-4, ...
, 1 (an edge), 0 (a vertex). We know that a cells are 1 2 .A 2 3 (a 3D Euclidean polytope) is a polyhedron.
The polyhedron’s (a 2D Euclidean polytope) is a polygon.
The polygon’s cells are (a 1D Euclidean polytope) is a segment.
Finally, the segment’s cells are vertices.
The cells of a 4 (a 4D Euclidean polytope) are 3 0 1 .A , a set of (polyhedra, also called volumes in the context of 4). [Aguilera,98] defines Orthogonal Polyhedra (3D-OP) as polyhedra with all their edges and faces oriented in three orthogonal directions.
Orthogonal Pseudo-Polyhedra (3DOPP) will refer to regular and orthogonal polyhedra with non-manifold boundary. Similarly, 4D Orthogonal Polytopes (4D-OP) are defined as 4D polytopes with all their edges, faces and volumes oriented in four orthogonal directions and 4D Orthogonal 49 Pseudo-Polytopes (4D-OPP) will refer to 4D regular and orthogonal polytopes with nonmanifold boundary.
Because the 4D-OPP's definition is an extension from the 3D-OPP's, is easy to generalize the concept to define n-dimensional Orthogonal Polytopes (nD-OP) as n-dimensional polytopes with all their n-1, n-2,..., 1 oriented in n orthogonal directions. Finally, n-dimensional Orthogonal Pseudo-Polytopes (nD-OPP) are defined as ndimensional regular and orthogonal polytopes with non-manifold boundary [Aguilera, 02]. 5.2 Adjacency Analysis For 2D, 3D And 4D-OPP's 5.2.1 Adjacenc...