# Pérez Águila, Ricardo - Capítulo 5. 4D Orthogonal Polytope- 4D Orthogonal Polytopes

Pérez Águila, Ricardo
- Capítulo 5. 4D
Orthogonal Polytope-
4D Orthogonal Polytopes
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Pérez Águila, 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...