# Pérez Águila, Ricardo - Appendix A. Unraveling The 4D Simple- 4D Orthogonal Polytopes

Pérez Águila, Ricardo
- Appendix A. Unraveling The 4D
Simple-
4D Orthogonal Polytopes
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Pérez Águila, Ricardo

- Appendix A. Unraveling The 4D

Simple-

4D Orthogonal Polytopes

-- Licenciatura en Ingeniería

en Sistemas Computacionales. - Departamento de Ingeniería en

Sistemas Computacionales. - Escuela de Ingeniería, - Universidad de las Américas

Puebla.

## Teaser

Appendix A
Unraveling The 4D Simplex
A.1 Introduction
In chapter 4 was presented the methodology used by Aguilera and Pérez in
[Aguilera, 01] to unravel the 4D hypercube which is based in the unraveling of the cube to
obtain the analogous method.

We have not found any references that mention any methods
or results (as the tesseract is the result of the hypercube's unraveling) about the unraveling
process for other 4D regular polytopes as the 4D simplex which corresponds to the 4D
equivalent of the tetrahedron (Figure A.1).

As the hypercube's unraveling process, we will
visualize a projection onto our 3D space of the volumes (tetrahedrons) on the 4D simplex's
boundary through its unraveling and raveling processes.
FIGURE A.1
The 4D simplex [Aguilera, 02c].
A.2 The 3D Simplex (Tetrahedron) Unraveling Methodology
Although the tetrahedron's unraveling process is trivial, we will consider here some
key points that will be extended later in the 4D simplex unraveling:
102
1
Identify a face that is -naturally embedded- into the plane where all the tetrahedron's
faces will be positioned.

This face will be called -central face-.

Because the central face
is -naturally embedded- in the selected plane, it will not require any transformation.
2
Each of the remaining faces shares an edge with the central face.

These faces will be
called -adjacent faces-.
3
The adjacent faces will rotate around those edges that share with the central face.
4
When the central and adjacent faces are identified, it must be determined the rotating
angles and their directions.

The rotating angle is the supplement of the tetrahedron's
dihedral angle.
TABLE A.1
Unraveling the 3D simplex [Aguilera, 02c].
1
2
4
7
5
3
6
8
Table A.1 presents some snapshots from the 3D simplex's unraveling sequence.

In
snapshots 1 to 4, the applied rotations are 0, 10.94°, 27.35° and 43.76° (the rotation's
sign depends of the adjacent face).

In snapshots 5 and 6, the applied rotations are 54.7°...