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

Pérez &Aacute;guila, Ricardo
- Capítulo 4. Unraveling
the 4D Hypercub-
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 4 Unraveling the 4D Hypercube 4.1 Introduction In section 1.4 were discussed the methods for visualizing 4D polytopes.
One of them is the visualization through the unravellings.
We remember that a cube can be unraveled as a 2D cross.
The six faces on the cube's boundary will compose the 2D cross (Figure 4.1).
The set of unraveled faces is called the unravellings of the cube. FIGURE 4.1 Unraveling the cube [Aguilera, 02b]. In analogous way, a hypercube also can be unraveled as a 3D cross.
The 3D cross is composed by the eight cubes that forms the hypercube's boundary (the hypercube’s properties were discussed in chapter 2).
This 3D cross was named tesseract by C.
H. Hinton in the XIX century (Figure 4.2) FIGURE 4.2 The unraveled hypercube: the tesseract [Aguilera, 02b]. 36 We also discussed that a flatlander will visualize the 2D cross, but he will not be able to assembly it back as a cube (even if the specific instructions are provided).
This fact is true because of the needed face-rotations in the third dimension around an axis which are physically impossible in the 2D space.
However, it is possible for the flatlander to visualize the raveling process through the projection of the faces and their movements onto the 2D space where he lives. Analogously, we can visualize the tesseract but we won't be able to assembly it back as a hypercube.
We know this because of the needed volume-rotations in the fourth dimension around a plane which are physically impossible in our 3D space. [Kaku, 94] and [Banchoff, 96] describe with detail the representation model for the hypercube through their unravellings.
They also mention the physical incapacity of a 3D being to ravel the hypercube back, because the required transformations are not possible in our 3D space (Figure 4.3). Y Z ? W Y X Z X -X -Z -Y FIGURE 4.3 The hypercube's unraveling process [Aguilera, 02c]. 37 [Kaku, 94] and [Banchoff, 96] also describe that if we witness the raveling process, s...