# Slowly Rotating Hypercube

The hypercube, also known as a tesseract, is the four-dimensional analog to the cube. I made this projection of a hypercube with Stella 4d, which is available at http://www.software3d.com/Stella.php.

## Four Views of a Tesseract, Rotating in Hyperspace

### Image

These tesseract views are all of the perspective projection-type, with the first one, above, being done in cell-first fashion.

The next one is projected edge-first.

The third one is projected vertex-first.

Lastly, face-first:

Although all of these are rotating in the same direction in hyperspace, the different projection-choices make the second and fourth images appear to be rotating in different directions. Why? I’m still trying to figure that out!

These animations were created with Stella 4d, software available at www.software3d.com/Stella.php.

## Rotating Compound of the Tesseract and Its Dual

### Image

Blue figure: a projection of the tesseract, or hypercube; also known as the 8-cell or octachoron — a four-dimensional figure composed of eight cubic cells in a regular arrangement.

Red figure: its dual, the 16-cell or hexadecachoron, which is composed of sixteen tetrahedral cells.

To buy (or just try) the software used to make this image, Stella 4d, please visit http://www.software3d.com/Stella.php.

# The Two Simplest Polychora

The most familiar polychoron, to those who have heard of any of them, is the hypercube, or tesseract. It is analogous to the cube, but in four dimensions. All polychora are four-dimensional. With numbers of spatial dimensions above four, only the term “polytope” is used. Polyhedra are 3-polytopes, and polychora are 4-polytopes.

This is a three-dimensional projection of a tesseract, as it rotates in hyperspace, casting a “shadow” into our space:

In three dimensions, a cube is not the simplest polyhedron. A tetrahedron (a regular triangle-based pyramid) is simpler.

The simplest polychoron is composed of five tetrahedral cells, and is analogous to the tetrahedron, but in hyperspace. Here is a rotating “hypertetrahedron.”

There are even more names for these two polychora, based on the number of cells (cubes or tetrahedra). The tesseract/hypercube is composed of eight cubes, so it is called the 8-cell, as well as the octachoron. The preferred names for the hypertetrahedron are the 5-cell and the pentachoron, as it is composed of five (tetrahedral) cells.

Just as there are other Platonic solids not mentioned here, there are other regular polychora as well. The others will be subjects of upcoming posts, and one has already appeared here once (the 120-cell, or hyperdodecahedron), just three posts back.

Software note:  these .gifs were made using Stella 4d, which may be purchased, and/or tried for free (on a trial basis), at http://www.software3d.com/Stella.php.