# Projections of the Four-Cube

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Rotate a four-dimensional cube and project it into three-dimensional graphics. The rotation is the composite of rotations in the *w*-*x*, *w-y*, and *w-z* planes, in that order. Opposite cubes (3-cells) of the hypercube have similar colors. Play with the opacity to see how the hypercube is bounded by eight cubes.

Contributed by: Michael Rogers (Oxford College/Emory University) (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The rotation of the hypercube, whose vertices have coordinates each ±1, is the composite of rotations in the *w-x*, *w-y*, and *w-z* planes, in that order. Combined with *Mathematica*'s built-in rotation of 3D graphics, this gives all six degrees of freedom of four-dimensional rotations. The bounding cubes are drawn in the order from least *w* coordinate to greatest. If all the opacities are set to one, the display shows the cubes "visible" looking from out on the positive *w* axis. The projection is accomplished by dropping the *w* coordinate; therefore it is a parallel projection, although the projection onto the screen has the default perspective of *Mathematica *3D graphics. The positive *x*, *y*, *z*, *w* axes of the hypercube may be shown; these rotate with the hypercube, which helps illustrate how the rotations work.

Snapshot 1: a pair of opposite cubes

Snapshot 2: two pairs of opposite cubes

Snapshot 3: a variation on a classic projection of the hypercube

## Permanent Citation

"Projections of the Four-Cube"

http://demonstrations.wolfram.com/ProjectionsOfTheFourCube/

Wolfram Demonstrations Project

Published: March 7 2011