Tesseract Rotation

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A tesseract carrying out all the possible independent 4-D rotations about its axes of symmetry. Since rotation interchanges two axes, the number of ways to rotate in 4-D is (4 choose 2), or 6. Each one holds two perpendicular axes constant. These axes form a stationary plane, just as the single axis of a cube rotating in 3-D forms a stationary line. The plane is rendered in purple. The purple tees are at the centers of cubes about which the rotation is occurring, and the purple corners are at the centers of faces shared by two such cubes.

The four dimensions are rendered using color as well as conventional perspective. Since color forms a three-space, the fourth dimension is projected into it at an angle. The four dimensions are rendered as (Red, Green, Blue, White).

Of course, it's possible for the tesseract rotate lots of different skewed ways that aren't pictured individually here - but any such motion can be expressed as a linear combination of these 6 rotations.

You might also enjoy the companion video (), which shows construction of the tesseract from a point by multiple extrusions.