
If we take a vector from the origin to a point, and rotate that vector about the origin, then the point traces a circular path. If we do the same to another vector, about a different axis this time, and add the two together, then the resulting endpoint will spiral around a donut or "torus'', which has a twodimensional curved surface in 3d space.
By adding a third vector, rotating about yet a different axis, we find that the endpoint spirals around the 3d surface of a "hypertorus'' in fourdimensional space.
Note: The projection into three dimensional space was achieved here by selecting a point in 4space and "casting shadows'' from that point onto a 3dimensional slice of the space. The fourth dimension is then made more apparent as extra depth in the image (objects that are closer to the camera in 4space appear to bulge outward towards us, whereas objects further away in the 4th direction appear to shrink away).

