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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 end-point will spiral around a donut or "torus'', which has a two-dimensional curved surface in 3-d space.
By adding a third vector, rotating about yet a different axis, we find that the end-point spirals around the 3-d surface of a "hypertorus'' in four-dimensional space.
Note: The projection into three dimensional space was achieved here by selecting a point in 4-space and "casting shadows'' from that point onto a 3-dimensional 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 4-space appear to bulge outward towards us, whereas objects further away in the 4th direction appear to shrink away).
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