I didn't know about these --- though perhaps I should've --- so maybe they're new to a few more people out there. http://en.wikipedia.org/wiki/Villarceau_circles http://mathworld.wolfram.com/VillarceauCircles.html are reasonably informative pages on the subject [the Mathworld page on "torus" currently has captions transposed for "spindle" and "horn" tori]. Imagine a torus in canonical position, generated by rotating about the z-axis a circle in the xz-plane with centre on the x-axis. Now tilt the xy-plane about (say) the y-axis, until it is tangential to the torus at their (two) furthest points of contact. Their intersection comprises two intersecting loops, each of which spirals around the torus tube once as it travels around the circumference; and rotating the plane about the z-axis generates two complete systems of such loops, each linked to all the others in its system. Well now, I came to study these loops as stereographic projections of Clifford parallels in elliptic space (see also Hopf fibration) --- see e.g. Berger, Cole & Levy sect 18.9 p 305 --- and spent some time wrestling with a conundrum: how can these spirals be correct, when (for various technical reasons) it's necessary that such projections be plane circles? And in the course of time it dawned on me that they're both spirals and circles! [Immediately following which, of course, I also discovered that everybody else knew this long ago.] Incidentally, once you know they're circles, its easy to see that their radius must be the same as that described by the generating circle around the origin. The web pages give rather sordidly computational proofs of these facts, along with an incomplete sketch of a more elegant argument (ascribed to Hirsch) using algebraic geometry; my own argument, though but one line in length, relies similarly on a theorem from geometric algebra. Can anybody supply an simple, elementary, synthetic proof that the intersection is (are?) circles? Also, Berger, Cole & Levy mentions in passing that there are Dupin cyclides (Moebius inversions of tori) with not only 4 but 6 systems of circles. I can't find anything out about these chimaeras --- indeed, I don't believe in them, but then, I didn't believe in Villarceau circles either! Fred Lunnon