Cool Stuff: Fractals
Alt.Fractals paperback - out January 2011

  • Cutting up doughnuts

    A cool thing about doughnuts that I didn't know until a few months ago was that you can always cut a doughnut so that its cross-section is two interlocking circles. 

    Vilarceau circles, in a sliced bagel


    First, what's the magical cutting angle that we have to use to get this cross-section, if we know the proportions of the doughnut
    We-ell, after a quick visit to the local bakery and the procurement of some ring doughnuts, and the use of a kitchen knife, it turns out that the cross-section cuts across the central throat at an angle, so that it just touches at two points which means that it has to be a common tangent for two opposing parts of the throat surface. 

    And that's enough for us to be able to work out the angle. After a few scribblings to work backwards from a couple of simple cases (angle 45 degrees, 60 degrees), it seems that if we know how fat the torus is (its minor radius, "r"), and we know how wide it is (measured from the centre to the centre of the ring limb, the major axis, "R"), the equation for the angle "A" for our cut turns out to be pretty simple. It's just :

    SIN A = r / R

    So as we transition between the two extremal shapes for the torus (from maximum skinny, r / R = 0, to maximum fatty, r / R = 1, r = R), the angle of the cut rotates nicely from zero degrees to 90 degrees.

    "Skinny torus" limit

    If the thickness of the torus limb shrinks towards zero, we know that the two intersecting shapes MUST be perfect circles of radius R, because they don't have enough room to be anything else. At the limit of r / R = 0, when the surface shrinks to a one-dimensional line, curved around into a circle, we know that the overlap region for the two interlocking shapes has to be 100% ... because there's no room on the surface for the circles NOT to be totally overlapping. 

    "Fat torus" limit

    As "r" approaches the size of the major radius, R, the torus throat closes off. Our cross-section for this extremal case is simply a vertical slice through the torus centre, like so:

    We have two circles touching, of radius r = R ... so the two circles are each exactly the same size as they were when they in the skinny torus limit.
    But the overlap is now zero rather than 100% 


    So we can define the proportions of a torus by the special angle A (between 0 and 90 degrees) needed to cut it to get the double-circle. We also know that for  0<A<90, the two circles each slide around the shape so that they cross the outside equator of the torus on one side, and cross the opposing side of the inner equator on the other. These two points are always offset from the torus' inner circular minor axis, by the same distance (r), so we know that the width of the two individual intersecting shapes has to be constant.  

    So if we keep R constant, and sweep the torus proportions over the full range, from r = 0 to r = R, and as the "special" cross-section angle tilts in sympathy from 0 to 90 degrees, the cross section is just a fixed-size pair of circles moving apart, from total overlap of 100%, to an overlap of zero. 

    3D Plots

    Here are a few plots of half-tori, all with different thicknesses and viewing angles, all sliced  perpendicularly to the observer's view, to give the two-circle "linked rings" cross-section:

    Unlike the case of the chopped cube, I can't think of any wonderfully exciting practical applications for this yet, but, hey, maybe something'll hit me six months from now. Perhaps the result has special applications for building multiple intersecting toroidal particle accelerator arrays, or perhaps there's an application for using a Villarceau cage as an unconventional fusor frame. Or maybe not. I think I have too many other things that need doing to have the time to check this one out properly. 

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