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 crosssection is two interlocking
circles.
Angles
First, what's the magical cutting angle
that we have to use to get this crosssection, if we know the
proportions of the doughnut
Weell, 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
crosssection 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 onedimensional 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 crosssection 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%
Intermediates
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 doublecircle. 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" crosssection angle tilts in sympathy from 0 to 90
degrees, the cross section is just a fixedsize pair of circles
moving apart, from total overlap of 100%, to an overlap of
zero.
3D Plots
Here are a few plots of halftori, all with different thicknesses and
viewing angles, all sliced perpendicularly to the observer's
view, to give the twocircle "linked rings" crosssection:
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.
External Links
copyright © Eric
Baird 2009
