It is not hard to cut a bagel into two
equal halves which are linked like two links of a
chain.
To start, you must visualize four key points. Center the bagel at
the origin, circling the Z axis.
A is the highest point above the +X axis. B is where the +Y axis
enters the bagel.
C is the lowest point below the -X axis. D is where the -Y axis
exits the bagel.
These sharpie markings on the bagel are just to help visualize the
geometry
and
the points. You
don't
need to
actually write on the bagel to cut it properly.
The line ABCDA, which goes smoothly through all four key points, is the
cut
line.
As it goes 360 degrees around the Z axis, it also goes 360 degrees
around the bagel.
The red line is like the black line but is rotated 180 degrees (around
Z or through the hole).
An ideal knife could enter on the black line and come out exactly
opposite, on the red line.
But in practice, it is easier to cut in halfway on both the black line
and the red line.
The cutting surface is a two-twist Mobius strip; it has two sides, one
for each half.
After being cut, the two halves can be moved but are still linked
together, each passing through
the hole of the other. (So when you buy your bagels, pick
ones with the biggest holes.)
If you visualize the key points and a smooth curve connecting them, you
do
not need to draw on the bagel. Here the two parts are pulled
slightly apart.
If your cut is neat, the two halves are congruent. They are of the same
handedness.
(You can make both be the opposite handedness if you follow these
instructions in a mirror.)
You can toast them in a toaster oven while linked together, but
move them around every
minute or so, otherwise some parts will cook much more than others, as
shown in this half.
It is much more fun to put cream cheese on these bagels than on an
ordinary bagel. In additional to
the intellectual stimulation, you get more cream cheese,
because there is slightly more surface area.
Topology problem:
Modify the cut so
the cutting surface is a one-twist Mobius strip.
(You can still get cream cheese into
the cut, but it doesn't separate into two parts.)
Calculus problem:
What is the ratio
of the surface area of this linked cut
to the surface area of the usual
planar bagel slice?
For future research:
How to
make
Mobius lox...
Note: I have had my students do this
activity in my
Computers and
Sculpture class. It is very successful if the students
work in pairs, with two bagels per team. For the first bagel, I
have them draw the indicated lines with a "sharpie". Then they
can do the second bagel without the lines. (We omit the schmear of
cream cheese.) After doing this, one can better appreciate the stone
carving
of
Keizo Ushio,
who makes analogous cuts in granite to produce monumental sculpture.
