## Cutting a Bagel in Half

My wife’s niece send us the address of a website  that  demonstrates how to cut a bagel in half with the halves linked.  The video at the end of the webpage shows how it works.  I couldn’t quite visualize how the halves were linked so I decided to work it out using a rectangular disk with the edges identified.  Here is my “proof” that the halves are linked.

Consider a bagel as just a solid torus.  The torus can be constructed out of a “flexible” rectangular disk with the upper edge “glued” to the lower edge with no twists and with the left edge glued to the right edge with no twists.  Here glue upper edge I-I to lower edge I-I and right edge I-O-I to left edge I-O-I.

Rectangular Disk with Opposite Edges Identified

With this scheme  let the edge I-I be the innermost horizontal circle of the torus (lying flat) and O-O the outermost circle.  To cut the bagel in half in the classical way, slice horizontally through the two circles.  Here is a section view.

Torus Section with Cutting Line Indicated

On the rectangular disk, the classical slicing looks like this.

Rectangular Disk - Classically Sliced

Now to make the linked halves as demonstrated on George W. Hart’s website, slice the torus as shown on this rectangular disk with the edges identified as above.

Rectangular Disk - Linked Halves Slicing

Consider the center lines of the two slices.  One is red and the other is blue.  Think of the torus as a translucent cylinder with the ends identified and look from above.

View of Center Lines from Above the Torus

The center lines are two linked loops, thus proving that we now have a sliced bagel that is hard to eat.

Center Lines of the Two Halves Linked