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Mathematically Correct Breakfast -- Linked Bagel Halves

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Original source (www.georgehart.com)
Tags: math geometry bagels www.georgehart.com
Clipped on: 2018-01-26

Mathematically Correct Breakfast

How to Slice a Bagel into Two Linked Halves

George W. Hart
Image (Asset 1/9) alt= 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.
Image (Asset 2/9) alt= and the points.  You don't need to actually write on the bagel to cut it properly.
Image (Asset 3/9) alt= As it goes 360 degrees around the Z axis, it also goes 360 degrees around the bagel.
Image (Asset 4/9) alt= 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.
Image (Asset 5/9) alt= the hole of the other.   (So when you buy your bagels, pick ones with the biggest holes.)
Image (Asset 6/9) alt= not need to draw on the bagel.  Here the two parts are pulled slightly apart.
Image (Asset 7/9) alt= (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.
Image (Asset 8/9) alt= 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.

Addendum: I made a video showing how to do this.