Trisecting a Pizza

Date: 05/13/2002 at 15:58:32
From: John A. Chunko
Subject: dividing a pizza into 3 equal slices??

Without the use of a ruler, protractor or other measuring device, is 
there a simple way to divide a pizza into 3 equal sized slices?  Use
of a straightedge is okay.  If not possible with only a straightedge,
then what would you need? A compass?  Something else?

Date: 05/13/2002 at 17:11:14
From: Doctor Peterson
Subject: Re: dividing a pizza into 3 equal slices??

Hi, John.

I can't think of a way to do it with just a straightedge, and a 
compass would get pretty messy on a pizza, though it can be done with 
compass alone, at least if you know the center. Of course, since a 
pizza is not an exact circle, you can't really be mathematically 
precise anyway.

I would suggest using a piece of paper. Cut it to the shape of the 
pizza, and you should have no trouble folding it to get the right 
angle with reasonable precision. Is that allowed? Or, there is a nice 
way to fold a rectangular piece of paper to get an exact 60 degree 
angle, and you could use that to mark out the necessary angles.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/14/2002 at 10:35:08
From: John A. Chunko
Subject: dividing a pizza into 3 equal slices??

Thanks very much for your time and solution.  

How do you fold a rectangular piece of paper to produce a 60 degree angle?  

Date: 05/14/2002 at 12:12:28
From: Doctor Peterson
Subject: Re: dividing a pizza into 3 equal slices??

Hi, John.

Here is one way (which I learned from an origami book):

  http://mathforum.org/dr.math/problems/oliver4.26.01.html 

I'd be interested to know the origin and outcome of your question; it 
sounds like the paper solution is acceptable. I thought about the 
question more last night, and had these thoughts:

There are two parts to the problem: (a) finding the center of the 
circle, and (b) finding the right angle. With compass and 
straightedge, both are easy. With compass only, it has been proved 
that you can do anything you can do with both (Mascheroni 
construction), so it can be done somehow, probably with great 
difficulty. It is also possible to do any such construction with only 
a straightedge AND a fixed circle with its center! I suspect that 
means you couldn't find the center of your circle with only the 
straightedge, but given the center there is probably a way to find 
the angle. Finally, it has been shown that with a few defined origami 
moves, you can do any compass-and-straightedge construction, and 
more, including trisecting an angle!

With a paper 60 degree angle, there's a neat way to divide the 
circumference of the pizza evenly without having to know the center 
first. Finding the center may actually be harder.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/14/2002 at 16:18:38
From: John A. Chunko
Subject: dividing a pizza into 3 equal slices??

Thanks... you are terrific!  At last, no more fights over the pizza 
slices!  :-)

Date: 05/14/2002 at 16:28:08
From: John A. Chunko
Subject: dividing a pizza into 3 equal slices??

BTW... your paper folding solution works perfect... the by-product of 
creating the 60° angle (on the sheet of paper) is a 120° angle formed 
adjacent to it (i.e., 120° template produced is exactly the correct 
angle needed for slicing up the pizza into equal thirds.)

One thing I couldn't figure out, however... you mentioned, "With a 
paper 60 degree angle, there's a neat way to divide the circumference 
of the pizza evenly without having to know the center first."  How 
would that work?... what is the "neat way?"

In any case... many thanks for the time and thought you put into the 
problem!

Regards... JC

Date: 05/14/2002 at 16:44:09
From: Doctor Schwa
Subject: Re: dividing a pizza into 3 equal slices??

I have a guess what Dr. Peterson had in mind...

If you put the vertex of the 60 degree angle on the
edge of the pizza, then the arc that it cuts off on the other
side will be 120 degrees of pizza!

The inscribed angle is half the arc.  Have you heard of
that theorem?

Of course, you're then left with the problem of cutting
the pizza into three equal pieces once you have three
points, 120 degrees apart, marked on the edge of the
pizza.  If you don't know where the center is, you may
still be in trouble.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/14/2002 at 16:47:41
From: Doctor Peterson
Subject: Re: dividing a pizza into 3 equal slices??

Hi, John.

One of my favorite theorems is that an angle inscribed in a circle 
cuts an arc whose central angle is twice the inscribed angle's 
measure.

So just place the 60 degree angle's vertex on the circumference of 
the circle, without worrying about what direction it points, and the 
two legs will divide the circle in a perfect 120 degree arc:

                      *********
                ******        /******
             ***             //      ***
           **               / /         **
         **                / /            **
        *                 /  /              *
       *                 /  /                *
      *                 /   /                 *
     *                 /   /                   *
     *                /    /                   *
    *                /    / 120                 *
    *               /     +                     *
    *              /       \                    *
     *            /          \                 *
     *           /             \               *
      *         /                \            *
       *       /                   \         *
        *     /                      \      *
         **  / 60                      \  **
           *+---------------------------**
             ***                     ***
                ******         ******
                      *********

Repeat the process, and you'll have marked the three points on the 
circumference where you have to cut, even though you don't know 
exactly where the cuts should meet!

There are probably lots of ways to find the center now; one would be 
to fold the 60 degree angle in half and use the 30 degree angle 
(measuring inward from the chord between two of the three points) to 
draw a radius from each of them. These will meet at the center.

You probably never knew that origami could be so useful!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/