Conference rusure::math

Well, you can't do it by dividing the circumference into 7 equal parts
and cutting to the center because that would mean you could construct
a regular heptagon.  Galois proved that the only regular n-gons that
can be constructed with straight-edge and compasses are of the form

			     k
			n = 2  X F

where F is a Fermat prime. (A Fermat prime is a prime of the form

			  m
			 2
			2   + 1 ,

e.g. 3, 5, 17, 257, etc.)

I assume that when you want the pizza divided up into 7 equal parts that
you mean 7 parts that have the same area (not 7 congruent parts).

Anyhow, here's a solution:

Let us assume that the radius of the pizza is R.  We will cut a circular
piece of radius r out of the center that has an area equal to 1/7 of the
area of the pizza.  The remaining annulus (with area 6/7 of the pizza)
can then easily be divided into 6 equal pieces by inscribing a regular
hexagon in the circle and joining their vertices to the center of the
circle.

To find r, we note that pi r^2 = 1/7 pi R^2, so that R/r=sqrt(7).
But sqrt(7) is quadratic and can therefore be constructed with straight-
edge and compasses.  So we are done.

(An easy way to construct sqrt(7) is to draw a circle with diameter 8.
 Let P be a point on a diameter AB that divides the diameter into two pieces
 with lengths 1 and 7.  Erect a perpendicular at P until it hits the circle
 again at Q.  Then PQ=sqrt(7) because ABQ is a right triangle with altitude
 PQ and an altitude is the mean proportion between the segments it forms
 on the hypotenuse.)

Right.

The initial source of this problem was someone who wanted a clearer proof
that you could not 7-sect an angle with straight edge and compass (apparently
he had called a pizza place and asked for a pizza cut in 7 pieces and was
given a proof of why it was impossible over the phone!).  I saw a loop
hole in the way the question was asked and came up with the same solution
you did.

A refignment (suggested by the person at Bell Labs who submitted the initial
problem) is to divide the pizza in to 7 rings rather than one circle and
six truncated wedges.  If you do this than the solution can be generalized
to divide a pizza into any number of pieces.  However, this only works if
you party consists of exactly one person who likes eating the crust.
						B.J.

A correction on 17.1:

A regular n-gon is constructible with straight-edge and compasses
if and only if n is of the form

		 k
		2  F  F  ... F
		    1  2      r

where each F  (if any) denotes a Fermat prime.
            i

My previous statement was incorrect.  For example, a regular
15-gon is constructible since 15=3 X 5, the product of two Fermat primes.

    Hi,
    	Didn't Gauss develop a constuction that would divide the
    circumference of a circle into 7 equal segments (or was it 17)?
    
    steve M	(beware the fractal dragon)