Cutting a Sphere

Date: 02/18/99 at 11:19:53
From: Nick Prodahl
Subject: Cutting a Sphere

I want to know how to cut a sphere into an odd number of pieces. I have 
tried it, but it is very difficult to divide it into an odd number of 
pieces with each piece being even with the others. Can you help me?              

Date: 02/19/99 at 08:51:06
From: Doctor Rob
Subject: Re: Cutting a Sphere

How about three pieces? Consider a plane passing through the center O 
of the sphere, and intersecting it in a circle. Construct a regular
hexagon PAQBRC inscribed in that circle using a compass set to the
radius of the sphere to mark six equidistant points P, A, Q, B, R, and
C along the circumference of the circle. Connect O to P, Q, and R,
three of these points, which are not adjacent along the circle. These
line segments OP, OQ, and OR will form angles of 120 degrees with each
other at O. Now erect a line XOY perpendicular to the first plane and
intersecting it at O. The line XOY and the points P, Q, and R will
determine three planes. The half of each of these planes on the same
side of XOY as P, Q, or R will be your three cuts. Each cut will
intersect the sphere in a semicircle, and the three cuts of the sphere
will divide it into congruent thirds. Here is a top view:

                      Q  _..-----.._ A/
                      .+'           `+.
                    ,'  \           /  `.
                  ,'     \               `.
                 /        \                \
                /          \                \
               .            \                .
              B|             \               |
              -+-             +--------------+ P
               |             / O,X,Y         |
               .            /                .
                \          /                /
                 \        /                /
                  `.     /               ,'
                    `.  /           \  .'
                      `+._         _.+'
                      R   ''-----''  C\

Here is a side view:

                              X
                         _..--+--.._
                      .-' ,'  |     `-.
                    ,'   /    |        `.
                  ,'    /     |          `.
                 /     ,      |            \
                /     ,       |             \
               .      ,       |              .
               |      |       |              |
             B -      +-------+------+-------+ P
               |   R,Q|       |O    C,A      |
               .      `       |              .
                \     `       |             /
                 \     `      |            /
                  `.    \     |          ,'
                    `.   \    |        .'
                      `-._`.  |    _.-'
                          ''--+--''
                              Y

Write back if this does not help.

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