Cutting a SphereDate: 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/ |
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