# Problem 23

• May 25th 2007, 10:56 AM
CaptainBlack
Problem 23
Consider a tunnel (not necessarily straight) of length 101 units through a

Prove that the sphere can be cut into two congruent pieces one of which
contains the tunnel in its interior.

RonL
• May 25th 2007, 12:38 PM
ecMathGeek
Quote:

Originally Posted by CaptainBlack
Consider a tunnel (not necessarily straight) of length 101 units through a

Prove that the sphere can be cut into two congruent pieces one of which
contains the tunnel in its interior.

RonL

How can these two pieces be congruent if one has a tunnel in it?
• May 25th 2007, 01:31 PM
CaptainBlack
Quote:

Originally Posted by ecMathGeek
How can these two pieces be congruent if one has a tunnel in it?

The two pieces of the sphere ignoring the hole are congruent

RonL
• May 28th 2007, 09:14 AM
CaptainBlack
Solution:

Let the entry and exit points be A and B. Then all posible paths of length
101 lie in an ellipsoid such that the sum of the distance from A and that from
B is less than 101 units. But the centre of the sphere cannot be inside this
ellipsoid as the sum of the distance from A to the centre and from B to the
centre is 102.

Now the plane through the centre normal to the plane containing the points
A, B and the centre does not intersect the ellipsoid, and so divides the sphere
(ignoring the hole) into two equal hemisphere.

RonL