The Conjecture
Let us start with the statement of the Poincare conjecture. The terms will
be defined later....
If a three dimensional surface is simply connected, compact,
and without boundary, then it is topologically equivalent to the three dimensional
sphere.
Although it sounds simple, this has been unproven for centuries. The result
is that the Clay Institute has posted a $1,000,000 US reward for any valid
solution.
Three dimensional?
Hopefully the reader is familiar with the term 'dimension'. A line
is 1-dimensional, because you can define any point by a single number (ie
length from one specific point). A sheet of paper is two dimensional,
since you need to define a point by two numbers (ie horizontal and vertical
distance from one corner). So a three dimensional surface requires that each
point be defined by three numbers. Examples would be a solid cube, a solid
sphere, or space itself. (It should be noted that there are surfaces for
which dimension cannot be defined, but they do not affect this conjecture)
We should also point out that the Earth is not a three dimensional 'sphere'
in the mathematical sense. A 3D sphere is by definition the set of all points,
which can be defined by three numbers, and which are equal distance from
some other point. All points on the Earths surface are approximately the
same distance from the core, but the surface is two dimensional. If we include
the third dimension, then clearly a point near the core is closer to the
core than we are.