extra dimension. Poincaré conjectured that certain closed 3-manifolds which had
all the algebraic hallmarks of being three-dimensional spheres actually had to be
spheres. It was as if you walked around a giant ball and all the clues you received
indicated it was a sphere but because you could not see the big picture you
wondered if it really was a sphere.
No one could prove the Poincaré conjecture for 3-manifolds. Was it true or
was it false? It had been proven for all other dimensions but the 3-manifold case
was obstinate. There were many false proofs, until in 2002 when it was
recognized that Grigori Perelman of the Steklov Institute in St Petersburg had
finally proved it. Like the solution to other great problems in mathematics, the
solution techniques for the Poincaré conjecture lay outside its immediate area, in
a technique related to heat diffusion.

the condensed idea

From donuts to coffee cups