Can the Poincare Conjecture Simplify 3D Objects for Mathematical Calculations?

[MarekS]

After reading the article on Poincare's conjecture in the Economist, I became curious about simplified 3-dimensional objects.

Excerpt:

To understand the Poincaré conjecture, start by thinking of any object existing in a three-dimensional world. Although it is usual to think of the object as three-dimensional, mathematicians consider only the surface of these objects—which are two-dimensional. All objects in a three-dimensional world can be simplified by smoothing out their shape to look like either a two-dimensional sphere (otherwise known as a circle) or a two-dimensional torus with however many holes necessary. To mathematicians, a chair is equivalent to an apple; a mug—at least, one with a handle—is like a doughnut.

Let's take a cube and simplify it into a circle. Could we then use equations ment for circles for the simplified shape, ie calculate the cube's surface area using S=pii*radius²?

How would the math look like for such calculations?

MarekS

 

[matt grime]

Whilst you could possibly work out how areas change under certain transformations, this is not what the poincare conjecture (or topology) is about. You're just doing some complicated sums (and undoubtedly integrals) to work out something quite trivial.

 

[CRGreathouse]

Topological onjects don't have area (hypervolume) in any well-defined sense. You can distort them in ways that change their volume without changing their topoogical properties.

 

[mathwonk]

but the idea of peremans proof, or hamiltons idea, was to impose a metric on the manifold, and show how to deforkm the metric until it became flat. manifolds with a metric do have "area" or volume, of course