Poincare was certainly one of the premier mathematicians at the beginning of the 20th century. This famous idea regarding spherical topology (which, when first published was considered merely a conjecture, but which has since graduated into the realm of a mathematical theorem) arose from Poincare's mind in 1904, and has become a key element in multi-dimensional geometry.
What is the Poincare Conjecture
Simply put, the Poincare Conjecture is meant to mathematically define the concept of a "sphere." While may seem both simple and trivial in a non-mathematical sense ("A sphere is merely a circle amplified into three dimensions," one might say), there is great importance in a mathematical sense in discovering a basic definition of what, exactly, separates a sphere from a non-sphere.
In terms of a simple, three-dimensional sphere, the conjecture arrived at by Poincare is rather simple: Any continuous line drawn around the outside of a sphere (a line of points at a constant distance from the shape's "origin"), can be "shifted" in such a way that the "loop" becomes smaller and smaller until finally coming together at a single point.
The Poincare Conjecture In Simpler Terms
What does this rather abstract explanation actually mean? As an example, take any sphere, be it a baseball, golf ball, bowling ball, etc. A rubber band wrapped around that ball can be moved along the edges of the ball without breaking or passing through itself until it is brought together to a single point (though a rubber band, being fully dimensional, cannot actually be contracted to the infinitesimally small single point).
This same "trick" cannot be accomplished if the object is a non-sphere. A torus (donut), for instance, will not allow this, as the rubber band would have to be broken in order for it to come together; likewise, a double-torus (figure eight) will make matters even more complicated.
What Does the Conjecture Actually Mean?
Essentially, what Poincare established in his conjecture is one of the basic truths of mathematical topology: A sphere (or a torus or a double torus or any other fundamental topological shape) is merely what it is, and cannot ever become anything else.
Because the Poincare Conjecture holds true no matter how big, small, twisted, curved, flat, or bent a sphere is, it will always be considered topologically identical to every other sphere. The same mathematical truths hold, the same defining equations can be used, and all is right with the world.
By this same token, a sphere can never be turned into a torus or any other topological entity. In this, the Poincare conjecture actually illustrates one of the essential features of topology, without which this entire fascinating area of mathematics would be practically useless.
References:
Ornes, Stephen. "What is the Poincare Conjecture". Seed Magazine, August 25, 2006.
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