I do have a small objection with that last post. A citizen of Flatland wouldn't really see circles as the sphere crosses his world. Circles are figures we in the 3-dimensional world can see, because we see them from above (Draw a circle on a piece of paper and look at it - it's a two-dimensional figure, drawn on a plane; but you're looking at from above, a third dimension).

But now imagine you lived in the piece of paper - try looking at the circle now. All you would see is a line. So our Flatlandian would see a point that grows into a large line, then decreases in size until it's again a point and disappears. Unless he happens to be within the general area where the sphere goes through his world. Then, depending on where he is exactly, he would see himself surrounded by a line - the distance to the line would grow, then shrink and disappear.

Ain't math fun?

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Now, after reading more about the Poincaré Conjecture, I've been trying to understand the notion of 3-spheres. A 3-sphere is basically a sphere that lives in four dimensions. Following the same logic, a 2-sphere would be our regular plain old sphere or soccer ball, and a 1-sphere would be a circle.

Generalizing, an n-Sphere consists of all points an equal distance away from a single point in (n+1)-dimensional space.

Imagine you live in Flatland, a flat, two-dimensional world. The only dimensions you can grasp are North-South and East-West. The concept of Up-Down is incomprehensible to you. If a 3-dimensional sphere (a ball) went through your world, what would you see? Remember, your world is flat, so all you would see are slices of the sphere as it goes through your flat world. First, you would see a point, then ever larger circles - once the sphere is half-way through you would begin to see smaller circles, until finally a small point would be your last image of the sphere.

Likewise, if a four-dimensional sphere (a 3-sphere) were to intersect our three dimensional world, we would simply see a point, followed by ever larger spheres, growing in time until it's halfway through our world - then, the spheres would shrink and disappear in a small point.

Once you're familiar with these explanations, take a look at these wonderful animated images.

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Russian mathematician Dr. Grigori Perelman is reporting that he has proven Poincaré famous conjecture.

According to this NYTimes article, "It will be months before the proof can be thoroughly checked. But if true, it will verify a statement about three-dimensional objects that has haunted mathematicians for nearly a century, and its consequences will reverberate through geometry and physics...

Formulated by the French mathematician Henri Poincaré in 1904, the Poincaré Conjecture is a central question in topology, the study of the geometrical properties of objects that do not change when the object is stretched, twisted or shrunk."

Additional Poincaré links:

• Jules Henri Poincaré biography
• Wolfram Research
• For a very simple and clear explanation of what the conjecture is all about, check out the Clay Mathematics Institute.

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