Saturday, July 11, 2009

The Poincaré Conjecture

I've been reading this fascinating book "The Poincaré Conjecture - in search of the shape of the Universe" by Donal O'Shea.

In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere among three-dimensional manifolds.

In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold.

Around 300 BC, the Greek mathematician Euclid undertook a study of relationships among distances and angles, first in a plane (an idealized flat surface) and then in space. An example of such a relationship is that the sum of the angles in a triangle is always 180 degrees. Today these relationships are known as two- and three-dimensional Euclidean geometry. This is the type of geometry we learn in the high school.

The Poincaré conjecture concerns a space that locally looks like ordinary three dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is just a three-dimensional sphere.

A sphere can be represented by a collection of two dimensional maps; therefore a sphere is a two dimensional manifold.

An important scientific application of 3-manifolds is in physical cosmology, as models for the Shape of the Universe – the surface of the earth is locally approximately flat – it is roughly a 2-manifold, and globally the surface of the earth is a sphere. The universe, likewise, looks locally approximately like 3-dimensional Euclidean space, so the universe may be modeled as a 3-manifold, and one may ask which 3-manifold it is. More precisely, the universe is better described as 4-dimensional space-time, so this description is of a 3-dimensional spatial section.

Our genetic code allows us to see in 3 spatial dimensions. Most animals possess perspective vision evolved to enable them estimating distance, a vital tool for survival to avoid predators and approach prays in great efficiency. Simply said we have no provisions to see more than 3 spatial dimensions, therefore we have difficulty comprehending concepts such as 4-dimensional space-time. We have trouble picturing the shape of the universe as a whole.

It is not possible to get outside the universe. This is an important difference between the Earth and the universe. A rocket can leave the surface of the Earth, and we can look at the Earth from outside it. Since we see in three dimensions and since the surface of the Earth is two-dimensional, we can see our planet is bending in a third dimension and visualize its whole shape easily. However even if we could get outside the universe in an attempt to see what shape it had, since the universe is three-dimensional, we would need to be able to see in at least four dimensions to visualize the universe as a whole.

The conjecture in question is a claim made by the mathematician Henri Poincaré over 100 years ago. He said that all four dimensional spaces in which a loop in that space can be narrowed to a point are four-dimensional spheres -- there are no other possible shapes for the universe.

Until 2002, no one had been able to prove or disprove this claim. The framework for this book was the announcement by Russian mathematician Grigori Perelman that he had proven it.

If I travel on a 2-dimensional manifold or surface like Earth, I will always come back to the same point. There are no edges that I would fall. The proof of Poincaré Conjecture supported the idea of Universe is finite (a closed 3-manifold) and has no boundaries. So if I traverse the universe in one direction I will always come back to where I started.

I will have more posts on the shape of the universe, so stay tuned.

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