To infinity and where?

By Steve Rensberry

 
The success of the movie Toy Story 3, along with the character Buzz Lightyear's "To infinity and beyond" statement, reminded me of a book by Eli Maor that has sat on my shelf now for several years. Published by Princeton University Press, the book "To Infinity and Beyond: A Cultural History of the Infinite," delves into a number of complex mathematical theories that I have yet to fully comprehend, but it's a book I've cherished for the depth of the subject matter alone and the fact that I can pick it up after just about any length of time and learn something from it. See: Princeton University Press. (1)
   Just the phrase, "To infinity and beyond" is a remarkable one in itself, if you think about it. How does one go "beyond" infinity? And where does one end up if he or she does accomplish the feat? The writers of Toy Story deserve credit for attaching such a crafty phrase to one of their star characters.
   The preface of Maor’s book begins with an interesting story which he attributes to the mathematician David Hilbert. The story involves a man who walks into a hotel one night looking for a room. He is at first told that they don’t have any available rooms, however, the owner thinks about it and tells him that maybe they have one after all. He then reluctantly wakes up each of his guests and asks them, one by one, to move one room over. The guest in room one would move to room two. The guest in room two would move to room three. And so on.
   Amazingly, the man is then shown to room number one, which of course is now vacant. In fact, every guest has a room because, as the story goes, the man had unknowingly checked into Hilbert's Hotel -- the "one hotel in town known to have an infinite number of rooms!" Maor writes.
   An idea of just how strange things can become when dealing with concepts like infinity is seen in Chapter 10, entitled simply "Beyond Infinity." Using studies first published by George Cantor at the University of Halle in Germany in 1874, Maor discusses the concept of denumeration, and certain sets which contain elements so dense that it is impossible to count every element, one being the "set of points along an infinite line, the number line."
   Such points -- which reflect the real number system and all their corresponding decimal forms -- form what Cantor called the infinity of the continuum. "They are not denumerable; they contain more elements -- vastly more- - than a denumerable set," Maor writes. (2)
   Then comes the remarkable opening statement of Chapter 10:
   "To show that the real numbers cannot be counted, Cantor first established a fact which, if anything, seems to be almost beyond belief: There are as many points along an infinite straight line as there are on a finite segment of it." Think about that for minute.
   The rest of the chapter is used to expound on Cantor's Continuum Hypothesis, which Maor says remained unsettled for 60 years, until 1963. "The  hypothesis turned out to be both true and false -- depending on what assumptions one starts from," he writes. In other words, the hypothesis sets apart from the standard axioms of set theory and can be rejected or accepted accordingly.

(1) Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540. Copyright 1987 by Birkhauser Boston, 765 Massachusetts Ave., Cambridge, Mass. 02139.
(2) "To Infinity and Beyond," page 60.

Life in four dimensions

By Steve Rensberry

Russian mathematician Grigori Perelman's proof of the famous Poincare conjecture in 2002-03 and the subsequent confirmation in 2006 of his efforts drew an enormous amount of interest - and rightly so considering that it has been an entire century since Henri Poincare first made his famous conjecture.

In simple terms the conjecture asserts that spherical objects defined in terms of three dimensional Euclidian space have the same type of surface connectivity as do spherical objects defined in terms of four dimensional Euclidian space.

Stephen Ornes in the Aug. 26, 2006 edition of Seed Magazine referred to it as something which "gives mathematicians a short and easy way to identify a deformed blob as a sphere in disguise."

I was among those intrigued by it, but what I found hard to get my head around beyond just the terminology and concepts was envisioning precisely what kind of shape a three-dimensional sphere would have in four dimensions, or what anything in four dimensions would look like for that matter. We're talking about dimensions in Euclidian space mind you and not that dealing with space-time construction, or Minkowski space.

But visualizing things in four dimensions is a nearly impossible feat. Even in our familiar three-dimensional way of looking at things we don't really see things in three dimensions. We don't see both sides, the inside and every angle of an object all at the same time. What we see is the two-dimensional surfaces of three-dimensional objects.

In three dimensions things are defined using three pairs of cardinal directions, represented by altitude, latitude and longitude. In four dimensions there is an addition set of cardinal directions which are orthogonal (at right angles) to each of the others.

Furthermore, in Euclidian geometry a two-dimensional sphere is defined by a set of three different points in three-dimensional space and referred to as a 2-sphere or 2-manifold. A three-dimensional sphere is defined by a set of four different points in four-dimensional space and referred to as a 3-sphere or 3-manifold.

As Ornes explains: "When most people think of a sphere, they generally consider the space that a sphere occupies—a ping-pong ball, for example. When topologists talk about a sphere, they are talking exclusively about its surface."

What Poincare suggested and Perelman proved was that all three-dimensional, finite, simply-connected manifolds which do not have holes are spheres.

A coffee cup with a handle is not a sphere, neither is a doughnut, a car tire or a pair of pants. A flattened saucer? A dinner plate? A square box which doubles as a coffee table? Well, those indeed are spheres for the simple reason that their surface, in what form it can be transformed into without ripping it apart, is a closed, simply connected, three-dimensional manifold.

Perelman's solution involved performing a type of mathematical surgery on the singularities or sections of a three-dimensional sphere which are malformed or crinkled, essentially creating two separate, topologically identical spheres - and lending another proof of William Thurston's geometrization conjecture in the meantime. It was a solution to the problematic areas in manipulated three-manifolds that no one had thought of before.

As with a lot of things, the importance of the conjecture isn't so much the idea that it would help us understand the universe itself - which some believe may be a 3-sphere - but the simple way in which Perelman arrived at his solution.

As Dennis Overbye of the New York Times wrote in an Aug. 18, 2006 article: "Everybody agrees that it is no surprise that the conjecture is true. What is surprising is the way it was proved, using mathematics far removed from traditional topology, establishing links no one had suspected between disparate fields and techniques.

Hmmmm. Isn't that the way it always is?