Cubical Progression

Small Infinity, Big Infinity | Numbers | Science News

Web edition: January 8, 2008

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Infinity is bigger than any number. But saying just how much bigger is not so simple. In fact, infinity comes in infinitely many different sizes—a fact discovered by Georg Cantor in the late 1800s.

Now a mathematician has come up with a new, different proof. Based on a simple game, the proof uses a strategy that might someday shed light on one of the great unsolved questions in mathematics.

The smallest infinity is the one you'd get to if you could count forever. The numbers 1, 2, 3, 4… are called the natural numbers, and they are the most obvious example of this size of infinity. In honor of them, anything that has this size of infinity is called "countable."

Lots of infinite things are countable. For example, suppose you take just the even numbers. They're countable, just like all the natural numbers are. You can prove it by counting them:

This leads to the peculiar result that, in some sense, there are the same number of even numbers as there are of all the natural numbers—even though there are only half as many. Infinity is an odd beast.

Cantor also came up with a clever way to count all the fractions, proving that they're also countable. So there are (in a sense) just as many fractions as whole numbers, too! (To see why, go to http://www.homeschoolmath.net/.)

Not everything infinite is countable, though. Take, for example, all the real numbers—all the counting numbers plus all the fractions plus all the irrational numbers. A real number is any number that can be expressed in decimals, though the expression might continue forever. For example, pi is a real number: It can be written as 3.14159….

What Cantor showed was that no matter what clever counting scheme you come up with, you'll never manage to count every last real number. He did it with a challenge: Just try it. Come up with a counting scheme, any counting scheme, and he'll find a real number you missed.

He reasoned this way. He would write down all your numbers in a list, including only the portion after the decimal point. If the number was rational, he would write it with an infinite number of zeros at the end, or the numbers would start repeating in an infinite loop. The list would look something like this, though it probably wouldn't be these particular numbers:

.48859283404162…

ENLARGE

.23190734486346…

.23987932750000…

.34576128733518…

.23758093475639…

etc.

Now Cantor would cook up a new number that wouldn't be anywhere in your list. To find the first digit, he would look at the first digit of the first number, which in this case is a 4. Cantor would make the first digit of his new number something different, say 5.

Now he would look at the second digit of the second number, in this case, 3. He'd make the second digit of his new number something different, say 4. And he'd make the third digit of his new number different from the third digit of the third number in the list (which is 9, so he could make his 0).

By keeping this up forever, he'd get a new real number that was different from every other number in your list. After all, it can't be the same as the first number, because it has a different first digit. And it can't be the same as the second number, because it has a different second digit. For the same reason, it can't be the same as any of the numbers in the list. So he's got you! You didn't count every last real number after all.

Cantor's discovery raised a question that hasn't been fully answered even today: Is there a "medium" size of infinity—bigger than the natural numbers but smaller than the real numbers? The supposition that nothing is in between the two in size is called the "continuum hypothesis," after the continuum of numbers. The question is so puzzling that it led to a genuine crisis in mathematics, and mathematicians still aren't sure of the answer.

When mathematicians get stuck on a problem like this, they usually need a new approach. Using a method entirely different from Cantor's, Matthew H. Baker of the Georgia Institute of Technology in Atlanta recently came up with a new proof that the real numbers aren't countable. Baker started by considering a little mathematical game. Let's say that Alice and Bob decide on some subset of the real numbers between 0 and 1. Alice starts by picking any number she likes between 0 and 1, and Bob follows by choosing a number that is bigger than Alice's but still less than 1. Then they take turns picking new numbers that are always between the last two numbers chosen.

If we call Alice's numbers A₁, A₂, A₃, etc., and Bob's B₁, B₂, B₃, etc., we can draw a picture that would look something like this:

Notice that Alice and Bob's picks get closer and closer to each other over time. An important theorem in calculus says that if they were to keep picking numbers this way forever, Alice's numbers would get closer and closer to just one single number, which will be bigger than all the A_n's and smaller than all the B_n's.

So that brings us to the point of the game. Remember at the beginning that Alice and Bob agreed on some subset of the interval. If the number Alice converges toward is in that subset, Alice wins, and if it's outside, then Bob wins.

Baker found a strategy that Bob can use to win anytime the subset is countable. Bob can make a list of all the numbers in the subset: S₁, S₂, S₃, etc. Bob has to make sure that Alice's numbers can't converge toward any number on that list.

For his first pick, he looks at S₁. If S₁ is smaller than A₁, he doesn't have to worry about Alice's numbers converging toward it, because he knows Alice's numbers have to converge toward something bigger than any of them. So he just picks any number he feels like that is allowed.

If S₁ is bigger than A₁, he can just pick S₁! After all, he also knows Alice's numbers can only converge toward something smaller than all of his numbers. By making each of his choices this way, he can make sure that Alice's numbers can't converge toward any point in the subset.

But then Baker noticed something else. If the subset that Alice and Bob agreed on at the beginning was the entire interval from 0 to 1, Bob obviously couldn't win. But if the subset were countable, Bob could win. So that implies that the whole interval can't be countable.

Baker says his proof doesn't show anything Cantor didn't, but that it's a nice example of how two areas of mathematics that don't seem to have much to do with one another—in this case, game theory and set theory—are in fact tightly linked. Hard problems in mathematics are often solved by bringing together fields of math that seem only distantly related.

Indeed, set theorists are pursuing game theory approaches. "People take some of these games seriously because they can give non-trivial insights into the nature of real numbers," Baker says. "They may be able to shed light on things like the continuum hypothesis."

Cubical Progression is based on a concept which is positioning sculptural object over the surface of the planet in a cartographic network. This network is established within the geographical coordinate system through a numerical progression. Starting from the center of the system (0°, 0°) the network is progressing into the four directions of the coordination plane (x, y, -x, -y) and designating specific positions, of which each is to hold one part of a cube divided. The division of the cube takes four parts and resembles a spiral dynamic when view from top, where each piece equals a L-shape. The scale of the cube follows as well a numerical progression from the theoretically infinite small to infinite big. All pieces and positions are building together one piece. This means that although the pieces are located at many points, they can be only understood as one work, and therefor have to be combined in the recipients mind.

Cubical Progression is translating a two dimensional idea onto a three dimensional body, as well as a cubical form onto a spherical one. The content of this work is its location, with which it occupies space on the physical scale as well as the numerical dimension. It is to exist in its physical presence on the location parallel to the verbalized idea which is manifesting its existence in mental space and elevating the planets physicality and scale within the recipients mind. Cubical Progression is a project that aims to follow the urge to enfold the whole of earth´s physical extension into a work of art and is understanding at the same time the all but impossible attempt to touch upon the wholeness of earth. It intents to neutralize existing projections onto certain areas of the earth and to create a bridge between remote places. It is also a concept that can be applied onto other planets, with other geographical coordinate systems. But carried out on earth, when using e.g. the fibonacci sequence, positions are found in places such as: Senegal, Chad, Angola, Georgia, Siberia, north Canada, Antarctica and the Ocean.

Cubical Progression

Project Abstract

fibonacci progression