Math Journey 4, Part 3
To Infinity via Geometry
In previous legs of the journey we learned that Georg Cantor used one-to-one correspondence to equate infinite sets of numbers with the infinite counting numbers. For a down to earth example of one-to-one, we visited the schoolyard during a winter recess and asked students to hold up their left hands. There was a mitten on each hand. Next they held up right hands, all covered by a mitten, and we concluded that the number of left and right mittens, though uncounted, were equal.
Cantor discovered that many sets of numbers – the evens, the odds, the integers, and also the fractions – could be matched up one-to-one with the counting numbers. Cantor also found that the irrational numbers and the reals seemed to be more numerous than counting numbers. Was there a hierarchy of infinities?
Geometry is another area rich in infinite sets. We are all taught that a line is made of a set of points and it stretches to infinity in two directions. Cantor showed that the full line has no more points than a semicircle. In the figure above, each point on the line is matched with a point on the semicircle by a connecting ray from the center B, in the manner that A is matched with C. Each point on the semicircle has a mate on the line as well. If a correspondence can be found which leaves no point unmatched, then the sets must be equal.
Many surprising results are obtained. The next drawing shows a correspondence between the points on line segments of unequal length. Let E be the point of intersection of lines through endpoints B and D and through A and C. Any point G on AB, when connected to E will correspond to a point G' on CD. Likewise any point F on CD can be shown to correpond to an F' on AB.
Also in the category of "Believe It Or Else" we have Albert of Saxony (1316-1390) contemplating the infinite and proving that infinity is equal to a piece of itself. He worked with the infinite rod and the infinite shell, whereas I'll give you the ceramic version:
Imagine a potter with an infinite supply of clay. The potter knows what all children do their first time working with clay – they love to roll out those long skinny worms. Our potter makes a worm, oh, about 9" long. She then reforms the worm into a ball. The worm and the ball, having been made of the same amount of clay, are equal in volume. Using more clay the potter makes a longer worm. She shapes it into a bigger ball. Suddenly, an idea hits her – thwack – in just the way that wet clay smacks down on her wheel. If she uses her infinite supply of clay and makes a worm infinitely long, it will be infinite in volume. If she then rolls it into a ball, the ball will be infinite in size. The question? If you had a ball infinite in size, what would you have?
I created the following sketch to show the same principle in 2-D. A strip of paper is made to have the same area as a circle. As the strip stretches to an infinite length, it and the circle both grow infinite in area, but how do they compare in the space they take up on an infinite sheet of paper? The circle engulfs it all, the strip does not.
The graphic artist Maurits Escher was fascinated by the concept of infinity and he looked for ways to model it in his works. Often working within a circle, he tessellated (filled) it with pieces growing infinitely smaller, either toward the edge or toward the center. I've recreated the framework for one of his entitled "Butterflies." Not being an artist, I stopped at the frame. The jpeg below ends this article.
In our last leg of this journey, we will do some more work with infinity within a circle. We show the way to an infinite number of stars! Stay tuned.
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