How Infinity Can Fit Inside the Finite
This summer, sprinters from around the world will gather in Rio to compete in the 100-meter dash. Should you choose to tune in, you'll be treated to electrifying race after electrifying race. With each crossing of the finish line, you'll also witness something seemingly impossible: a runner completing an infinite number of tasks in roughly ten seconds flat. Compared to such a monumental achievement, who cares about a gold medal? An athlete has just made infinity occur within a finite frame!
How is such a thing possible? To find out, we must first travel back to around 470 BC, when the great Greek Philosopher Zeno of Elea was wowing his compatriots -- including a young Socrates -- with his keen intellect, and in particular, his playful paradoxes. In one of these paradoxes, Zeno described a race and a runner, noting that before the runner completes his goal, he must first travel half the distance. Once halfway, he must then travel halfway again, and again, and again. If this was applied to a 100-meter race, our sprinter would run 50 meters (1/2), 25 meters (1/4), 12.5 meters (1/8), 6.25 meters (1/16), and so on until he passes the finish line.
Since one can technically always travel half of some set distance, that would mean the sprinter completes an infinite number of tasks! Zeno argued that this is impossible, and thus concluded that movement must be an illusion.
Since each leg of the 100-meter dash is exactly half the remaining distance to the finish line, it makes sense that the more legs we add up, the closer we'll get to the full 100 meters. So we would expect S1000, say, to be bigger than S10, and therefore closer to 100, but not quite equal to 100. Pushing this reasoning a step farther, in some sense the sum of all the numbers in the sequence must be equal to 100.
Here we have found a point of contact between the finite and the infinite: the sum of infinitely many numbers adding up to something finite. In the right context it seems to make perfect sense: if you split up 100 meters into infinitely many shorter pieces, then of course the sum of the lengths of all the pieces should be equal to the total length of 100.
Outside of fancy philosophical musings, there's a far simpler way to make something infinite fit within a finite space: Make a fractal, a mathematical set that exhibits a repeating pattern at every scale to infinity!
Perhaps the most basic example of a fractal is the Koch snowflake, an extrapolation of Swedish mathematician Helge von Koch's curve, in which a straight line is divided into three equal segments and the middle segment is replaced by two sides of an equilateral triangle of the same length as the segment being removed. This is then repeated for all of the straight lines an infinite number of times.
Zoom in on an edge of the fractal, and this is what you'll see!
So as counterintuitive as it may sound, it is quite possible to contain an infinite number of things within a finite space!