Problem: A space bug has a volume of 1 mm3 and can travel around at the speed of light. After enjoying its life for a century, a space bug splits into two bugs, which quickly grow to their parent’s size. If we release a single space bug into the wild, what will happen to its descendants in the long run?
Solution: Let’s see. After a hundred years, the first bug splits up. Suppose its kids don’t like each other and decide to fly in opposite directions. After another century, they are separated by whooping 200 light-years. They split up again, and now we have four bugs. By the end of the next century, they can also be as far as 200 light-years from each other.
By this point, the pattern seems pretty clear. The bug colony will form an ever-expanding diffuse cloud, and unless these bugs decide to throw a party, you will probably never see more than two of them close together…
Okay, Luke, if you insist, we will do it the proper way. If we measure distance in meters and time in years, the total volume of all bugs at time t is
Now imagine this: when we are just about to release the first bug, we flash a bright light to tell everyone about our great deed. The light from this flash will form an ever-expanding sphere, and since bugs cannot travel faster than light, they will always stay within that sphere. Thus, the volume available to bugs at time t is
Now, how do we figure out which of these functions grows faster? We could try plugging in small values of t, but that’s basically the same as what we did at the very beginning, and Luke said it was wrong.
We could try plugging in large values of t. Given that we are interested in long-term behavior, plugging in large values makes a bit more sense, but at large values of t both of the functions become pretty huge, and our calculator won’t be too happy about it.
Instead, let’s use a great mathematical truth that says, exponentials dominate polynomials.
Let’s break it down a bit. An exponential is, unsurprisingly, any function with the variable in the exponent. In our case, the total volume of bugs grows exponentially. A polynomial, on the other hand, is a function where the variable is raised to a constant power – like the volume available to bugs.
The statement that exponentials dominate polynomials means that if we wait long enough, any given exponentially growing function will become larger than any given polynomial.
Therefore, even though the bugs are so small and initially so far apart, they will eventually occupy all space available to them, after which they will crush each other and die.