In my time as a puzzle enthusiast, one of the puzzles I encountered was called the ant and the rubber band. It was only later that I realized that this puzzle had some cosmic significance.

**Problem Statement**

We have an ant that is trying to crawl from one end of a rubber band to the other. But as the ant crawls, the rubber band also stretches out. The ant crawls one centimeter per second. The rubber band starts out one meter long, and stretches out one meter per second. This is one of those magical math rubber bands that can stretch indefinitely. Let’s just say the ant is mathemagical too. Will the ant ever reach the end?

At first glance, it looks bad for the ant. The ant crawls crawls one centimeter closer, but falls a whole meter back. So the ant is losing about 99 cm per second. That doesn’t sound like a path to victory.

But it’s not quite as bad as it sounds. Once the ant is 1% of the way across the rubber band, it’s no longer losing 1 meter per second, it’s losing 99 cm per second. If the ant is halfway, it’s losing 50 cm per second. So the ant may be losing ground, but it loses less and less ground as time goes on. Perhaps at some point the ant may finally start to gain ground, and reach the end of the rubber band.

What do you think?

**Solution**

Our preliminary analysis suggests that it’s very important to keep track of how much progress the ant has made as a percentage of the total rubber band length. So our solution will involve a *change of variables*. Instead of looking at the absolute distance, we’ll look at fractional progress.

First, let’s solve a simplified version of the puzzle. Rather than continuously stretching the rubber band, we stretch it in discrete steps. For one second, it’s a constant 1 meter. Then, the rubber band instantly stretches to 2 meters. One second later, and it instantly stretches to 3 meters.

So in the first second, the ant makes 1% progress. In the second second, it makes 0.5% progress (1%/2). In the third second, it makes 0.333…% progress (1%/3). If we add up all these numbers, we can calculate the total progress by the Nth second:

1%*(1 + 1/2 + 1/3 + 1/4 + 1/5 + … + 1/N)

This is the “harmonic series”. If you know a little math trivia, you know that this series diverges to infinity. That implies that given enough time, the ant can cross a stretching rubber band of any length. But, if you don’t believe me, here is a simple proof.

1%*(1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + … )

≥ 1%*(1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + … )

= 1%*(1 + 1/2 + 2*(1/4) + 4*(1/8) + 8*(1/16) + …)

= 1%*(1 + 1/2 + 1/2 + 1/2 + … ) -> diverges to infinity

The original puzzle involves a continuously stretching rubber band. Our approach is the same, but the solution involves some calculus. I leave the solution as an exercise to the reader who knows calculus—not because I’m too lazy to write out the solution, but because I’d like to spare my readers who don’t know calculus! Anyway, the percentage progress by time t is:

1% * ln(1+t)

That means that the ant will complete its journey by e^100 – 1 seconds, or about a billion billion billion billion years. Uh, recall that it’s a mathemagical ant, it lives forever.

**Ant Cosmology**

At the beginning of this article, I presented a puzzle. The puzzle is not, “Will the ant reach the end?”, but “Why does this puzzle have cosmic significance?”

The ant and the rubber band is mathematically analogous to the expansion of the universe (aka big bang theory). The ant represents light from distant galaxies. The rubber band represents the stretching fabric of space.

There are a lot of misconceptions about big bang theory. For further reading on those misconceptions, I recommend a Scientific American article “Misconceptions about the Big Bang”, or a journal article by the same authors. Here I’ll address one common point of confusion.

According to the big bang theory, there are some objects that are so far away that they are receding faster than the speed of light. Many people think that this contradicts relativity theory, and even scientists have incorrectly asserted that the expansion of the universe never exceeds the speed of light.

But faster-than-light recession is not the same as faster-than-light travel. Rather, it’s caused by the expansion of space itself. The stretching of the rubber band.

The “Hubble distance” refers to the distance past which objects recede faster than light. Counterintuitively, we can see objects beyond the Hubble distance. Recall, the ant was receding at 100 times the rate of the ant’s crawl-speed, but was still able to reach its destination. Likewise, even if a faraway galaxy is receding at the speed of light, its light will still eventually reach us–although it may take a very long time.

I remember reading about these misconceptions as a teenager, and being shocked. But soon I realized, oh, it’s just like the ant and the rubber band. Familiar territory.

**Distance and Dark Energy**

I’ve said what I wanted to say, but there are just a couple bonus topics to squeeze in.

First question, what does it mean to say that a galaxy is a million lightyears away?

Going by the ant analogy, we can define (at least) three distances: 1) The distance between the two ends of the rubber band when the ant reaches its destination (a billion billion billion billion meters). 2) The distance between the two ends of the rubber band when the ant started (1 meter). 3) The distance that the ant walked (a billion billion billion billion *centi*meters. These three distances are vastly different from one another. So when we say a galaxy is a million lightyears away, which do we mean?

This is a case where a little bit of knowledge helps you realize that you don’t know anything. How far away really is anything? Have I ever understood an astronomer, ever? But then you go look it up, and you realize that astronomers are usually using the third definition. If a galaxy is a million lightyears away, that refers to the distance travelled by the light. If you want to know how far the galaxy is right this moment (definition 1), you have to do some maths, which I leave as an exercise to the reader.

Second topic: one real world complication is that the expansion of the universe is not constant. There are two countervailing forces which cause the expansion to decelerate and accelerate. The expansion naturally decelerates because of the gravitational forces pulling matter together. However, expansion has been observed to accelerate, leading to the theory of dark energy.

But what does it mean to accelerate or decelerate? In the rubber band puzzle, the rubber band can be said to expand at meter per second, which is a constant rate. However, from another perspective, the rubber band expands 100% in the first second, 50% in the second second, 33% in the third second, and so on. That sounds a bit like the rubber band is slowing down.

Again, this is a matter of convention. By convention, we would say that the rubber band is expanding at a constant rate. If the rubber band instead expanded 100% each second (doubling in length each time), then we would describe that as accelerating expansion. Left to its own devices, dark energy would cause the expansion of the universe to accelerate, which is to say that it would double in size at fixed time intervals.

In a universe with dark energy (that is, our universe), light from some far away galaxies would never reach us. It wouldn’t be able to keep up with the expansion of the universe. The universe would eventually reach a maximum size (under the second definition of distance). Imagine, some places you can never go, because it’s geometrically impossible.

Pierce R. Butler says

Instead of looking at the absolute distance, we’ll look at fractional progress.A Mister Zeno the Elder on line one for you, Siggy.

Rob Grigjanis says

Like the cosmologists, I like to use comoving coordinates; so x=0 corresponds to one end of the band (the end the ant starts at), x=1 corresponds to the other end, for any time. The actual distance between any two comoving points separated by Δx would be a(t)Δx, where a(t) expresses the expansion. In this case a(t) = 1 + t for t ⋝ 0. Any comoving observer (one standing still on the band) would see the ant walk past them at the same constant speed (v=0.01), analogous to the

localspeed of light. At any time t ⋝ 0, we would then havev = a(t)dx/dt

So the comoving distance travelled by time T is (with the integration limits from 0 to T)

x(T) = v ∫dt/a(t)

= v ∫dt/(1+t)

= vln(1+T)

So, if x(T) = 1 (ant reaching other end),

ln(1 + T) = 1/v = 100

and

T = −1 + exp(100)

Siggy says

@Rob Grigjanis,

Yeah, I thought to say that the new coordinates that we use to solve the ant problem are known as comoving coordinates. I couldn’t find a place to fit it in.

Rob Grigjanis says

Might be worthwhile to add the calculation for exponential expansion (as in a dark energy dominated universe). Then a(t) = exp(kt), where k is some constant (analogous to the Hubble parameter), and

x(T) = v ∫exp(-kt)dt

= (v/k)[ 1 − exp(-kT)]

As T goes to ∞, x(T) reaches a maximum of (v/k), so it could never reach any comoving object with a larger value of x than that.

Rob Grigjanis says

@4:

That’s been bugging me since I posted it. Of course, the ant never reaches x=v/k, but it’s worse than that. The expression above might look like the ant gets arbitrarily close to comoving point v/k, but that’s not the case.

To get the actual distance d(T) from the starting point, we have to multiply the expression by a(T). That gives

d(T) = (v/k)[ exp(kT) − 1 ]

So in fact the ant is always a distance (v/k) short of the comoving point x=v/k, throughout its journey.