# The Infinite Properties of Space

When we consider the distance between two objects of mass and move the two objects closer together by 1/2 the distance, and continue to do that, wouldn’t it seem that the two objects would never touch each other? As there will always be a distance between them that can be halved.

There is nothing profound about setting arbitrary limits in a hypothetical and then marveling at the results. Yes, if you only close the distance by half each time, they will never touch. This is the same if you only move 99% of the remaining distance each time. Or 10%. Or not move them at all.

When I was a kid it was a frog hopping to the edge of the pond - half the distance each time. It’s in the same class as the ‘grain on a chessboard’ (1 on the first square, 2 on the next and so on) problem.

“Tell Zeno I’m willing to meet him halfway.”

My HS geometry teacher claimed that when he was in college, the dean lined up all of the guys who couldn’t make up their minds whether they wanted to be math or engineering majors along one wall and their girlfriends along the opposite wall. Then the dean declared that when he blew a whistle they could close half the distance to their girlfriends. When they were nicely lined up, he’d blow the whistle again and they could close half the remaining distance. Repeat until they reached their girlfriends.

Some guys instantly saw that they would never get to their girlfriend and left; they were shunted into the math program. The rest saw that they would get close enough for all practical purposes and stayed; they became engineers.

After a long pause, the smart girl in the class commented, “So you’re a bachelor to this day, then.”

ISTR that calculating limits like this was week 1 of calculus class.

Is this your first time doing a marijuana?

As others mentioned, standard Zeno paradox.

however, also consider that at a certain point, the distance separation would be within the distance between two atoms of any two solid masses. At that point, they are “touching”.

The physics answer is to ask a question. Are these masses true point particles? If the answer is no, then their boundaries will overlap even if their mathematical centers don’t touch.

As you might imagine, this creates issues in the real world.

Is that available on a t-shirt? 'Cause I want that.

I doubt it. I made it up.

But doubtless you can get someone to print it for you.

As I recall, Zeno made it a race between the warrior Achilles and a tortoise, and he definitely gave the tortoise a wi-i-i-i-d-d-d-e head start.

Yeah. We have to assume this is the case here, Achilles must get his toes to touch the tortoise’s heels to “win.” And for arbitrary reasons, “Zeno, I have better things to do today, so I’m just gonna lean over and grab now that I’m close,” is not an option left open to Achilles.

And Zeno knew that this wasn’t real. He was simply making the point that we can conceive of things that are impossible when we apply arbitrary restrictions that seem not to be the problem at first glance.

Consider: the tortoise and Achilles are 10 ft apart. They are not allowed to move. An earthquake opens a sinkhole, and they fall in, the tortoise landing on Achilles. They should never have met, and yet they did. Solve for x.

Did I just do someone’s philosophy homework?

“Space is big. *Really *big. I mean, you may think it’s a long way down to the chemist’s, but that’s peanuts compared to space.” - Douglas Adams

But if they take the first halfing after a minute, the second halfing after another half minute, the third after a quarter minute, etc., then although it takes an infinite number of halfings to get there, they manage to reach after two minutes.

I think you’re slightly off there. It doesn’t take 2 minutes, it takes 1.999…
I’ll see myself out.