Square tiling problem

[gaunt lynx]

Can any rectangle with real, positive side lengths be filled in with a finite number of squares? Me and my dad were thinking about and working on this and we got to seeing if you could tile a rectangle with side lengths 1 and π. We couldn’t figure it out but this was the first notes page that we went through.

[snow storm]

Is pi the first irrational number you tried?

[gaunt lynx]

Yes

We chose it because it's also transcendental.

[snow storm]

Yeah, that was gonna be what I was gonna say.

[gaunt lynx]

We started by showing and doing our best to prove that you couldn't do it if all the squares were the same size.

[snow storm]

Are you familiar with the concept of a golden rectangle?

[gaunt lynx]

No, enlighten me.

[snow storm]

A golden rectangle is a rectangle with aspect ratio 1:phi, where phi is the golden ratio, (1 + sqrt(5))/2.

What makes a golden rectangle significant is that you can divide it into a square and another golden rectangle.

[gaunt lynx]

Ok yes I have encountered this before.

[snow storm]

Right. Which means that you can divide a golden rectangle into squares forever, getting infinitesimally small.

[gaunt lynx]

Yes

[snow storm]

In fact, the same is true for any irrational aspect ratio.

[gaunt lynx]

Oh cool

So this disproves the conjecture.

[snow storm]

Or proves it, depending on what the conjecture is.

[gaunt lynx]

ok

Thanks!

[snow storm]

No problem.

[snow storm]

An interesting video on the problem:https://www.youtube.com/watch?v=ubHVK71F01M

[gaunt lynx]

Could you devide it up in a different way to tile it without going into the loop?

[snow storm]

No, you can't. Think about it like this; take a golden rectangle and divide it into the square and the smaller golden rectangle. Now, the square needs to be tiled over, right? But then you divide the second rectangle into a square and the third rectangle, and the second square also needs tiling over, etc. ad infinitum.

The largest square you can use to tile a rectangle is the smallest square that a descent like this one yields.

[gaunt lynx]

I was thinking something more along the lines of this.

[snow storm]

I think the video said that it's the square with the side length of the greatest common factor of the rectangle side lengths.

[gaunt lynx]

(ms paint mockup)

[snow storm]

I don't... understand.

[gaunt lynx]

Like using a smaller square and other squares to make a rectangle.

[snow storm]

It doesn't matter that its a "loop", what matters is that it's an infinitely repeating process.