From triangles to lines: two Erdős theorems

Speaker 10:00

The podcast from Tangente magazine, welcome. We've crafted a really cool custom-tailored deep dive for you today. And we're actually going to start it off with a bit of a thought experiment.

Speaker 20:11

I love a good thought experiment. Let's hear it.

Speaker 10:13

Okay, so imagine you step into a room. Only instead of a normal square room, this one is shaped like a perfect triangle.

Speaker 20:22

Okay. A triangular room.

Speaker 10:24

Right. And instinctively, just by standing there and looking around, you'd probably realize that you are generally closer to the flat walls, like the sides of the triangle, than you are to those sharp corners.

Speaker 20:37

Oh, yeah, definitely. I mean, your brain just intuitively grasps the basic geometry of that space. You know, the walls create this perimeter right in front of you while the corners kind of, well, they recede away into these narrow pockets.

Speaker 10:51

Exactly. It feels like a very simple, almost completely trivial observation about how physical space works.

Speaker 20:58

It does. But you know, when you take that seemingly trivial observation and actually bring it into the world of theoretical mathematics, it turns into this 60-year quest for a hidden truth.

Speaker 11:12

Which is wild. And that's exactly what we are exploring today. The invisible rules that govern the space right around you. We're going to unpack how asking the most basic questions about shapes and scattered dots can reveal this rigorous, unbreakable architecture hidden underneath our physical reality.

Speaker 21:31

Yeah, and our guide for this exploration is the legendary mathematician Paul Erdős.

Speaker 11:36

Oh, Erdős.

Speaker 21:37

He's fascinating.

Speaker 11:38

He really is.

Speaker 21:39

He had this totally unique genius for looking at things that seemed incredibly basic, like literally the physical distances inside a triangle. And then posing these conjectures that were so deeply woven into the fabric of mathematics.

Speaker 11:53

Right, conjectures that took decades and, you know, some of the most brilliant minds on Earth just to finally explain.

Speaker 21:59

Exactly. So we have two of his most beautiful puzzles on the table for this deep dive today. We've got the Erdos-Mordell theorem and the Erdos-De Bruyne theorem.

Speaker 12:08

Nice. Let's go right back to that visual of the triangular room then. So you are standing inside it. And it doesn't even have to be perfectly symmetrical, right?

Speaker 22:18

Right. It could be skewed, stretched out, completely irregular, whatever.

Speaker 12:24

Okay, so you're in there. You know you are generally closer to the sides than the corners. But in 1935, Erdős asked a very specific question. He asked exactly how much closer.

Speaker 22:35

And the constraint he proposed is just fascinating. Erdős stated that if you calculate your distance to all three walls and average it out, and then you calculate your distance to all three corners and average that out, a permanent ratio emerges.

Speaker 12:51

A permanent ratio, no matter what?

Speaker 22:54

No matter what. No matter where you are standing in that room, your average distance to the walls is always less than or equal to exactly half of your average distance to the corners.

Speaker 13:04

Okay, so let's make that math really tangible for you listening. If you sum up your distances to the three sides and then divide by three, that number is at most half of what you get if you sum up your distances to the three vertices and divide by three.

Speaker 23:19

Spot on.

Speaker 13:20

To visualize the mechanics of this, I like to think about a rubber band. Like, imagine you have a rubber band attached to your waist.

Speaker 23:27

Okay, I'm picturing it.

Speaker 13:28

If you try to stretch that rubber band all the way out to the three sharp corners of the room, you are really pulling it into the deepest recesses of the space. It takes a ton of tension.

Speaker 23:39

Oh, right. Because the corners are so far away.

Speaker 13:41

Exactly. But if you let the rubber band just snap to the nearest point on the flat walls, the distance is dramatically shorter because, you know, the shortest distance to a line is always a perpendicular drop.

Speaker 23:52

That is a phenomenal analogy.

Speaker 13:54

Thanks. So Erdős is essentially proving that the snap to the wall distance is universally capped at half the stretch to the corner distance on average.

Speaker 24:04

Yeah, that perfectly captures the physical tension of the geometry there. And I mean, there was only one specific scenario where the distance to the walls is exactly half the distance to the corners rather than strictly less than.

Speaker 14:17

All right. The tipping point. What's the exception?

Speaker 24:20

That equality only happens if you are standing in a perfect equilateral triangle. All sides are totally equal. And you are standing in the exact dead center of the room.

Speaker 14:31

The ultimate balance point. But mathematicians didn't need Erdős to figure out that specific perfect scenario, did they? Like that ground had already been covered.

Speaker 24:40

Oh yeah, long before Erdős was even born. Way back in 1649, Vincenzo Viviani proved a theorem showing that in an equilateral triangle, the sum of the perpendicular distances from any interior point to the sides is actually constant.

Speaker 14:55

Wait, constant? Like it never changes?

Speaker 24:57

Exactly.