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May 12, 2025 · 22 min

Erdős distances: an open conjecture, a proof of alignment

In this episode, follow an open conjecture, then explore a theorem: an infinite set of points with integer distances between every pair must lie on a straight line.

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  1. Speaker 1

    The podcast from Tangente magazine. Welcome, everyone. We are thrilled to have you here for this deep dive.

  2. Speaker 2

    Absolutely. Thanks for having me.

  3. Speaker 1

    You know, when you look up at a clear night sky, you see thousands of distinct stars just scattered all across the darkness.

  4. Speaker 2

    Just endless points of light.

  5. Speaker 1

    Right. And if you were to take a ruler and measure the distance between every single pair of those stars, you would naturally expect to find, I don't know, a totally random, chaotic mess of numbers.

  6. Speaker 2

    Oh, for sure. It should be pure astronomical noise. I mean, there is no reason to assume any underlying order. It's just an infinite, messy list of random distances.

  7. Speaker 1

    But what if the universe was actually playing a trick on you? Like, what if there was a hidden mathematical cage limiting exactly how many different distances could possibly exist up there?

  8. Speaker 2

    That's a crazy thought.

  9. Speaker 1

    It really is. And today's deep dive is an exploration of the legendary Hungarian mathematician, Paul Erdős, specifically his deeply fascinating geometric problems. We are looking at a man who had this uncanny knack for asking questions that sound like simple riddles.

  10. Speaker 2

    Yeah, things you could easily explain to a child.

  11. Speaker 1

    Exactly. But which actually took decades and the collective effort of mathematicians all over the globe to finally solve.

  12. Speaker 2

    It is truly remarkable. And the math we are exploring today doesn't require you to have advanced calculus degrees to understand the premise. It's really just dots and lines.

  13. Speaker 1

    Dots and lines.

  14. Speaker 2

    Yeah. But the journey to the solution is staggering. We are going to explore two major ideas from our source material today. First, we'll look at how many distinct distances you can actually find between a given set of dots on a page.

  15. Speaker 1

    Okay.

  16. Speaker 2

    And second, we'll look at the bizarre geometric phenomena that happen when you try to force all of those distances to be perfect whole numbers.

  17. Speaker 1

    It is genuinely mind-bending stuff. But before we get to the whole universe of points, we have to start with a single simple shape. There is a puzzle from our sources that Erdős posed back in 1996. Let's try to visualize this.

  18. Speaker 2

    Okay, let's do it.

  19. Speaker 1

    Picture a nine-sided shape in your head, an enneagon, and let's make it a standard convex shape, meaning it's pushed outwards. So no internal dented angles like a star might have.

  20. Speaker 2

    So you basically have nine vertices or corner points forming this general circular outline.

  21. Speaker 1

    Right. Now imagine you draw straight lines connecting every single point to every other point. So you trace all nine of the outer sides and then you draw all the internal diagonals criss-crossing through the middle of the shape.

  22. Speaker 2

    That's a lot of lines.

  23. Speaker 1

    It is. And if you were to take a ruler and measure all those lines, what would you find? Well, best case scenario, you drew a perfectly symmetrical, regular nine-sided shape where all the sides are exactly equal.

  24. Speaker 2

    Right, like a perfect stop sign, but with nine sides.

  25. Speaker 1

    Exactly. Because of that perfect symmetry, you'd only find exactly four distinct lengths among all those crisscrossing lines.

  26. Speaker 2

    And on the flip side, the worst case scenario is that you drew a highly irregular, just completely lopsided shape. Because there are nine outer sides and 27 internal diagonals, you have 36 lines in total.

  27. Speaker 1

    Which is a huge mess.

  28. Speaker 2

    Yeah. If you drew it messily enough, you could end up with 36 completely different lengths.

  29. Speaker 1

    Okay, so here's the riddle Erdős posed to the mathematics community. How can you arrange nine points in a plane so that you get exactly five distinct lengths?

  30. Speaker 2

    Five.

  31. Speaker 1

    Right. Not the highly symmetrical four, not the chaotic 36, exactly five distinct distances.

  32. Speaker 2

    And the answer to this riddle is hyper-specific. You cannot just, you know, nudge points around on a piece of paper hoping to strike gold. You actually have to select your nine points from the vertices of a larger, perfectly regular polygon.

  33. Speaker 1

    A template,

  34. Speaker 2

    basically. Exactly. Specifically, you have to borrow them from either a regular 10-sided shape, a decagon, or a regular 11-sided shape, a hendecagon.

  35. Speaker 1

    Wow.

  36. Speaker 2

    Yeah. If you pick nine points from those specific templates, and only those templates, you get exactly five distinct lengths.

  37. Speaker 1

    Okay, let's unpack this. Because if nine points are that strictly governed by these rigid templates, just to find five lengths, what happens when we just throw all the rules out?

  38. Speaker 2

    What do you mean?

  39. Speaker 1

    Well, couldn't you just scatter a bunch of dots totally randomly across a piece of paper and guarantee they all have completely different distances? I mean, just make it

  40. Speaker 2

    chaotic. It's a very natural assumption to make. If you want variety, just introduce chaos, right?

  41. Speaker 1

    Yeah.

  42. Speaker 2

    But that fundamentally misunderstands the game Erdős is playing here. This brings us to his famous 1946 minimal distances conjecture. Erdős wasn't interested in the maximum number of distinct distances you can create by being messy.

  43. Speaker 1

    I see.

  44. Speaker 2

    He was hunting for the minimum number of distinct distances you're absolutely guaranteed to get, no matter how maliciously someone tries to range the points to reduce that number.

  45. Speaker 1

    Ah, so I should think of it like an adversarial game.

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