Pop Video: A Curious Pattern Indeed | Nagwa Pop Video: A Curious Pattern Indeed | Nagwa

Pop Video: A Curious Pattern Indeed

Grant Sanderson • 3Blue1Brown • Boclips

A Curious Pattern Indeed


Video Transcript

Pick two points on a circle and draw a line straight through. The space which was encircled is divided into two. To this point add a third one, which gives us two more chords. The space through which these lines run has been fissured into four. Continue with a fourth point and three more lines drawn straight. Now the count of these joined regions sums, in all, to eight. A fifth point and its four lines support this pattern gleaned. Counting sections, one divines that there are now 16.

This pattern here of doubling does seem a sturdy one. But one more step is troubling is the sixth gives 31. Wait! What? One, two, four, eight, 16, 31? What’s going on here? Why does the pattern start off as powers of two, only to fall short by one at the sixth iteration?

That seems arbitrary. Why not one, two, four, eight, 16, 32, 63 or one, two, four, eight, 15? If you keep going, the number of sections deviates even further from powers of two, except when it hits 256.

But this just begs the question of “what the pattern really is and why it flirts with powers of two? In my next few videos, I’ll explain what’s happening, which will include one of my all-time favourite proofs. But interesting problems deserve to be shared, pondered, and discussed, before their secrets are hastily revealed. So while I work on animating my explanation, I encourage you to think of your own.

To be clear, the question is this: if you take some set of points on a circle, you connect every pair of them with a line, how many pieces do these lines cut the circle into? Does it matter where these points are? And why does the answer coincide with powers of two, for fewer than six points?

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