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3Blue1Brown - "A Curious Pattern Indeed" [CC by Yuval Dolev]
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 this lines run, has been fissured into four.
Continue with a fourth point, and three more lines drawn straight,
Now the count of disjoint regions, sums, in all, to eight.
A fifth point and its four lines, support this pattern glean,
Counting sections, one divines, that there are now sixteen.
This pattern here of doubling, does seem a sturdy one,
But one more step is troubling, as the sixth gives thirty-one.
Wait.. what?
1, 2, 4, 8, 16, ...
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 - 1, 2, 4, 8, 16, 32, 63?
Or - 1, 2, 4, 8, 15?
If you keep going, the number of section 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 favorite proofs.
But interesting problem 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 coencide with powers of two, for fewer than six points?