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?