Poem (cr. Arthur)

Take two points on a circle,
And draw a line straight through.
The space that was encircled
Is now split up into 2.

To these points add a third one,
Which gives us two more chords.
The space through which these lines run
Has been fissured into 4.

Continue with the fourth point,
And three more lines drawn straight,
Now the count of disjoint regions
Sums in all to 8.

A fifth point and its four lines
Support this pattern glean,
Counting sections, one divines
That there is now 16.

This pattern here of doubling
Does seem a sturdy one,
But one more step is troubling
As the sixth gives 31.