Here's the "Troll Pi" or "Pi equals 4" image.
Here's the breakdown, as simple as I can make it.
All of the following facts are true:
- Panels one to four describe a sequence of curves. (Here, "curve" is a generic term referring to any continuous line, be it straight or crooked or curved.)
- Each curve in the sequence has a well-defined length of exactly 4.
These facts are also true:
- The sequence of curves converges uniformly on a limit.
- As panel five correctly states, the limit of the sequence is a circle.
- It is not a sawtoothed curve.
- It is not an "infinitely jagged" sawtoothed curve.
- It is not a "polygon with an infinite number of sides" or "infinigon".
- It is not a fractal.
- The limit is an ordinary, perfectly smooth perfect circle.
- Thus, the length of the limit is exactly π (3.1 or so). (Because it is a perfect circle with diameter 1.)
- It's not 4!
And so is this final fact:
- None of these facts contradict each other.
The limit of a sequence isn't necessarily a member of that sequence.
Because of this, the limit of a sequence need not necessarily share any properties with the members of that sequence.
Here, you've seen a sequence of curves of length 4, whose limit does not have length 4.
You've also seen a sequence of jagged, right-angled curves whose limit is not jagged or right-angled at all, but smooth.
This is not a problem. It's not a contradiction. It's just the way it is.
Breathe in. Breathe out. Carry on with whatever you were doing.