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:
- The picture describes a sequence of curves. (Here, "curve" is a generic term referring to any continuous line, be it straight or crooked or curved.)
- This sequence of curves converges on a limit curve.
- The limit curve is a circle.
- It is not a sawtoothed curve.
- It is not an "infinitely jagged" sawtoothed curve.
- It is not a fractal.
- The limit curve is a perfectly smooth perfect circle.
- The length of the limit curve is exactly π (3.1 or so). (Because it is a perfect circle with diameter 1.)
These facts are also true:
- Each curve in the sequence has a well-defined length.
- Each curve in the sequence has a length of exactly 4.
- Thus, the lengths of the curves also form a sequence: 4, 4, 4, 4....
- This sequence also converges on a limit.
- The limit is 4, not π.
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 curve does not have length 4. You've also seen a sequence of jagged, right-angled curves whose limit curve 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.