Take a look at this picture:
Why is it that Pi /= 4?
I think the issue is no matter how many corners you cut, you still don’t have a circle (the sawteeth will always form a 45 degree angle)
(ETA: I thing the final shape may be an octogon)
A wavy line is longer than a non-wavy line. The perimeter of a circle is a smooth line, not a line that is made artificially longer by putting an infinite number of tiny kinks in it.
Manhattan norm vs Euclidean norm
You could do the same construction with any regular polygon, and you’d get a different result each time. The limit of any “sawtooth” line formed in this way might be the circle, but there’s no guarantee that taking that limit will result in the line having the same length. Just because each element of the sequence has the same length does not automatically tell you that it’s true for the limit.
You could do it with a line. Imagine a diagonal line going from the origin to the point (1,1) in the x-y plane. The length is square root of 2. Now you can approximate it as closely as you like by a zigzag of horizontal lines whose total length is always exactly two. What is wrong is assuming that arc length is continuous in this sense.
To put a finer point on it, the mathematical definition of smooth exactly accounts for this sort of situation. To be smooth, it must be continuously differentiable (no jagged edges/abrupt turns, essentially). This approximation will “look like” a circle, but at the limit it is not differentiable at any point, and it is not equal to, at least not in this sense of length, to a smooth curve. (That a circle is smooth is left as an exercise for the reader, i.e. “it looks obvious to me but I’m not sure if I can come up with a proof for it off the top of my head”).
Na. The sawteeth don’t have to sides of equal length so they don’t have to make a 45 degree angle, you can approximate a circle.
In the example, they should be using the diagonals to approximate the circumference of the circle. For example, in the picture in the top right corner the circumference would be estimated by drawing four lines between the four main compass points (i.e., from the top of the circle to the right of the circle, from the right of the circle to the bottom of the circle, etc.). These will be inside the circle (not outside), and will sum to 2.83 (= 4 lines, each of length = square root of (0.5 squared + 0.5 squared)). Eventually, using more and more smaller lines (i.e., going to infinity) this will approach pi (or 3.14).
But isn’t the formula for the area of a circle based on this same ability to treat a curve as straight line?
No, I think you mean circumference. And it’s not treating a curve as a straight line; it is an arc which can be measured.
I think the OP’s question was to also explain why the drawing [incorrectly] states that pi = 4. Here’s why:
pi = C/d
If you are to believe the picture, the perimeter of the stepped-square gets infinitely close to the circumference of the circle.
Therefore, pi = P/d
Thus pi = 4/1 = 4
Here’s a clue: the penultimate panel is incorrect.
You can apply the same argument to the length of a straight line. For example, take the diagonal of a unit square (a square with sides of length 1). You can approximate that diagonal as closely as you like with a jagged line of length 2. Therefore, following the same reasoning, the length of the diagonal is 2, and not the square root of 2. So it’s not a useful way to define length even for a straight line, and that’s why it’s not used for smooth curves like a circle.
You can treat curves as made up of lots of straight line segments in some sense, and calculate their length in this manner, but you have to use line segments that are oriented tangent to the curve. Length is an odd duck, and it cares a lot about that sort of thing. It won’t do to just hang around in the vicinity of the curve moving whichever way you like.
After all, you could imagine a single point, being approximated in as small an interval around that point as you want by as much repeated buzzing around as you want, so that the approximations have as much total length as you want, despite the single point itself having only length 0.
Yeah. I’ve forgotten much of my pure maths degree so I couldn’t really explain why. It just seemed like a hand wave with no mathematics behind it.
Since the last panel mentions Archimedes, I wonder what Archimedes himself would have made of this argument. Would he have seen the flaw?
Well he’d certainly have known it was wrong. I doubt he’d have explained the error in the same way any mathematician post Newton would have.
The error in the diagram has already been pointed out, but more importantly, Archimedes would’ve seen the flaw because this isn’t how he measured pi. What he did to get pi was make a similar argument with an “infinitely” refined polygon (he didn’t actually believe in infinity and taking limits), but also inscribe a polygon inside the circle, and show that pi would have to be somewhere between the perimeters of these two shapes.
In this diagram, to get pi you’d have to draw a square inside the circle as well, which will have a slightly smaller perimeter. Then you do the same argument, cutting corners and adding more sides, until the shape becomes more and more circular, but never quite getting there. The perimeters of these two shapes would form upper and lower bounds between which pi would have to be. From this method Archimedes determined that pi must be somewhere between 3.141 and 3.142, pretty good for his time. Unfortunately, pi can never actually be described by a relation between two integers, so this method can only give you upper and lower bounds for pi but not the exact value.
I should note that this isn’t exactly how it was done, since using squares isn’t too accurate (while it’s close near the top, bottom, and sides, beyond that it fails). I think Archimedes actually used hexagons to start out with.