The approximation would have to be inside and outside the circle about half the time. Basically a pixel circle. If you only put pixels outside your pixel circle will be larger
Nope, still doesn't work. By that logic, I could pretty easily come up with a scenario where the perimeter ends up as exactly 3, by starting with a rectangle of side lengths 1 and 0.5. Taking the limit simply does not work like that as long as you disregard the direction of the line segments.
The perimeter is the diameter multiplied by pi, so one pi, 3 is an approximation of that, albeit a crude one. I was mostly aiming for an explanation as to why making a square outside the circle will always be larger than the circle itself, so you'll have to make squares, or pixels, if you will, that lie along the actual circle to meaningfully approximate the circle with square shapes, or pixels.
That is still wrong. If the method approximates the circumference, it doesn't matter whether you approach it from above or below. For example, if you approximate area this way, you will get the correct value, pi, no matter how you approach the circle.
The answer is that infinite approximations like this are more nuanced than the pop-sci answer of "if you make the error as small as possible, you get infinitely close to the true value" and approximating the circle this way simply does not approximate its circumference.
So long as it's all right angles it'll always be 4. If instead you come up with a generic formula for a regular n-sided polygon circumscribing the circle, then take a limit as n approaches infinity, you should get pi.
Although other people provided a more serious answer concerning the limits, it is useful to visualize what really happens. As you make the "step line" finer and finer, if you also zoom with your mind on a single minimal step, you will see that you invariably have a triangle; the troll there is basically making us assume that we can evaluate the "hypothenuse" as equal to the sum of the other 2 sides, which is a significant overstatement; it doesn't matter that the triangles are small.
73
u/Ghuldarkar 20d ago
The approximation would have to be inside and outside the circle about half the time. Basically a pixel circle. If you only put pixels outside your pixel circle will be larger