Happy Pi Day! mouse or arrow keys to move scroll or w/s to zoom (only the y axis scales) a (hold) to show the key The 1 second wait is not really necessary, its just so the graphs have enough data to reach the edge of the screen. The x axis is the number of iterations (every blob is one iteration), and the y axis is the approximation. The methods are (please correct me if I made any mistakes): π = ∫0->1 √(1 - x2) I don't know who discovered this but its obvious, so nobody cares :\ π = ∫0->1 4/(1+x2) I don't know anything about this, or remember where I learnt about it :\ π2/6 = 1 + 1/4 + 1/9 + 1/16 + ... 3blue1brown has a great video on this, it has something to do with Euler, probably :\ π/4 = 1 - 1/3 + 1/5 - 1/7 + ... This is the Leibniz method, but I think Madhava discovered it, I don't know anything about it :\ π = 4∑[n=0->∞] ((-1)^n)/(2n+1)(2n+2)(2n+3) This is the Nilakantha series but for some reason wikipedia doesn't have an article on it :\ it’s the one used to calculate pi in the bottom right corner. The Monte Carlo method is to pick random points on a square x = -1 to 1 and y = -1 to 1 and check if they are in the unit circle, and use the ratio of points in the unit circle and total points to get π/4 The orange, dark blue, light green and white methods, were invented entirely by my own brain derived from this method: π/4 = 1 -1/3 +1/5 -1/7 +1/9 ... it doesn't converge quickly but it jumps around pi very regularly, so my first method (the orange one) is the average of consecutive iterations. That is also an alternating series which jumps around pi, and my second method (dark blue) is the average of consecutive iterations of that method. There is also a light green one which is the average of consecutive terms of that average of consecutive terms of that average of consecutive terms. If you do this z times, you get the white equation, where the parameter z (set to 10) adjusts how many time you look at the average of the average of the average ... 1 - 1/3 + 1/5 - 1/7 + ... I came up with this myself, but I don't know if anyone else has already, if you have already heard of this process please tell me. the grey line is pi to 3288 decimal places from: https://www.piday.org/million/ (I could have use 1,000,000 but even 3288 is extremely unnecessary, 30 would be enough) If you scroll to the right really quickly you will see that the integral methods are much slower to calculate, but that is because it calculates them with many different accuracies, but unlike the other methods you don't need to know the previous value to find a value, so they are actually about as fast. If you zoom in a lot, you will find a little difference between the grey line and the white method I think its scratch's rounding, also movement using the arrow keys will stop working, its because the movement speed is divided by the zoom, which gets so big that scratch just decides that dividing by it gets zero :\ Well done for reading through the whole description, I just kept adding to it so got huge.
