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Learning Opportunities

This puzzle can be solved using the following concepts. Practice using these concepts and improve your skills.



The Mandelbrot set is the set of all complex numbers c that do not diverge (don't get bigger and bigger) when you repeatedly apply the function:

z(n+1) = z(n)^2 + c

with z(0) = 0.
For example, if c = 1 then z(0)=0, z(1) = 0^2+1 = 1, z(2) = 1^2 + 1 = 2, z(3) = 2^2 + 1 = 5, etc. - so for c = 1, it diverges (it gets bigger each time).

c can be any complex number, which can be simplified as a 2-column vector [x,y] where 'x' is the 'real' part and 'y' is the 'imaginary' part. [x,y]^2 is defined as [x^2-y^2, 2*x*y] (if your coding language doesn't natively support complex arithmetic).

Given an input n which is the number of rows of output to give in your answer, with the number of columns being 1.5*(n-1)+1, each having the same step size - step = 2/(n-1). Approximate the Mandelbrot set by checking whether each point on the 2D region from x = -2 to +1 and y = -1 to +1 is still 'small' after 10 iterations of the function. 'Small' is defined here by whether the absolute value of z(10) (the 10th iteration) is <= 1.

e.g. for c = [0.5,0.5], z(0) = [0,0], z(1) = [0.5,0.5],
z(2) = [0.5,0.5]^2+[0.5,0.5] = [0.5^2-0.5^2, 2*0.5*0.5]+[0.5,0.5] = [0,0.5]+[0.5,0.5] = [0.5,1]
z(3) = ...
z(10) ≈ [9.746656569875421e+17,1.5363420963186726e+17i]
|z(10)| ≈ 9.866998597750816e+17 <- definitely not in the set

Print the 2D grid with n evenly spaced points in -1≤y≤1 and 1.5*(n-1)+1 points in -2≤x≤1 and display points that diverge as a space ' ' and points in the set as a '*'. (Leave trailing spaces so the grid is a square)

Note: the step size will be the same in x and y.
Note: the absolute value of a complex number is the square root of the sum of the two components squared √(x^2+y^2).
n: the (integer) number of rows for your 2D grid, number of columns = 3*(n-1)/2+1.
n lines: the approximated Mandelbrot set with points in the set shown as '*' and points that diverged shown as a space ( ).
2 ≤ n ≤ 25 (25 rows and 37 columns at most)
n is always odd

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