Sort of beyond that.

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Different Resolution
Yielding Different Results**

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Don't ask my why the 500px blurs out slightly ahead of the 1000px series. Probably just luck of the draw (e.g. where the last significant digit lands).

All those colourful pictures of the Mandelbrot Set? They're based on what pixels tend toward infinity (are greater than 2) at each level of iteration.

This is the base iteration:

return x*x + c

So, like a sculptor, higher resolutions narrow the field.

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And then, as much as I'd like to push 10,000px images to the web, as of 2015, modern browsers just aren't up to the job yet. Give 'em a year or two.

Code be code.

And this be the code.

#Copyright Brett Paufler #Released Under the Exclusive #For Information Purposes ONLY (ALL RIGHTS RESERVED) License import numpy as np #because that's how the cool kids do it import skimage.io def base_iter(x,c): '''This is the Heart of solving the Mandelbrot Set Full numpy arrays will be fed into this function ''' return x*x + c def main_iter(c,numIt=50): '''Returns the Mandelbrot Set c is a numpy array of complex numbers defined over the area of interest numIt is the number of iterations, more, better but there is a point of diminishing returns ''' #This x (and this c) are conceptually the same, # as per base_iter above x = np.zeros_like(c) for _ in range(numIt): x = base_iter(x, c) #If the absolute value of (a+bi) ever exceeds 2 #The point is not part of the Mandelbrot Set #And tends toward infinity x = abs(x) x[x>2] = np.nan #Most of these values are already np.nan x[x<=2] = 1 x[np.isnan(x)] = 0 return x def i_coord(s, (uL,lR)): '''returns a complex coordinate grid (aka c in the above) size = (s,s) = s high, s wide min_max is passed in a tuple of two complex numbers uL is the min (Upper Left) lR is the max (Lower Right) Odd fact, I am right/left dyslexic Often heard around my house, 'No! Your other right!' ''' #real part, as a row r = np.linspace(np.real(uL), np.real(lR), s, dtype=complex) #imaginary part, as a column im = np.linspace(np.imag(uL), np.imag(lR), s, dtype=complex) im = np.reshape((im * 1j), (s,1)) #turns into a column, essentially #add a row and a column together in Numpy #get a Matrix, row wide, column high return (r+im) def next_coor(s, mandebrot, c): '''Returns the area of 'Most Interest' in the Mandelbrot Array Most interest being defined as relatively even numbers of 0's & 1's s = size (s,s) of image out, as it does everywhere else mandebrot (misspelled, of course) is the picture image array c is the starting complex array (for this go round, this image) ''' #t being the value of 50-50 0's & 1's t = s*s/8.0 m = [] #This is messy #It's the type of hell that's usually easier for me # to rewrite than debug m.append((abs(t - sum(sum(mandebrot[:s/2, :s/2]))), (np.min(c[:s/2, :s/2]), np.max(c[:s/2, :s/2]) ))) m.append((abs(t - sum(sum(mandebrot[:s/2, s/2:]))), (np.min(c[:s/2, s/2:]), np.max(c[:s/2, s/2:]) ))) m.append((abs(t - sum(sum(mandebrot[s/2:, :s/2]))), (np.min(c[s/2:, :s/2]), np.max(c[s/2:, :s/2]) ))) m.append((abs(t - sum(sum(mandebrot[s/2:, s/2:]))), (np.min(c[s/2:, s/2:]), np.max(c[s/2:, s/2:]) ))) #sort, take the first (lowest t-sum), # return the second part of the tuple m.sort() _, (min_max) = m[0] return min_max def auto_zoom(s, n, min_max=(-2.0 +2.0j, 2.0 -2.0j)): '''Successive Mandelbrot's with Magnification s = image size n = number of successive images (blurs out at 20 or so min_max is the starting area of interest for a bunch of size options (20 images each) for s in [100, 250, 500, 1000]: min_max = (-2.0 +2.0j, 2.0 -2.0j) auto_zoom(s,20,min_max) ''' #1 loop per image out for n in range(1,n): print "\nStarting Loop" print "Min Max: %d, %r" % (s, min_max) c = i_coord(s, min_max) #c array for current coordinates mandebrot = main_iter(c) #the main computation sN = "mandebrot_{:0>4}_{:0>4}.png".format(s,n) print "Saving %s" % sN skimage.io.imsave(sN, mandebrot) #save the image min_max = next_coor(s, mandebrot, c) #reset coordinates def main_seq_picture_iter(s, numPics=12, min_max=(-2.0 +2.0j, 2.0 -2.0j), intermediate=True): '''saves a series of images, whereas autozoom focuses in (drills ever deeper) into the Mandelbrot Set, This function outputs a series of ever refined images focusing on a single area of the set s = size of pictures to save numPics = number of sequential images (iterations, refinement of the Mandelbrot set) min_max is tuple of area of interest so by manually (or by other means) the images may be obtained for any section of the set intermediate = whether to save the 'part way' image progressions (rather than just the final product) Probably rather than a function call, one should think of this as a script. It solves a problem (pictures for this webpage). But to solve another problem, it would need to be rewritten or tweaked (and not just passed different parameters) ''' #c = array of complex numbers #target of Mandelbrot image area c = i_coord(s, min_max) #g = for the compound image out # green and grayscale g = np.zeros((s,s)) #x is the incremented array # the increased resolution copy of the Mandelbrot set # as per other functions above (same x, essentially) x = np.zeros_like(c) for n in range(1, numPics + 1): x = base_iter(x, c) #p = binary (black/white) (0,1) # color scale image for each iteration # Note the logic, similiar to main_iter() p = x.copy() p = abs(p) p[p>2] = np.nan p[p<=2] = 1 p[np.isnan(p)] = 0 #increments the soft output, grayscale array # Since this is an image effect # p*n, p+n, p by itself, whatever g += p * n if intermediate: #save a PNG sN = "mandebrot_resolve_{:0>4}_{:0>4}.png".format(s,n) skimage.io.imsave(sN,p) print sN #After the Loop, Output some Compound Images z = np.zeros_like(g, dtype=int) o = np.ones_like(g, dtype=int) * 255 sN = "mandebrot_full_resolve.png" #white circle, gray Mandelbrot skimage.io.imsave(sN,g) #one layer, so grayscale sN = "mandebrot_full_resolve_green.png" #green circle skimage.io.imsave(sN,np.dstack((z,g,z,g))) #(R,G,B, Alpha) g = g / np.max(g) * 255 print np.max(g), np.min(g) g = np.array(g, dtype=int) sN = "mandebrot_full_resolve_green_%s.png" % s skimage.io.imsave(sN, np.dstack((z,g,z,o))) #not used print sN sNa = "mandebrot_full_resolve_green_alphatrack_%s.png" % s skimage.io.imsave(sNa, np.dstack((z,g,z,g))) #Green Mandelbrot print sNa return s main_seq_picture_iter(1000, 12) ''' Originally, I had it on my TODO list to do a colourized version of the Mandelbrot Set (Using some sort of compound logic), but I'm not convinced the colours mean anything (are just random), so I've sort of lost my enthusiasm for the project. Still, someday... '''Code as Text File: mandelbrot_numpy.txt

And that's all I have to say about that...

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