displaying Mandelbrot set in python using matplotlib.pyplot and numpy

Question

I am trying to get a plot of a Mandelbrot set and having trouble plotting the expected plot.

As I understand, the Mandelbrot set is made up of values c, which would converge if are iterated through the following equation z = z**2 + c. I used the initial value of z = 0.

Initially, I was getting a straight line. I look for solutions online to see where I went wrong. Using the following link in particular, I attempted to improve my code:

https://scipy-lectures.org/intro/numpy/auto_examples/plot_mandelbrot.html

Here is my improved code. I don't really understand the reason of using np.newaxis and why I am plotting the final z values that converge. Am I misunderstanding the definition of the Mandelbrot set?

# initial values 
loop = 50 # number of interations
div = 600 # divisions
# all possible values of c
c = np.linspace(-2,2,div)[:,np.newaxis] + 1j*np.linspace(-2,2,div)[np.newaxis,:] 
z = 0 
for n in range(0,loop):
      z = z**2 + c

plt.rcParams['figure.figsize'] = [12, 7.5]
z = z[abs(z) < 2] # removing z values that diverge 
plt.scatter(z.real, z.imag, color = "black" ) # plotting points
plt.xlabel("Real")
plt.ylabel("i (imaginary)")
plt.xlim(-2,2)
plt.ylim(-1.5,1.5)
plt.savefig("plot.png")
plt.show()

and got the following image, which looks closer to the Mandelbrot set than anything I got so far. But it looks more of a starfish with scattered dots around it. Image

For reference, here is my initial code before improvement:

# initial values 
loop = 50
div = 50
clist = np.linspace(-2,2,div) + 1j*np.linspace(-1.5,1.5,div) # range of c values 
all_results = []

for c in clist: # for each value of c
    z = 0 # starting point
    for a in range(0,loop): 
        negative = 0 # unstable

        z = z**2 + c 

        if np.abs(z) > 2: 
            negative +=1
        if negative > 2: 
            break

    if negative == 0:
        all_results.append([c,"blue"]) #converging
    else:
        all_results.append([c,"black"]) # not converging

Answer 1

Alternatively, with another small change to the code in the question, one can use the values of z to colorize the plot. One can store the value of n where the absolute value of the series becomes larger than 2 (meaning it diverges), and color the points outside the Mandelbrot set with it:

import pylab as plt
import numpy as np
# initial values 
loop = 50 # number of interations
div = 600 # divisions
# all possible values of c
c = np.linspace(-2,2,div)[:,np.newaxis] + 1j*np.linspace(-2,2,div)[np.newaxis,:] 
# array of ones of same dimensions as c
ones = np.ones(np.shape(c), np.int)
# Array that will hold colors for plot, initial value set here will be
# the color of the points in the mandelbrot set, i.e. where the series
# converges.
# For the code below to work, this initial value must at least be 'loop'.
# Here it is loop + 5
color = ones * loop + 5
z = 0
for n in range(0,loop):
      z = z**2 + c
      diverged = np.abs(z)>2
      # Store value of n at which series was detected to diverge.
      # The later the series is detected to diverge, the higher
      # the 'color' value.
      color[diverged] = np.minimum(color[diverged], ones[diverged]*n)

plt.rcParams['figure.figsize'] = [12, 7.5]
# contour plot with real and imaginary parts of c as axes
# and colored according to 'color'
plt.contourf(c.real, c.imag, color)
plt.xlabel("Real($c$)")
plt.ylabel("Imag($c$)")
plt.xlim(-2,2)
plt.ylim(-1.5,1.5)
plt.savefig("plot.png")
plt.show()