What will we cover in this tutorial?
In this tutorial you will learn what the Julia set is and understand how it is calculated. Also, how it translates into colorful images. In the process, we will learn how to utilize vectorization with NumPy arrays to achieve it.
Step 1: Understand the Julia set
Juila set are closely connect to the Mandelbrot set. If you are new to the Mandelbrot set, we recommend you read this tutorial before you proceed, as it will make it easier to understand.
Julia sets can be calculated for a function f. If we consider the function f_c(z) = z^2 + c, for a complex number c, then this function is used in the Mandelbrot set.
Recall the Mandelbrot set is calculated by identifying for a point c whether the function f_c(z) = z^2 + c , for which the sequence f_c(0), f_c(f_c(0)), f_c(f_c(f_c(0))), …., does not diverge.
Said differently, for each point c on the complex plane, if the sequence does not diverge, then that point is in the Mandelbrot set.
The Julia set has c fixed and and calculates the same sequence for z in the complex plane. That is, for each point z in the complex plane if the sequence f_c(0), f_c(f_c(0)), f_c(f_c(f_c(0))), …., does not diverge it is part of the Julia set.
Step 2: Pseudo code for Julia set of non-vectorization computation
The best way to understand is often to see the non-vectorization method to compute the Julia set.
As we consider the function f_c(z) = z^2 + c for our Julia set, we need to choose a complex number for c. Note, that complex number c can be set differently to get another Julia set.
Then each we can iterate over each point z in the complex plane.
c = -0.8 + i*0.34 for x in [-1, 1] do: for y in [-1, 1] do: z = x + i*y N = 0 while absolute(z) < 2 and N < MAX_ITERATIONS: z = z^2 + c set color for x,y to N
This provides beautiful color images of the Julia set.
Step 3: The vectorization computation using NumPy arrays
How does that translate into code using NumPy?
import numpy as np import matplotlib.pyplot as plt def julia_set(c=-0.4 + 0.6j, height=800, width=1000, x=0, y=0, zoom=1, max_iterations=100): # To make navigation easier we calculate these values x_width = 1.5 y_height = 1.5*height/width x_from = x - x_width/zoom x_to = x + x_width/zoom y_from = y - y_height/zoom y_to = y + y_height/zoom # Here the actual algorithm starts x = np.linspace(x_from, x_to, width).reshape((1, width)) y = np.linspace(y_from, y_to, height).reshape((height, 1)) z = x + 1j * y # Initialize z to all zero c = np.full(z.shape, c) # To keep track in which iteration the point diverged div_time = np.zeros(z.shape, dtype=int) # To keep track on which points did not converge so far m = np.full(c.shape, True, dtype=bool) for i in range(max_iterations): z[m] = z[m]**2 + c[m] m[np.abs(z) > 2] = False div_time[m] = i return div_time plt.imshow(julia_set(), cmap='magma') # plt.imshow(julia_set(x=0.125, y=0.125, zoom=10), cmap='magma') # plt.imshow(julia_set(c=-0.8j), cmap='magma') # plt.imshow(julia_set(c=-0.8+0.156j, max_iterations=512), cmap='magma') # plt.imshow(julia_set(c=-0.7269 + 0.1889j, max_iterations=256), cmap='magma') plt.show()
Python for Finance: Unlock Financial Freedom and Build Your Dream Life
Discover the key to financial freedom and secure your dream life with Python for Finance!
Say goodbye to financial anxiety and embrace a future filled with confidence and success. If you’re tired of struggling to pay bills and longing for a life of leisure, it’s time to take action.
Imagine breaking free from that dead-end job and opening doors to endless opportunities. With Python for Finance, you can acquire the invaluable skill of financial analysis that will revolutionize your life.
Make informed investment decisions, unlock the secrets of business financial performance, and maximize your money like never before. Gain the knowledge sought after by companies worldwide and become an indispensable asset in today’s competitive market.
Don’t let your dreams slip away. Master Python for Finance and pave your way to a profitable and fulfilling career. Start building the future you deserve today!
Python for Finance a 21 hours course that teaches investing with Python.
Learn pandas, NumPy, Matplotlib for Financial Analysis & learn how to Automate Value Investing.
“Excellent course for anyone trying to learn coding and investing.” – Lorenzo B.