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Mandelbrot set

<table class="tfo-notebook-buttons" align="left"> <td> <a target="_blank" href="https://colab.research.google.com/github/tensorflow/docs/blob/master/site/en/r1/tutorials/non-ml/mandelbrot.ipynb">Run in Google Colab</a> </td> <td> <a target="_blank" href="https://github.com/tensorflow/docs/blob/master/site/en/r1/tutorials/non-ml/mandelbrot.ipynb">View source on GitHub</a> </td> </table>

Note: This is an archived TF1 notebook. These are configured to run in TF2's compatibility mode but will run in TF1 as well. To use TF1 in Colab, use the %tensorflow_version 1.x magic.

Visualizing the Mandelbrot set doesn't have anything to do with machine learning, but it makes for a fun example of how one can use TensorFlow for general mathematics. This is actually a pretty naive implementation of the visualization, but it makes the point. (We may end up providing a more elaborate implementation down the line to produce more truly beautiful images.)

Basic setup

You'll need a few imports to get started.

# Import libraries for simulation
import tensorflow.compat.v1 as tf

import numpy as np

# Imports for visualization
import PIL.Image
from io import BytesIO
from IPython.display import clear_output, Image, display

Now you'll define a function to actually display the image once you have iteration counts.

def DisplayFractal(a, fmt='jpeg'):
  """Display an array of iteration counts as a
     colorful picture of a fractal."""
  a_cyclic = (6.28*a/20.0).reshape(list(a.shape)+[1])
  img = np.concatenate([10+20*np.cos(a_cyclic),
                        30+50*np.sin(a_cyclic),
                        155-80*np.cos(a_cyclic)], 2)
  img[a==a.max()] = 0
  a = img
  a = np.uint8(np.clip(a, 0, 255))
  f = BytesIO()
  PIL.Image.fromarray(a).save(f, fmt)
  display(Image(data=f.getvalue()))

Session and variable initialization

For playing around like this, an interactive session is often used, but a regular session would work as well.

sess = tf.InteractiveSession()

It's handy that you can freely mix NumPy and TensorFlow.

# Use NumPy to create a 2D array of complex numbers

Y, X = np.mgrid[-1.3:1.3:0.005, -2:1:0.005]
Z = X+1j*Y

Now you define and initialize TensorFlow tensors.

xs = tf.constant(Z.astype(np.complex64))
zs = tf.Variable(xs)
ns = tf.Variable(tf.zeros_like(xs, tf.float32))

TensorFlow requires that you explicitly initialize variables before using them.

tf.global_variables_initializer().run()

Defining and running the computation

Now you specify more of the computation...

# Compute the new values of z: z^2 + x
zs_ = zs*zs + xs

# Have we diverged with this new value?
not_diverged = tf.abs(zs_) < 4

# Operation to update the zs and the iteration count.
#
# Note: We keep computing zs after they diverge! This
#       is very wasteful! There are better, if a little
#       less simple, ways to do this.
#
step = tf.group(
  zs.assign(zs_),
  ns.assign_add(tf.cast(not_diverged, tf.float32))
  )

... and run it for a couple hundred steps

for i in range(200): step.run()

Let's see what you've got.

DisplayFractal(ns.eval())

Not bad!