Python/fractals/julia_sets.py
Christian Clauss b75a7c77f8
pre-commit autoupdate: pyupgrade v2.34.0 -> v2.37.0 (#6245)
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Python

"""Author Alexandre De Zotti
Draws Julia sets of quadratic polynomials and exponential maps.
More specifically, this iterates the function a fixed number of times
then plots whether the absolute value of the last iterate is greater than
a fixed threshold (named "escape radius"). For the exponential map this is not
really an escape radius but rather a convenient way to approximate the Julia
set with bounded orbits.
The examples presented here are:
- The Cauliflower Julia set, see e.g.
https://en.wikipedia.org/wiki/File:Julia_z2%2B0,25.png
- Other examples from https://en.wikipedia.org/wiki/Julia_set
- An exponential map Julia set, ambiantly homeomorphic to the examples in
http://www.math.univ-toulouse.fr/~cheritat/GalII/galery.html
and
https://ddd.uab.cat/pub/pubmat/02141493v43n1/02141493v43n1p27.pdf
Remark: Some overflow runtime warnings are suppressed. This is because of the
way the iteration loop is implemented, using numpy's efficient computations.
Overflows and infinites are replaced after each step by a large number.
"""
import warnings
from collections.abc import Callable
from typing import Any
import numpy
from matplotlib import pyplot
c_cauliflower = 0.25 + 0.0j
c_polynomial_1 = -0.4 + 0.6j
c_polynomial_2 = -0.1 + 0.651j
c_exponential = -2.0
nb_iterations = 56
window_size = 2.0
nb_pixels = 666
def eval_exponential(c_parameter: complex, z_values: numpy.ndarray) -> numpy.ndarray:
"""
Evaluate $e^z + c$.
>>> eval_exponential(0, 0)
1.0
>>> abs(eval_exponential(1, numpy.pi*1.j)) < 1e-15
True
>>> abs(eval_exponential(1.j, 0)-1-1.j) < 1e-15
True
"""
return numpy.exp(z_values) + c_parameter
def eval_quadratic_polynomial(
c_parameter: complex, z_values: numpy.ndarray
) -> numpy.ndarray:
"""
>>> eval_quadratic_polynomial(0, 2)
4
>>> eval_quadratic_polynomial(-1, 1)
0
>>> round(eval_quadratic_polynomial(1.j, 0).imag)
1
>>> round(eval_quadratic_polynomial(1.j, 0).real)
0
"""
return z_values * z_values + c_parameter
def prepare_grid(window_size: float, nb_pixels: int) -> numpy.ndarray:
"""
Create a grid of complex values of size nb_pixels*nb_pixels with real and
imaginary parts ranging from -window_size to window_size (inclusive).
Returns a numpy array.
>>> prepare_grid(1,3)
array([[-1.-1.j, -1.+0.j, -1.+1.j],
[ 0.-1.j, 0.+0.j, 0.+1.j],
[ 1.-1.j, 1.+0.j, 1.+1.j]])
"""
x = numpy.linspace(-window_size, window_size, nb_pixels)
x = x.reshape((nb_pixels, 1))
y = numpy.linspace(-window_size, window_size, nb_pixels)
y = y.reshape((1, nb_pixels))
return x + 1.0j * y
def iterate_function(
eval_function: Callable[[Any, numpy.ndarray], numpy.ndarray],
function_params: Any,
nb_iterations: int,
z_0: numpy.ndarray,
infinity: float = None,
) -> numpy.ndarray:
"""
Iterate the function "eval_function" exactly nb_iterations times.
The first argument of the function is a parameter which is contained in
function_params. The variable z_0 is an array that contains the initial
values to iterate from.
This function returns the final iterates.
>>> iterate_function(eval_quadratic_polynomial, 0, 3, numpy.array([0,1,2])).shape
(3,)
>>> numpy.round(iterate_function(eval_quadratic_polynomial,
... 0,
... 3,
... numpy.array([0,1,2]))[0])
0j
>>> numpy.round(iterate_function(eval_quadratic_polynomial,
... 0,
... 3,
... numpy.array([0,1,2]))[1])
(1+0j)
>>> numpy.round(iterate_function(eval_quadratic_polynomial,
... 0,
... 3,
... numpy.array([0,1,2]))[2])
(256+0j)
"""
z_n = z_0.astype("complex64")
for i in range(nb_iterations):
z_n = eval_function(function_params, z_n)
if infinity is not None:
numpy.nan_to_num(z_n, copy=False, nan=infinity)
z_n[abs(z_n) == numpy.inf] = infinity
return z_n
def show_results(
function_label: str,
function_params: Any,
escape_radius: float,
z_final: numpy.ndarray,
) -> None:
"""
Plots of whether the absolute value of z_final is greater than
the value of escape_radius. Adds the function_label and function_params to
the title.
>>> show_results('80', 0, 1, numpy.array([[0,1,.5],[.4,2,1.1],[.2,1,1.3]]))
"""
abs_z_final = (abs(z_final)).transpose()
abs_z_final[:, :] = abs_z_final[::-1, :]
pyplot.matshow(abs_z_final < escape_radius)
pyplot.title(f"Julia set of ${function_label}$, $c={function_params}$")
pyplot.show()
def ignore_overflow_warnings() -> None:
"""
Ignore some overflow and invalid value warnings.
>>> ignore_overflow_warnings()
"""
warnings.filterwarnings(
"ignore", category=RuntimeWarning, message="overflow encountered in multiply"
)
warnings.filterwarnings(
"ignore",
category=RuntimeWarning,
message="invalid value encountered in multiply",
)
warnings.filterwarnings(
"ignore", category=RuntimeWarning, message="overflow encountered in absolute"
)
warnings.filterwarnings(
"ignore", category=RuntimeWarning, message="overflow encountered in exp"
)
if __name__ == "__main__":
z_0 = prepare_grid(window_size, nb_pixels)
ignore_overflow_warnings() # See file header for explanations
nb_iterations = 24
escape_radius = 2 * abs(c_cauliflower) + 1
z_final = iterate_function(
eval_quadratic_polynomial,
c_cauliflower,
nb_iterations,
z_0,
infinity=1.1 * escape_radius,
)
show_results("z^2+c", c_cauliflower, escape_radius, z_final)
nb_iterations = 64
escape_radius = 2 * abs(c_polynomial_1) + 1
z_final = iterate_function(
eval_quadratic_polynomial,
c_polynomial_1,
nb_iterations,
z_0,
infinity=1.1 * escape_radius,
)
show_results("z^2+c", c_polynomial_1, escape_radius, z_final)
nb_iterations = 161
escape_radius = 2 * abs(c_polynomial_2) + 1
z_final = iterate_function(
eval_quadratic_polynomial,
c_polynomial_2,
nb_iterations,
z_0,
infinity=1.1 * escape_radius,
)
show_results("z^2+c", c_polynomial_2, escape_radius, z_final)
nb_iterations = 12
escape_radius = 10000.0
z_final = iterate_function(
eval_exponential,
c_exponential,
nb_iterations,
z_0 + 2,
infinity=1.0e10,
)
show_results("e^z+c", c_exponential, escape_radius, z_final)