fractals.julia_sets

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 https://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.

Attributes

c_cauliflower
c_exponential
c_polynomial_1
c_polynomial_2
nb_iterations
nb_pixels
window_size
z_0

Functions

eval_exponential(→ numpy.ndarray)	Evaluate $e^z + c$.
eval_quadratic_polynomial(→ numpy.ndarray)
ignore_overflow_warnings(→ None)	Ignore some overflow and invalid value warnings.
iterate_function(→ numpy.ndarray)	Iterate the function "eval_function" exactly nb_iterations times.
prepare_grid(→ numpy.ndarray)	Create a grid of complex values of size nb_pixels*nb_pixels with real and
show_results(→ None)	Plots of whether the absolute value of z_final is greater than

Module Contents

fractals.julia_sets.eval_exponential(c_parameter: complex, z_values: numpy.ndarray) → numpy.ndarray

Evaluate $e^z + c$. >>> float(eval_exponential(0, 0)) 1.0 >>> bool(abs(eval_exponential(1, np.pi*1.j)) < 1e-15) True >>> bool(abs(eval_exponential(1.j, 0)-1-1.j) < 1e-15) True

fractals.julia_sets.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

fractals.julia_sets.ignore_overflow_warnings() → None

Ignore some overflow and invalid value warnings.

>>> ignore_overflow_warnings()

fractals.julia_sets.iterate_function(eval_function: collections.abc.Callable[[Any, numpy.ndarray], numpy.ndarray], function_params: Any, nb_iterations: int, z_0: numpy.ndarray, infinity: float | None = 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, np.array([0,1,2])).shape
(3,)
>>> complex(np.round(iterate_function(eval_quadratic_polynomial,
... 0,
... 3,
... np.array([0,1,2]))[0]))
0j
>>> complex(np.round(iterate_function(eval_quadratic_polynomial,
... 0,
... 3,
... np.array([0,1,2]))[1]))
(1+0j)
>>> complex(np.round(iterate_function(eval_quadratic_polynomial,
... 0,
... 3,
... np.array([0,1,2]))[2]))
(256+0j)

fractals.julia_sets.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]])

fractals.julia_sets.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, np.array([[0,1,.5],[.4,2,1.1],[.2,1,1.3]]))

fractals.julia_sets.c_cauliflower

fractals.julia_sets.c_exponential

fractals.julia_sets.c_polynomial_1

fractals.julia_sets.c_polynomial_2

fractals.julia_sets.nb_iterations = 56

fractals.julia_sets.nb_pixels = 666

fractals.julia_sets.window_size = 2.0

fractals.julia_sets.z_0