Drawing Julia sets

[alexn11]

Describe your change:

Drawing Julia sets

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Documentation change?

Checklist:

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized.
• I know that pull requests will not be merged if they fail the automated tests.
• This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
• All new Python files are placed inside an existing directory.
• All filenames are in all lowercase characters with no spaces or dashes.
• All functions and variable names follow Python naming conventions.
• All function parameters and return values are annotated with Python type hints.
• All functions have doctests that pass the automated testing.
• All new algorithms have a URL in its comments that points to Wikipedia or other similar explanation.
• If this pull request resolves one or more open issues then the commit message contains Fixes: #{$ISSUE_NO}.

[ghost]

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[algobytewise]

Hi alexn11, I couldn't find the images produced by your program neither on the Wikipedia article nor through a short google search. Do you know of such a website to have a visual confirmation that the program does what it is supposed to do?

[alexn11]

Hi algobytewise,
The first two images are of the Cauliflower Julia set: it is visible on this wikipedia article https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension (9th row of 1st table), specifically this image:https://en.wikipedia.org/wiki/File:Julia_z2%2B0,25.png)
For the two other ones, there are similar images: http://www.math.univ-toulouse.fr/~cheritat/GalII/galery.html, or p.4 of this article:https://ddd.uab.cat/pub/pubmat/02141493v43n1/02141493v43n1p27.pdf. All these sets are proved to be ambiently homeomorphic. If you think it's better I could change the parameter c to match one of these examples exactly.
Thanks for reviewing my PR.

[algobytewise]

Thanks for the prompt response. Visually it looks fine. Correct me if I'm wrong, but I think for the examples you picked, it's not important to have two plots per example (one for the absolute value & one whether it escapes) because there is not much of a color gradient for the absolute value. Since it's in the directory fractals, you could add one example that looks a little more "fractal-like", for example, something from https://en.wikipedia.org/wiki/Julia_set#Quadratic_polynomials. For them, the two plots should make sense, I guess.

You still need to add the type-hints for mypy. https://github.com/TheAlgorithms/Python/blob/master/CONTRIBUTING.md has some basic information on that.

By the way, I get the following warnings:
julia_sets.py:39: RuntimeWarning: overflow encountered in multiply return z * z + c
julia_sets.py:39: RuntimeWarning: invalid value encountered in multiply return z * z + c
julia_sets.py:27: RuntimeWarning: overflow encountered in exp return numpy.exp(z) + c
julia_sets.py:27: RuntimeWarning: invalid value encountered in exp return numpy.exp(z) + c

[alexn11]

Thanks a lot, your remarks are very helpful.

[algobytewise]

I think the type is not "numpy.array" but "numpy.ndarray".

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[algobytewise]

Thanks for implementing all the changes. I'm not sure that it's a good idea to just hide the RuntimeWarnings. I haven't looked at the code in detail, but I imagine that having an overflow might make the difference whether a pixel is categorized as above or below the escape_radius, resulting in a bug. Maybe that's the reason why the last example is dotted while the example on the website has solid colored sections.

An easy solution might be to change the array-type to 64 bit, see https://stackoverflow.com/questions/7559595/python-runtimewarning-overflow-encountered-in-long-scalars. Maybe reducing the number of iterations might also do the trick. Otherwise you may have to implement a check in the code whether the numbers are getting too big and maybe stop the iteration-process for the concerned values.

I'm also getting the following errors from flake-8 test. I guess some of the comment-lines are too long & need to be separated into several lines.
julia_sets.py:3:89: E501 line too long (388 > 88 characters)
julia_sets.py:6:89: E501 line too long (92 > 88 characters)
julia_sets.py:8:89: E501 line too long (208 > 88 characters)
julia_sets.py:12:1: F401 'typing.NewType' imported but unused
julia_sets.py:12:1: F401 'typing.Union' imported but unused
julia_sets.py:65:89: E501 line too long (108 > 88 characters)

[alexn11]

Yes you're right for the dotted graph. I've made a new version that replace nans and infinites with a large value. Reducing the number of iterations would decrease the quality of the output. I was looking for a way to have numpy replace overflows etc with a specific value without raising the warning (while continuing array computations) but couldn't find any. Given the way the loop is implemented, the runtime warnings are unavoidable. I think this is fine as long as we are aware of why these warnings occur.

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[algobytewise]

Your solution is definitely a progress but I'm not sure that it fully solves the problem. Have a look at the following example:

z_0 = numpy.array([[2, 3]])
z_final = iterate_function(
	eval_quadratic_polynomial,
	0,
	10,
	z_0,
)

The final value is [[ 0 -1876701183]], which does not make much sense: both values should be really high or infinite, not 0 or negative.
(my mistake, see below)

[algobytewise]

One solution that comes to my mind is to apply the eval_function not to the array as a whole but to individual values. While iterating you could check whether the value crosses the escape radius and then break the loop for this value. This way you should never get close to overflow.

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[poyea]

@alexn11 Please take a look:

FAILED fractals/julia_sets.py::julia_sets.eval_quadratic_polynomial
FAILED fractals/julia_sets.py::julia_sets.iterate_function

[alexn11]

@alexn11 Please take a look:

FAILED fractals/julia_sets.py::julia_sets.eval_quadratic_polynomial
FAILED fractals/julia_sets.py::julia_sets.iterate_function

Thanks for pointing these out. There was some typing issue. Should be resolved now.

[poyea]

@algorithms-keeper review-all

[poyea]

Thank you for your pull request!🤩Looks good to me. While we're waiting for the final checks, feel free to improve anything you want.

[alexn11]

Thank you for your pull request!star_struckLooks good to me. While we're waiting for the final checks, feel free to improve anything you want.

Great! Thank you very much for the edits!