Mandelbrot chapter

book/Maths/Mandelbrot/.ipynb_checkpoints/Mandelbrot-checkpoint.md

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+jupytext:
+  formats: md:myst
+  text_representation:
+    extension: .md
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+  display_name: Python 3 (ipykernel)
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+  name: python3
+---
+
+# Mandelbrot set
+
+<img src="https://quotepark.com/media/authors/benoit-mandelbrot.detail.jpeg" width="256px" align="left" margin="16px /">
+
+The [Mandelbrot set](https://en.wikipedia.org/wiki/Mandelbrot_set) is one of these iconic abstract object popular outside of mathematics for its aesthetic appeal.
+Along with LOL cats, YouTube is filled with psychedelic videos deep diving into the incredibly complicated fractal structure of the Mandelbrot set.
+Like many other fractal objects, the Mandelbrot set is often used in [Generative Arts](https://en.wikipedia.org/wiki/Generative_art), see for instance [Olbaid-ST](https://www.deviantart.com/olbaid-st/gallery/36049300/ultrafractal).
+
+Despite what its name suggests, the Mandelbrot set was not discovered by [Benoît Mandelbrot](https://en.wikipedia.org/wiki/Benoit_Mandelbrot), a Polish-born French-American mathematician.
+It finds its origin in the field of **complex dynamics**, first studied by [Pierre Fatou](https://en.wikipedia.org/wiki/Pierre_Fatou) and [Gaston Julia](https://en.wikipedia.org/wiki/Gaston_Julia) in the early 1900s.
+It was properly defined in 1978 by [Robert W. Brooks](https://en.wikipedia.org/wiki/Robert_W._Brooks) and Peter Matelski.
+They were also the firsts to publish a computer-generated visualization of the Mandelbrot set, shown below.
+[Adrien Douaby](https://en.wikipedia.org/wiki/Adrien_Douady) and [John H. Hubbard](https://en.wikipedia.org/wiki/John_H._Hubbard) were the first to really study the mathematical properties of the Mandelbrot set.
+They are also the ones who named the set in honor of Mandelbrot for his influential work in fractal geometry.
+
+<img src="https://upload.wikimedia.org/wikipedia/commons/d/d7/Mandel.png" width="512px" align="center" margin="16px" />
+
+**<center>The first published picture of the Mandelbrot set, by R. Brooks & P. Matelski in 1978. From [Wikipedia](https://en.wikipedia.org/wiki/Mandelbrot_set).</center>**
+
+Today, even a fairly standard laptop has enough computational power to generate visualizations of the Mandelbrot set with far more details than the 1978 image.
+Mathematicians and artists alike regularly come up with new ways to render this incredible mathematical object.
+One such example is shown below.
+
+```{code-cell} ipython3
+---
+render:
+  image:
+    align: center
+tags: [remove-input]
+---
+import numpy as np
+import numba as nb
+import matplotlib
+import matplotlib.pyplot as plt
+from matplotlib import colors
+import time
+
+# --> Compute the Mandelbrot set.
+@nb.jit(nopython=True)
+def mandelbrot_kernel(c, maxiter, horizon):
+    z = 0
+    for i in range(maxiter):
+        z = z**2 + c
+        
+        if np.abs(z) > horizon:
+            return z, i
+
+    return z, 0
+
+@np.vectorize
+def mandelbrot(c, maxiter=1000, horizon=2):
+    return mandelbrot_kernel(c, maxiter, horizon)
+
+# --> Mesh the complex plane.
+cr = np.linspace(-2.25, 0.75, 3000)
+ci = np.linspace(-1.25, 1.25, 2500)
+c = cr[:, None] + 1j*ci[None, :]
+
+# --> Computation.
+maxiter, horizon = 1000, 2**20
+z, n = mandelbrot(c, maxiter, horizon)
+
+# --> Render the Mandelbrot set.
+log_horizon = np.log2(np.log(horizon))
+with np.errstate(invalid="ignore"):
+    M = np.nan_to_num(n + 1 - np.log2(np.log(np.abs(z))) + log_horizon)
+    
+fig, ax = plt.subplots(1, 1, figsize=(8, 8))
+
+light = colors.LightSource(azdeg=315, altdeg=10)
+M = light.shade(M.T, cmap=plt.cm.gray, vert_exag=1.5, norm=colors.PowerNorm(0.3), blend_mode="hsv")
+
+ax.imshow(M, extent=[cr.min(), cr.max(), ci.min(), ci.max()], interpolation="bicubic")
+
+ax.axis(False)
+
+year = time.strftime("%Y")
+text = ("The Mandelbrot fractal set \n" "Rendered with matploblib %s, %s - https://matplotlib.org" % (matplotlib.__version__, year))
+ax.text(cr.min() + 0.025, ci.min() + 0.025, text, color="white", fontsize=12)
+```
+
+```{code-cell} ipython3
+
+```