Playing with complex numbers

Hello!

Another essay. This time let's play with complex numbers, with formulas and their visualization.

Idea

A complex number is a certain extension of a real number, in fact a vector for which a whole set of axioms is defined. Any complex (and hence real) number can be written as

z = a + b i

$z = a + bi$ , where a is the real part, b is the imaginary part, i is the root of the equation

x^{2} = - 1

$x ^ {2} = - 1$ . For him, many operations are defined that are defined for a real number, for example,

z_{1} + z_{2} = a_{1} + a_{2} + (b_{1} + b_{2}) i

$z_1 + z_2 = a_1 + a_2 + (b_1 + b_2) i$ . Interestingly, if you do various operations with them, raise to a power, multiply, etc., and then take

R e (z)

$Re (z)$ (real part) for the axis Ox, and

I m (z)

$Im (z)$ (imaginary part) for the Oy axis, you can get funny pictures.

By the way, I myself came up with all the following formulas.

Visualization function

Routine. The function, which according to this iterative function draws everything on the field:

import random import numpy as np def vis(A, f, step=1.0, c=None): x = [] y = [] for B in np.arange(0, A, step): v = f(A, B) x.append(v.real) y.append(v.imag) plt.figure(figsize=[8, 8]) mxabs = max([i[0] ** 2 + i[1] ** 2 for i in zip(x, y)]) ** 0.5 x = np.array(x) / mxabs y = np.array(y) / mxabs if c is None: plt.scatter(x, y) else: plt.scatter(x, y, color=[c(x[i], y[i]) for i in range(len(x))]) plt.show()

All our functions depend on two parameters A and B. Moreover, we iterate over B inside vis (), and A is the global parameter of the function.

The Curl Function

f (A, B) = B s i n (B) * e^{i B c o s (A)}

$f (A, B) = Bsin (B) * e ^ {iBcos (A)}$

Her declaration in python:

 def func_1(A, B): return math.sin(B) * B * math.e ** (1j * (B * math.cos(A)))

And run

 A = 121.5 vis(A, func_1, step=0.1)

And the result for A = 121.5:

And for A = 221.5:

Note that these numbers do not follow from the calculation of any definite integral on a smooth manifold and other smart words meaningless in this context. These are really random numbers, and there is still exactly infinity of different A, as a result of which beauty is obtained.

Need to paint

We declare a color function (such a function that returns a tuple of three numbers in coordinates):

 def sigm(x): #         0  1 return (1 / (1 + 1.2 ** (-x*50)) - 0.5) * 2 color_1 = lambda x, y: (0.2, sigm(x ** 2 + y ** 2) / 1.4, 1 - sigm(x ** 2 + y ** 2)) color_2 = lambda x, y: (sigm(x ** 2 + y ** 2), 0.5, 0.5)

Choose random parameter A, let it be 149:

 vis(149, func_1, step=0.1, c=color_1)

Geese Function

Geese are described as follows:

f (A, B) = c o s (B) s i n (B) * B * e^{i B c o s (A)}

$f (A, B) = cos (B) sin (B) * B * e ^ {iBcos (A)}$

Python declaration:

 def func_2(A, B): return math.cos(B) * math.sin(B) * B * math.e ** (1j * (B * math.cos(A)))

Its result for A = 106:

Focaccia function

f (A, B) = c o s (B (A + 1)) * e^{i B c o s (A)}

$f (A, B) = cos (B (A + 1)) * e ^ {iBcos (A)}$

 def func_3(A, B): return math.cos((A + 1) * B) * math.e ** (1j * (B * math.cos(A)))

 vis(246, func_3, step=0.1, c=color_2)

 vis(246, func_3, step=0.1, c=color_2)

Untitled Function

f (A, B) = B s i n (A + B) * e^{i B s i n (A)}

$f (A, B) = Bsin (A + B) * e ^ {iBsin (A)}$

 color_3 = lambda x, y: (0.5, 0.5, sigm(x ** 2 + y ** 2)) vis(162, func_4, step=0.1, c=color_3)

 vis(179, func_4, step=0.1, c=color_3)

Beauty formula

f (A, B) = c o s (B (A + 1))^{f r a c 32} * e^{i B c o s (A)}

$f (A, B) = cos (B (A + 1)) ^ {\ frac {3} {2}} * e ^ {iBcos (A)}$

 def func_5(A, B): return math.cos((A + 1) * B) ** 1.5 * math.e ** (1j * (B * math.cos(A)))

 color_4 = lambda x, y: (sigm(x ** 2 + y ** 2) / 2, 0.5, 1 - sigm(x ** 2 + y ** 2)) vis(345, func_5, step=0.1, c=color_4)

 vis(350, func_5, step=0.1, c=color_4)

That's all for now.

Whole code

 import numpy as np import random import matplotlib.pyplot as plt import math def vis(A, f, step=1.0, c=None): x = [] y = [] for B in np.arange(0, A, step): v = f(A, B) x.append(v.real) y.append(v.imag) plt.figure(figsize=[7, 7]) mxabs = max([i[0] ** 2 + i[1] ** 2 for i in zip(x, y)]) ** 0.5 x = np.array(x) / mxabs y = np.array(y) / mxabs if c is None: plt.scatter(x, y) else: plt.scatter(x, y, color=[c(x[i], y[i]) for i in range(len(x))]) plt.show() def func_1(A, B): return math.sin(B) * B * math.e ** (1j * (B * math.cos(A))) def func_2(A, B): return math.cos(B) * math.sin(B) * B * math.e ** (1j * (B * math.cos(A))) def func_3(A, B): return math.cos((A + 1) * B) * math.e ** (1j * (B * math.cos(A))) def func_4(A, B): return math.sin(A + B) * B * math.e ** (1j * B * math.sin(A)) def func_5(A, B): return math.cos((A + 1) * B) ** 1.5 * math.e ** (1j * (B * math.cos(A))) def sigm(x): return (1 / (1 + 1.2 ** (-x*50)) - 0.5) * 2 color_1 = lambda x, y: (0.2, sigm(x ** 2 + y ** 2) / 1.4, 1 - sigm(x ** 2 + y ** 2)) color_2 = lambda x, y: (sigm(x ** 2 + y ** 2), 0.5, 0.5) color_3 = lambda x, y: (0.5, 0.5, sigm(x ** 2 + y ** 2)) color_4 = lambda x, y: (sigm(x ** 2 + y ** 2) / 2, 0.5, 1 - sigm(x ** 2 + y ** 2)) colors = [color_1, color_2, color_3, color_4] funcs = [func_1, func_2, func_3, func_4, func_5] while True: col = random.choice(colors) func = random.choice(funcs) vis(random.random() * 200 + 100, func, step=0.1, c=col) if input() == "exit": break

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