Indras_pearls_Fig12_1.py

import collections
import mpmath as mp
import matplotlib.pyplot as plt


mp.dps = 100

alpha = beta = 2.2
sq2 = mp.sqrt(2)
gamma = (alpha * beta + mp.sqrt(alpha * alpha * beta * beta + 8 - 4 * (alpha * alpha+beta*beta)) )/ 2
delta = ((gamma - 2) * (beta + sq2)) / (beta * gamma - 2 * alpha + sq2 * 1j * gamma)
a11 = alpha / 2
a12 = (alpha * gamma - 2 * beta + 2j * sq2) / ((2 * gamma + 4) * delta)
a21 = (alpha * gamma - 2 * beta - 2j * sq2) * delta / (2 * gamma - 4)
a22 = alpha / 2

a = mp.matrix([[a11, a12], [a21, a22]])

b11 = (beta - sq2 * 1j) / 2
b12 = (beta * gamma - 2 * alpha - sq2 * 1j * gamma) / ((2 * gamma + 4) * delta)
b21 = (beta * gamma - 2 * alpha + sq2 * 1j * gamma) * delta / (2 * gamma - 4)
b22 = (beta + sq2 * 1j) / 2
b = mp.matrix([[b11, b12], [b21, b22]])

A = mp.matrix([[a22, -a12], [-a21, a11]])
B = mp.matrix([[b22, -b12], [-b21, b11]])
I = mp.matrix([[1, 0], [0, 1]])

table = {
    "I": ["a", "b", "A", "B"],
    "a": ["a", "ab", None, "B"],
    "b": ["ba", "b", "bA", None],
    "A": [None, "b", "A", "AB"],
    "B": ["Ba", None, "BA", "B"],
    "ab": ["ba", "b", "abA", None],
    "AB": ["ABa", None, "BA", "B"],
    "ba": ["a", "ab", None, "baB"],
    "bA": [None, "b", "A", "bAB"],
    "Ba": ["a", "Bab", None, "B"],
    "BA": [None, "BAb", "A", "B"],
    "abA": [None, "b", "A", "abAB"],
    "ABa": ["a", "ABab", None, "B"],
    "baB": ["Ba", None, None, "B"],
    "bAB": [None, None, "BA", "B"],
    "Bab": ["ba", "b", None, None],
    "BAb": [None, "b", "bA", None],
    "abAB": [None, None, "BA", "B"],
    "ABab": ["ba", "b", None, None],
}

keys = list(table.keys())

def build_dict(li):
    d = {}
    for index, symbol in enumerate(["a", "b", "A", "B"]):
        pattern = li[index]
        if pattern is not None:
            d[symbol] = keys.index(pattern)
    return d


AUTOMATON = {i: build_dict(table[keys[i]]) for i in range(len(keys))}
print(AUTOMATON)


def generate_group_elements(max_len):
    mats = []
    queue = collections.deque([("", 0, I)])
    while queue:
        word, state, M = queue.popleft()
        mats.append(M)

        if len(word) < max_len:
            for symbol, to in AUTOMATON[state].items():
                queue.append(
                    (word + symbol, to, M @ {"a": a, "b": b, "A": A, "B": B}[symbol])
                )
    return mats


mats = generate_group_elements(8)
xmin, xmax = -0.5, 0.5
ymin, ymax = -0.5, 0.5
image_width = 1000
image_height = 1000
super_sampling_level = 1
num_octaves = 4
num_steps = 8
zoom_factor = 10
grid_opacity = 0.5
shading_opacity = 0.16


def iterate(z):
    upper = mp.mpc(0)
    lower = mp.mpc(0)
    for m in mats:
        upper += 1 / ((m[0, 0] * z + m[0, 1]) * (m[1, 0] * z + m[1, 1]) ** 3)
        lower += 1 / (m[1, 0] * z + m[1, 1]) ** 4
    return upper / lower

mp.cplot(
    iterate, re=[-0.4, 0.4], im=[-0.4, 0.4], points=1000000, verbose=True, file="klein.png"
)