some plots for Imre Bárány about Victor Grinberg's balancing game

balancing_game.py

# https://users.renyi.hu/~geza/GeoSzem/imre24

import numpy as np
import matplotlib.pyplot as plt

def F(a, g):
    numerator = np.sin(g)**2 * ((a + 1)**2 * np.tan(g)**2 + a**2)
    denominator = (a + 1)**2 - (a + 1) * np.sin(g)
    return numerator / denominator


N = 1000
a = np.linspace(0, 10, N + 1)
# g = np.linspace(0.7966, 0.7968, N)
g = np.linspace(0.0, 0.85, N)

A, G = np.meshgrid(a, g)


# I don't really understand why .T is necessary, it's to make f_value = f[a_index, g_index] true.
f = F(A, G).T
print(f.shape)


agg = np.mean(f >= 1, axis=0)
solution = g[agg == 1].min()
print(solution)


a_values = []
f_values = []
for_every_g_argmin_of_f_over_a = np.argmin(f, axis=0)

assert len(g) == len(for_every_g_argmin_of_f_over_a)
for g_index, (g_value, a_index) in enumerate(zip(g, for_every_g_argmin_of_f_over_a)):
    a_value = a[a_index]
    f_value = f[a_index, g_index]
    a_values.append(a_value)
    f_values.append(f_value)

assert np.allclose(np.array(a_values), a[for_every_g_argmin_of_f_over_a])


fig, ax1 = plt.subplots()
plt.xlabel("$g$")

ax1.plot(g, a_values, label="optimal $a$ for each $g$", c="r")
ax1.set_ylabel("$a$")

ax2 = ax1.twinx()

ax2.plot(g, f_values, label="minimal $F(a, g)$ for each $g$", c="b")
ax2.set_ylabel("$F(a, g)$")
ax1.legend(loc="upper left")
ax2.legend(loc="center left")


# line showing where F'(g) = min_a F(a, g) goes above 1.
ax2.plot([solution, solution], [min(f_values), max(f_values)], c="black")
plt.show()



N = 1000
a = np.linspace(0, 10, N + 1)
g = np.linspace(0.6, 0.9, N)

A, G = np.meshgrid(a, g)

f = F(A, G)


plt.rcParams['text.usetex'] = True


plt.figure(figsize=(8, 6))
plt.pcolormesh(A, G, (f >= 1).astype(int), shading='auto', cmap='viridis')
plt.xlabel("$a$", fontsize=20)
plt.ylabel("$g$", fontsize=20)
plt.title(r"Yellow means $F(a, g) \ge 1$.", fontsize=20)
plt.show()