File:Daffodil lemma proof visualisation.png
Summary
| Description |
English: First five terms of the power series expansion of for as steps in a walk, with stride length and direction of step . This is used to visualise the proof of the Daffodil Lemma. |
| Date | |
| Source | Own work |
| Author | Dom walden |
| PNG development | |
| Source code | Python codeimport matplotlib.pyplot as plt
import numpy as np
def data(z, theta):
gf = np.array([z**2, z**7, z**12, z**22, 2*z**32, z**42])
gf_acc = np.array([0, z**2, z**2 + z**7, z**2 + z**7 + z**12, z**2 + z**7 + z**12 + z**22, z**2 + z**7 + z**12 + z**22 + 2*z**32])
x = np.real(gf_acc)
y = np.imag(gf_acc)
u = np.real(gf)
v = np.imag(gf)
return x, y, u, v
fig, ax = plt.subplots()
theta = 0/5 * 2 * np.pi
z = 0.9 * np.exp(1j * theta)
x, y, u, v = data(z, theta)
ax.quiver(x, y, u, v, angles='xy', scale=4, label='$\\theta = 0$')
theta = 1/5 * 2 * np.pi
z = 0.9 * np.exp(1j * theta)
x, y, u, v = data(z, theta)
ax.quiver(x, y, u, v, color='blue', angles='xy', scale=4, label='$\\theta = \\frac{2\\pi}{5}$')
theta = 1/3 * 2 * np.pi
z = 0.9 * np.exp(1j * theta)
x, y, u, v = data(z, theta)
ax.quiver(x, y, u, v, color='red', angles='xy', scale=4, label='$\\theta = \\frac{2\\pi}{3}$')
circle = plt.Circle((0, 0), 1.757521632206236, fill=False)
ax.add_patch(circle)
ax.set(xlim=(-2, 2), ylim=(-2, 2))
ax.legend()
ax.set_aspect('equal', adjustable='box')
plt.tight_layout()
plt.savefig("daffodil_lemma_proof_visualisation.png")
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