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Visualizing the Möbius Strip in Python…

The formulation of the Möbius strip involves a parameterization in terms of two variables: \(\theta\), which is an angle that varies from \(0\) to \(2 \pi\), and \(w\), which is a distance from the centerline of the strip and varies between -1 and 1.

Given these parameters, the \((x, y, z)\) coordinates for points on the Möbius strip can be determined using the following equations:

import numpy as np
import matplotlib.pyplot as plt

# Create a Möbius strip
theta = np.linspace(0, 2 * np.pi, 100)
w = np.linspace(-1, 1, 20)
theta, w = np.meshgrid(theta, w)
phi = 0.5 * theta
r = 1 + w * np.cos(phi)
x = np.cos(theta) * r
y = np.sin(theta) * r
z = w * np.sin(phi)

# Plot the Möbius strip
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(x, y, z, edgecolor='k', color='c')
ax.view_init(50, 30)  # Adjust view angle for better visualization
ax.set_title('Möbius Strip in 3D')

Visualization Strategy:

  1. Parameter Space Definition: We begin by defining our parameter space for θ and w. The variable θ spans the entire 360-degree range of the strip, while w varies between -1 and 1 to account for the width of the strip.
  2. Meshgrid Creation: Using NumPy’s meshgrid function, we create a grid of points in the (θ, w) parameter space. Each combination of (θ, w) corresponds to a unique point on the Möbius strip.
  3. Position Calculation: For every combination of (θ, w), we compute the x, y, and z coordinates using the above equations.
  4. 3D Plotting: Using Matplotlib’s 3D plotting capabilities, we then plot the computed (x, y, z) coordinates as a surface. The edgecolor is set to ‘k’ (black) to distinguish between individual segments on the strip, and color is set to ‘c’ (cyan) for the surface color.
  5. View Angle Adjustment: The line ax.view_init(50, 30) adjusts the view angle of the 3D plot, making it easier to perceive the Möbius strip’s twist.
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