Visualizing the Banach-Tarski Paradox

1. The Banach-Tarski Paradox in a Nutshell

The paradox states:

“A solid ball can be decomposed into a finite number of disjoint subsets, which can then be reassembled (using only rotations and translations) into two identical copies of the original ball.”

Key Ingredients

• Free Group Actions: The paradox relies on the free group of rotations (generated by two independent rotations, say ( a ) and ( b )).
• Non-Measurable Sets: The pieces used in the decomposition do not have a well-defined volume (they are “invisible” to classical geometry).

2. Interpreting the Given Expressions

The expressions you provided seem to describe subsets of the sphere under group actions:

1. $( S(a^{-1}) )$:

• Represents the set of points on the sphere that are mapped backwards under the rotation $( a^{-1} )$.
• Analogous to “all points that move under the inverse rotation.”

2. $( S(a) )$:

• Represents the set of points on the sphere that are mapped forwards under the rotation ( a ).
• Analogous to “all points that move under the rotation ( a ).”

3. $( aS(a^{-1}) )$:

• Represents the set $( S(a^{-1}) )$ rotated by ( a ).
• This is a shifted version of the first set.

Visualization

• Imagine a sphere divided into orbits (paths traced by repeated applications of ( a ) and ( b )).
• The sets $( S(a^{-1}) )$ and ( S(a) ) correspond to different “shifts” of these orbits under rotation.
• The paradoxical decomposition comes from carefully selecting points from these orbits (using the Axiom of Choice) to create two identical spheres.

3. How This Leads to Volume Doubling

1. Decompose the Sphere:

• Split the sphere into 5 pieces based on group actions $(e.g., ( S(a), S(a^{-1}), S(b), S(b^{-1}) )$, and a “fixed” set).
• These pieces are non-measurable (no volume).

2. Reassemble into Two Spheres:

• Rotate ( S(a) ) and $( S(a^{-1}) )$ to form one new sphere.
• Rotate ( S(b) ) and $( S(b^{-1}) )$ to form another new sphere.
• The leftover “fixed” set is negligible (it’s a sparse set).

Result

• You’ve turned one sphere into two of the same size!
• This works because the Axiom of Choice allows us to pick points in a way that “hides” the missing volume.

4. Python Simulation (Conceptual)

Since we can’t truly visualize non-measurable sets, here’s a discrete approximation of how group actions work in Banach-Tarski:

import numpy as np
import matplotlib.pyplot as plt

# Simulate points on a sphere (discrete approximation)
theta = np.linspace(0, 2*np.pi, 100)
phi = np.linspace(0, np.pi, 50)
theta, phi = np.meshgrid(theta, phi)
x = np.sin(phi) * np.cos(theta)
y = np.sin(phi) * np.sin(theta)
z = np.cos(phi)

# Define rotation matrices (simplified)
def rotate_x(points, angle):
    R = np.array([[1, 0, 0],
                  [0, np.cos(angle), -np.sin(angle)],
                  [0, np.sin(angle), np.cos(angle)]])
    return np.dot(points, R.T)

# Apply rotations (a and a^{-1})
points = np.vstack([x.ravel(), y.ravel(), z.ravel()]).T
rotated_a = rotate_x(points, np.pi/4)  # Rotation a
rotated_a_inv = rotate_x(points, -np.pi/4)  # Rotation a^{-1}

# Plot original and rotated sets
fig = plt.figure(figsize=(12, 4))
ax1 = fig.add_subplot(131, projection='3d')
ax1.scatter(*points.T, s=1, alpha=0.3)
ax1.set_title("Original Sphere")

ax2 = fig.add_subplot(132, projection='3d')
ax2.scatter(*rotated_a.T, s=1, alpha=0.3, color='red')
ax2.set_title("Rotated by a")

ax3 = fig.add_subplot(133, projection='3d')
ax3.scatter(*rotated_a_inv.T, s=1, alpha=0.3, color='green')
ax3.set_title("Rotated by a^{-1}")

plt.tight_layout()
plt.show()

Output:

• Shows how rotations ( a ) and $( a^{-1} )$ transform the sphere.
• The real Banach-Tarski uses infinitely precise versions of this.

5. Conclusion

The expressions $( S(a^{-1}), S(a), aS(a^{-1}) )$ represent group-theoretic “pieces” of the sphere that, when rotated and reassembled, allow volume doubling.