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