Colliding Blocks and Pi: The Hidden Connection to Quantum Computing

The Colliding Blocks Puzzle Revisited

In 2019, a video about how two colliding blocks can compute pi took the internet by storm. This seemingly simple physics problem captured the imagination of viewers worldwide, becoming one of the most popular online math demonstrations. Now, it's time to revisit this intriguing puzzle and uncover even more fascinating connections - including a surprising link to quantum computing.

The Basic Setup

Let's recap the original puzzle:

• Two blocks are placed on a frictionless plane
• The left block (smaller, 1 kg) is stationary
• The right block (larger) moves towards the left block
• A wall is placed to the left of both blocks
• The goal is to calculate the total number of collisions, including those with the wall

Mass Ratios and Collision Counts

The number of collisions depends on the mass ratio between the two blocks:

• Equal masses (1:1 ratio): 3 collisions
• 100:1 mass ratio: 31 collisions
• 10,000:1 mass ratio: 314 collisions

As the mass ratio increases by powers of 100, the number of collisions consistently displays digits of pi. This remarkable pattern forms the core of our investigation.

Idealized Physics and Real-World Limitations

Before diving deeper, it's important to acknowledge some idealized assumptions in this puzzle:

• Perfectly elastic collisions (no energy loss)
• No relativistic effects for extreme mass ratios
• Ignoring practical limitations of real-world experiments

While these assumptions may seem unrealistic, they allow us to explore the pure mathematical principles underlying the phenomenon. This idealization is crucial for uncovering hidden connections to other fields, such as quantum computing.

Problem-Solving Principles

To tackle this puzzle, let's employ some general problem-solving strategies:

1. List relevant equations and theorems
2. Draw pictures and diagrams
3. Respect mathematical symmetries
4. Break complex problems into simpler parts

Key Physical Laws

Two fundamental principles govern the behavior of our colliding blocks:

1. Conservation of Energy
2. Conservation of Momentum

Let's express these mathematically:

• Energy: ½m₁v₁² + ½m₂v₂² = constant
• Momentum: m₁v₁ + m₂v₂ = constant (except after wall collisions)

Where:

• m₁, v₁: mass and velocity of the larger block
• m₂, v₂: mass and velocity of the smaller block

Visualizing the Problem: State Space

To gain deeper insights, let's represent the system's state in a two-dimensional space:

• x-axis: velocity of the large block (v₁)
• y-axis: velocity of the small block (v₂)

This "state space" approach is a powerful tool in physics, allowing us to track the system's evolution as a single point moving through an abstract space.

The Energy Ellipse

Plotting our conservation of energy equation in this space yields an ellipse:

(m₁/2)x² + (m₂/2)y² = constant

This ellipse represents all possible velocity combinations that maintain the system's total energy.

Transforming to a Circle

To simplify our analysis, we can rescale our coordinates:

• x = √(m₁) * v₁
• y = √(m₂) * v₂

This transformation turns our energy ellipse into a perfect circle:

x² + y² = constant

This seemingly small change introduces valuable symmetry to our problem, making it easier to solve.

Analyzing the Circular State Space

Let's examine how our system evolves within this circular state space:

1. Initial condition: leftmost point of the circle (large block moving left, small block stationary)
2. After collision: point moves along a line with slope -√(m₁/m₂)
3. Wall collision: y-coordinate flips sign (moves vertically)
4. Repeat steps 2-3 until reaching the "end zone"

The "end zone" is the region where both blocks move right, with the larger block faster than the smaller one.

Geometric Interpretation

Our physics problem has now transformed into a geometric puzzle:

• Start at the leftmost point of a circle
• Move along lines with a specific slope, alternating with vertical movements
• Count the number of lines drawn before reaching the end zone

This abstraction reveals a key insight: the number of lines (collisions) depends on how many equal-sized arcs fit around the circle's circumference.

The Inscribed Angle Theorem

A crucial geometric principle helps us prove that these arcs are indeed equal:

• The inscribed angle theorem states that an angle inscribed in a circle is half the central angle subtending the same arc
• This theorem guarantees that the angle between our lines (θ) corresponds to a consistent arc length of 2θ on the circle

Connecting to Pi

The relationship between our collision count and pi emerges from this geometric interpretation:

• Total circumference: 2π radians
• Arc size: 2θ radians
• Number of collisions ≈ π / θ

When θ is a small power of 10 (due to our mass ratio), the collision count naturally displays digits of pi.

Small Angle Approximation

The exact relationship between our mass ratio and θ involves the arctangent function:

θ = arctan(√(m₂/m₁))

However, for small angles, arctan(x) ≈ x. This "small angle approximation" explains why we see pi's digits for mass ratios that are powers of 100.

Beyond Powers of 100

While powers of 100 yield familiar base-10 representations of pi, the underlying principle applies more broadly:

• In base 2 (binary), mass ratios of powers of 4 would yield binary digits of pi
• Other bases would have their own corresponding mass ratio patterns

This generalization highlights the fundamental nature of the relationship between colliding blocks and pi.

The Value of Idealized Models

Some might question the practical value of such an idealized model. However, there are two important justifications for this approach:

1. Simplification as a starting point: Idealized models provide a foundation for understanding complex systems. Real-world complications can be gradually introduced once the core principles are grasped.

2. Revealing hidden connections: Abstracting away real-world messiness can expose surprising links between seemingly unrelated fields. This puzzle connects classical mechanics, geometry, and even quantum computing.

The Quantum Computing Connection

Perhaps the most intriguing aspect of this puzzle is its unexpected relationship to quantum computing, specifically Grover's algorithm for database search. This connection, noticed by theoretical physicist Adam Brown, demonstrates how insights from one field can illuminate another.

While the full explanation of this quantum link requires a deeper dive, it's worth noting that the circular state space and the way our system evolves within it mirror certain aspects of quantum algorithms. This parallel showcases the power of abstract mathematical thinking in revealing connections across diverse scientific domains.

Practical Demonstrations and Limitations

Despite the idealized nature of our model, attempts have been made to demonstrate this phenomenon in practice:

• Students at the University of Bonn successfully showed 31 collisions for a 100:1 mass ratio
• Mathematician Matt Parker and colleagues attempted larger-scale demonstrations, highlighting practical challenges

These real-world experiments underscore both the validity of the underlying principle and the limitations of perfectly replicating idealized physics.

The Unsolved Pi Problem

Interestingly, there's a subtle mathematical nuance that makes this puzzle technically an unsolved problem:

• The small angle approximation introduces a tiny error
• This error could theoretically cause an off-by-one discrepancy in the collision count
• Such a discrepancy would occur if pi had a sequence of digits followed by an equal number of 9's
• While extremely unlikely, proving this impossible for all digits of pi is beyond current mathematical techniques

This quirk reminds us that even seemingly simple problems can harbor deep mathematical mysteries.

Conclusion: The Power of Abstract Thinking

The colliding blocks puzzle exemplifies the value of abstract mathematical thinking:

1. It transforms a physical problem into a geometric one, making it easier to analyze
2. It reveals unexpected connections between classical mechanics and pi
3. It hints at deeper links to quantum computing algorithms
4. It demonstrates how idealized models can provide insights into complex systems

By stripping away real-world complications, we uncover fundamental principles that bridge multiple fields of study. This approach not only aids in problem-solving but also exposes the beautiful interconnectedness of mathematics and physics.

As we continue to explore these hidden connections, we open doors to new understanding and potential breakthroughs across scientific disciplines. The journey from colliding blocks to pi to quantum computing reminds us that in the world of mathematics and physics, seemingly simple questions can lead to profound and far-reaching insights.

Article created from: https://www.youtube.com/watch?v=6dTyOl1fmDo