Why colliding blocks compute pi

Overview

3Blue1Brown revisits the colliding blocks problem, revealing its connection to pi and quantum computing, specifically Grover's algorithm. The video explains how idealizing collisions and using a state-space representation transforms the physics problem into a geometry puzzle involving circles and lines. This geometric interpretation, combined with a small angle approximation, demonstrates why the number of collisions for mass ratios that are powers of 100 yields digits of pi, though a rigorous proof remains an unsolved problem.

Key takeaways

The colliding blocks problem, when idealized, can be mapped to a geometric problem on a circle.
Conservation of energy in a two-body system naturally leads to a circular path in a rescaled state space.
The number of collisions in the idealized system is determined by how many equal angular segments fit within a full circle.
For mass ratios that are powers of 100, the angle involved is the arctangent of a small power of 10, which approximates a power of 10.
This approximation allows the number of collisions to have the same digits as pi, though a rigorous proof of this connection is an unsolved mathematical problem.
Abstracting complex physical systems into simpler mathematical models (like state spaces and geometric puzzles) can reveal hidden connections and facilitate problem-solving.

Chapters

0:00 Introduction to the Colliding Blocks Problem and Pi Connection

The 2019 video on colliding blocks computing pi is 3Blue1Brown's most popular.
This year's revisit explores a secret connection to Grover's algorithm in quantum computing.
The full connection to pi is technically an unsolved problem.

0:44 Recap of the Colliding Blocks Setup and Initial Observations

Two blocks on a frictionless plane: one stationary (smaller), one moving (larger).
A wall exists to the left of the stationary block.
Mass ratios of 1:1, 100:1, and 10,000:1 yield collision counts of 3, 31, and 314 respectively.
Mass ratios of 1,000,000:1 yield 3,141 collisions.

3:27 Idealizing Assumptions and Problem Transformation

Assumptions include perfectly elastic collisions (no energy loss) and ignoring relativistic effects.
Conservation of energy and momentum are key physics principles.
A state space is created where velocities (v1, v2) are coordinates, representing the system's state.
Conservation of energy forms an ellipse in the (v1, v2) state space.

14:59 Rescaling State Space to a Circle and Geometric Interpretation

Rescaling coordinates to sqrt(m1)*v1 and sqrt(m2)*v2 transforms the ellipse into a circle, simplifying the problem.
Conservation of momentum becomes a linear equation (a line) in this new coordinate system.
Collisions are represented by the state point moving along the circle and intersecting the momentum line.
The problem transforms into counting how many times a line with a specific slope intersects a circle before entering an 'end zone'.

23:24 Connecting Geometry to Pi and Unsolved Mathematical Aspects

The slope of the line is related to the negative square root of the mass ratio.
The inscribed angle theorem shows that arcs between intersection points are equal.
The number of collisions is related to how many arcs of angle 2*theta fit within 2*pi radians.
Using a small angle approximation, arctan(sqrt(m2/m1)) is close to sqrt(m2/m1) for small ratios.
The exact computation of pi via this method is an unsolved problem due to potential 'off-by-one' errors related to the digits of pi.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, 3Blue1Brown.