Coding Challenge #139: Calculating Digits of Pi with Collisions

The speaker fondly recalls January 2019 and a captivating video from the educational YouTube channel 3Blue1Brown that explained a unique counting puzzle involving two bouncing blocks. The video demonstrated how the number of times the blocks collided could reveal digits of pi, an intriguing concept that combined physics and mathematics. The speaker also references a 2003 paper by G. Galperin and a Numberphile video for further exploration of the phenomenon. The channel's focus on coding is then introduced, with the speaker's intention to code a simulation of the 3Blue1Brown video in JavaScript, aiming to replicate the appearance of pi through collisions.

The speaker begins the coding challenge by establishing a basic object-oriented system in JavaScript, defining a Block class with properties for position, size, and velocity. The initial setup includes drawing the blocks and loading a sound effect. The speaker then proceeds to write functions for updating block positions based on velocity and checks for collisions between the two blocks. A 'collide' function is implemented to determine if the blocks intersect, crucial for the simulation's core mechanics.

The coding challenge continues with the implementation of elastic collision physics between the two blocks. The speaker introduces the concept of mass and uses formulas from physics to calculate new velocities after a collision, ensuring the conservation of momentum and kinetic energy. The 'bounce' function is developed to update the velocities of the blocks based on their masses and initial velocities. The speaker also discusses the implications of equal versus unequal masses in the collision outcome.

The simulation becomes more realistic as the speaker adds a 'hitWall' function to simulate collisions with an immovable object, treating it as an object with infinite mass. The block's velocity is reversed upon hitting the wall, mimicking a perfectly elastic collision. Additionally, sound effects are integrated to enhance the auditory experience of the simulation, with the clacking sound playing upon collisions.

The speaker introduces a counting mechanism to tally the number of collisions, aiming to observe the digits of pi emerge from the simulation. The count is displayed on the screen, and the speaker experiments with adjusting the mass of the blocks to influence the count's progression. The speaker also contemplates the stopping criteria for the count and the limitations of Euler integration in the simulation's accuracy.

To improve the simulation's accuracy, the speaker implements a technique using smaller time steps, effectively increasing the number of updates per frame to more closely approximate continuous time. This approach helps in creating a more precise simulation, allowing for a better representation of the physical behavior of the blocks. The speaker also discusses the trade-off between accuracy and performance, especially as the number of time steps increases.

The speaker successfully generates multiple digits of pi through the simulation and contemplates the integration of sound effects that play upon collisions. The challenge of managing the sound playback within the increased number of time steps is addressed, ensuring that the clacking sound enhances rather than distracts from the simulation. The speaker demonstrates achieving up to seven digits of pi with a high number of time steps, showcasing the effectiveness of the refined simulation.

In an ambitious attempt, the speaker pushes the simulation to process up to ten million time steps per draw, resulting in a significant increase in computational demand but also in the accuracy of the simulation. The speaker successfully achieves 11 digits of pi through this method, demonstrating the potential of the approach despite the computational intensity and slowdown in the animation's frame rate.

The speaker concludes the coding challenge by reflecting on the process and outcomes, acknowledging the successful generation of 11 digits of pi through elastic collisions. They also consider alternative methods and optimizations for the simulation, such as using BigDecimal for higher precision or exploring different integration techniques. The speaker invites the audience to share their thoughts and solutions on thecodingtrain.com and hints at potential future live streams and coding attempts.

The speaker ends the video with a light-hearted celebration of Pi Day, expressing excitement about the successful coding challenge and the creative approach to generating digits of pi. They humorously mention Tau Day and the arrival of summer, signing off with a cheerful tune that encapsulates the joy of mathematical discovery and the fun of coding.