“Why colliding blocks compute pi” — 3Blue1Brown

Episode summary

Grant returns to his 2019 viral result — two frictionless blocks colliding elastically produce a total count of collisions whose digits match pi, when the mass ratio is a power of 100 — and delivers the full physical/mathematical explanation he left out of the original. The spine of the argument: translate the dynamics into a 2D state space where v1 and v2 become coordinates, rescale axes by sqrt(mass) so conservation of energy becomes a circle (not an ellipse), then exploit the inscribed angle theorem plus the small-angle approximation arctan(x) ≈ x to show that the number of collisions before the “end zone” is the largest integer n with n·arctan(sqrt(m2/m1)) < π. For mass ratios that are powers of 100, that arctan is a power of 10, and the answer literally reads off the digits of pi. The back third is a meta-essay: purity exposes hidden connections, and this puzzle is secretly mirrored in Grover’s algorithm — the subject of his next video.

Key arguments / segments

• [00:00:00] Recap of the 2019 result and the pi coincidence for mass ratios 1, 100, 10K, 1M
• [00:03:00] Idealizing assumptions: perfectly elastic, no sound, non-relativistic — and why the idealization is the point
• [00:04:00] Problem-solving principle #1: write down conservation laws you already know (energy, momentum)
• [00:06:00] Principle #2: draw pictures — package dynamic variables into a state space (v1, v2)
• [00:08:00] Principle #3: respect symmetries — rescale axes by sqrt(mass) so the energy ellipse becomes a circle
• [00:11:00] Momentum conservation becomes a line with slope -sqrt(m1/m2) intersecting the circle at two points; each collision jumps between them
• [00:14:00] Wall collision flips the y-sign; block collision steps down the diagonal — game until you enter the “end zone” (both velocities rightward, v2 < v1)
• [00:15:00] Inscribed angle theorem: every arc between consecutive collision points subtends the same angle 2θ on the circle
• [00:17:00] Collision count = floor(π/θ) where θ = arctan(sqrt(m2/m1))
• [00:19:00] Small-angle approximation arctan(x) ≈ x for small x: for mass ratio 100^k, θ is within 1e-3k of a power of 10, so the digits match pi
• [00:21:00] Taylor-series rigor: error is O(θ³); the only way this fails is if pi has a run of n nines starting at digit n — no known case
• [00:22:00] Strictly speaking, the “digits of pi” result is an unsolved problem gated on a conjecture about the decimal expansion of pi
• [00:23:00] Why abstract away messiness: (a) simpler variant first, (b) purity exposes hidden connections — e.g. Grover’s algorithm

Notable claims

• State-space + coordinate rescaling is the universal move: when dynamic variables are changing, package them into a higher-dimensional point and study its trajectory under the constraint surfaces imposed by conservation laws
• The appearance of pi is not coincidence — it falls out of the energy-conservation ellipse becoming a circle, and circles have 2π of circumference
• The full colliding-blocks → pi implication is technically an unsolved problem because it depends on a (universally believed but unproven) fact about pi’s decimal expansion
• A beam of light bouncing between two angled mirrors is a direct analogue of the colliding blocks — same geometry, different physics (Galperin 2003)
• The same state-space + arc-counting structure reappears inside Grover’s quantum search algorithm

Why this is in the vault

Two reasons. First, the meta-principle Grant closes on — “purity exposes hidden connections” — is the cleanest articulation of why Ray Data Co spends any time at all on theoretical exposition rather than only on applied case studies. The colliding-blocks → Grover’s isomorphism is the kind of unexpected bridge that would not exist if mathematicians had refused to idealize, and it’s the closest available rhetorical template for explaining why Sanity Check treats things like information theory and measure concentration as practically relevant to shipping data products. Second, the video is a reusable pedagogy reference for any future newsletter on state-space thinking — translating a tangled physical or business question into a geometric question via the right coordinate change is a move the newsletter returns to often, and this is the cleanest worked example in the vault so far.

Mapping against Ray Data Co

• Direct feed to the Grover’s / quantum hype-correction arc. The vault already has 2026-04-20-3blue1brown-but-what-is-quantum-computing-grovers-algorithm and 2026-04-20-3blue1brown-grovers-algorithm-clarification. This video is the prerequisite primer that Grant himself positions as “you can’t really understand Grover’s geometrically until you’ve seen the colliding-blocks version.” For a Sanity Check piece on why quantum hype is premature (see 2026-03-11-ark-invest-quantum-computing-bitcoin for the Ark counter-position we need to correct), the colliding-blocks state-space picture is the most accessible on-ramp to the “rotating-vector” view of Grover. Mapping strength: strong / prerequisite.
• State-space + coordinate-change as a business-problem tool. The move “identify the conserved quantities, pack the changing variables into a single point, rescale until symmetry appears” is exactly the frame we want for a future piece on dashboards that track the wrong thing. When a metric changes, the question isn’t “did this number go up,” it’s “what surface in state space does the system sit on, and did it jump to a different surface.” Mapping strength: medium / conceptual-reuse.
• “Unsolved problem” humility. The fact that the pi result is technically unsolved is a great rhetorical beat for any piece arguing against over-claiming in ML/AI research — even the most visually convincing results can rest on unproven conjectures.

Related

• 2026-04-20-3blue1brown-but-what-is-quantum-computing-grovers-algorithm
• 2026-04-20-3blue1brown-grovers-algorithm-clarification
• 2026-03-11-ark-invest-quantum-computing-bitcoin
• 2026-04-20-3blue1brown-how-and-why-to-take-a-logarithm-of-an-image