Approximating Pi via the Mandelbrot Set | Numberphile

The Mandelbrot Set and Julia Sets for the equation z² + c are deeply interconnected mathematical structures where each point in the Mandelbrot Set corresponds to a specific Julia Set, and vice versa; specifically, any point c in the Mandelbrot Set maps to the origin of the corresponding Julia Set with that same c value, while any point in a Julia Set maps to a specific point in the Mandelbrot Set when considering different starting offsets, creating a bidirectional relationship that explains why Julia Sets become increasingly chaotic as their corresponding c values approach the boundary of the Mandelbrot Set.

The Mandelbrot set serves as a map or catalog of all possible Julia sets. Each point in the Mandelbrot set corresponds to a specific Julia set obtained by fixing C at that point. Points inside the main cardioid of the Mandelbrot set produce connected Julia sets, while points outside produce disconnected 'dust' Julia sets. Zooming into specific regions of the Mandelbrot set reveals miniature versions of the overall structure, demonstrating self-similarity. Notably, the famous 'seahorse valley' region produces Julia sets that resemble seahorses, illustrating how local Mandelbrot structure maps to corresponding Julia set geometry.

Julia sets are closely related to the Mandelbrot set but typically produce boring audio results. For the Mandelbrot set, any point on a Julia set will converge to exactly the same orbit as the corresponding point on the Mandelbrot set. This means Julia sets of the Mandelbrot set do not offer additional sonic diversity beyond what is already present in the main set itself.

Every point on the Mandelbrot set generates a unique Julia set using its corresponding c value. A Julia promenade traces a path around the Mandelbrot set, creating sequential Julia sets that evolve and blend into each other. When c comes from colored (outside) regions, Julia sets are disconnected; when from black (inside) regions, they are connected with large black areas. This relationship shows how the Mandelbrot set acts as a map determining the fundamental nature of all possible Julia sets.

This section explains bifurcations (branching) as state changes in non-linear systems where the number or stability of fixed points changes as a parameter varies. Bifurcation diagrams plot fixed points or periodic orbits as a function of a parameter, revealing how system behavior changes. As parameters increase, systems transition from stable fixed points to periodic orbits and eventually to chaotic behavior through period doubling bifurcations. The Mandelbrot set bifurcation diagram shows how fixed points split and how the system transitions to chaos. The section emphasizes that the Mandelbrot set is not only a symbol of fractal self-similarity but also represents bifurcation and chaotic systems, demonstrating the deep connection between fractal geometry and chaotic dynamics.

Bifurcation theory studies how qualitative changes occur as system parameters vary. Local bifurcations involve changes in stability or emergence/disappearance of equilibria as single parameters cross critical values. The discrete logistic map x(n+1) = rx(n)(1 - x(n)) demonstrates bifurcation phenomena: for r < 1, populations die out; for 1 < r < 3, stable fixed points emerge; at r ≈ 3.0, period-doubling bifurcation creates 2-cycles; further increases produce period-4, period-8 cycles, eventually leading to chaos for r > 3.57. Bifurcation diagrams plot parameter values against system states, revealing extinction regions, stable fixed points, periodic windows, and chaotic regimes. Chaos exhibits extreme sensitivity to initial conditions - tiny differences produce vastly different long-term outcomes despite deterministic rules. Bifurcation conditions relate to eigenvalue locations: crossing the imaginary axis indicates stability loss/gain in continuous systems, while magnitude crossing 1 indicates period-doubling in discrete systems.

In chaos theory, a bifurcation occurs when a system far from equilibrium experiences a perturbation that causes it to transition from having a single stable solution to having multiple possible solutions. This concept, developed by Nobel laureate Ilya Prigogine in the 1970s, explains how systems can suddenly branch into different evolutionary paths when they are disturbed from their stable state.

This section presents comprehensive bifurcation theory governing system transitions. Five main types exist: saddle-node (two fixed points collide and annihilate), pitchfork (one splits into three), transcritical (two exchange stability), period-doubling (period doubles repeatedly), and Hopf (limit cycle emerges from fixed point). Classification rules: continuous systems bifurcate when eigenvalues cross imaginary axis; discrete systems when eigenvalue modulus equals 1. The logistic map xₙ₊₁ = μxₙ(1-xₙ) demonstrates universal route to chaos: extinction for μ < 1, stable fixed point for 1 < μ < 3, period-doubling cascade for 3 < μ < 3.57, and chaos for μ > 3.57. Cobweb diagrams visualize iterations by alternating between curve and line y=x. Feigenbaum constant (~4.669) marks chaos onset, demonstrating universality across diverse physical and biological systems.

Bifurcation theory studies how the qualitative behavior of a dynamical system changes as parameters vary. A bifurcation occurs when a small change in a parameter causes a sudden qualitative change in the system's behavior, such as the appearance of new equilibrium points or periodic orbits. This concept is essential for understanding complex systems and transitions in physical, biological, and social phenomena.

This video presents Galperin's method for calculating pi by counting collisions between two objects. When a blue object is moved toward a red object, and the blue object's mass is progressively increased (1kg, 100kg, 10000kg), the number of collisions approaches pi (31, 314, 3141...). Three key assumptions enable this: objects are perfectly rigid, no external forces act, and collisions are perfectly elastic. Two conservation laws apply: Conservation of Momentum and Conservation of Kinetic Energy. When expressed in transformed coordinates (X = √(m1)V1, Y = √(m2)V2), kinetic energy conservation becomes X² + Y² = constant, revealing a deep connection between physics laws and geometry. A coordinate transformation (new X = √(m1) × x, new Y = √(m2) × y) makes incidence angles equal reflection angles. The number of collisions C satisfies π < Cα < π + α, where α = arctan(√(m1/m2)). When √(m1/m2) = 10^(-N), C/10^N ≈ π. An unresolved conjecture states that C/10^N approximates pi to N decimal places, equivalent to saying pi's decimal expansion has no sequence of N-1 consecutive 9s before the Nth decimal place. This is verified for N up to 100 million but not proven.

This section presents four methods for approximating pi and introduces fractional derivatives. First, Archimedes' geometric method uses inscribed/circumscribed polygons: with diameter 50 cm, the inscribed hexagon has perimeter 150 cm and circumscribed hexagon has perimeter ~173 cm, giving 3 < pi < 3.46. Using dodecagons narrows this to 3.1 < pi < 3.22, and 96-gons give 3.1408 < pi < 3.1429. Second, Buffon's Needle Problem gives probability P = 2L/(πd) for needle intersections. Third, the Monte Carlo method approximates pi by randomly throwing points into a square containing a circle, where the ratio of points inside the circle to total points approximates π/4. Fourth, fractional derivatives are derivatives of non-integer order. The half-derivative of x^a is D^(1/2)(x^a) = Γ(a+1)/Γ(a+1/2) x^(a-1/2), where Γ is the Gamma function. The Gamma function generalizes factorials: Γ(n+1) = n! for integers.

In an idealized physics system with two blocks and a wall, where collisions are perfectly elastic and frictionless, the total number of collisions before the blocks move away indefinitely equals the digits of Pi. When the first block is 100 times heavier than the second, there are 3 collisions (first digit of Pi); when 10,000 times heavier, there are 31 collisions (first two digits); when 1,000,000 times heavier, there are 314 collisions (first three digits). This elegant but highly inefficient algorithm, discovered by mathematician Gregory Galperin in 1995, demonstrates how Pi emerges from the law of conservation of energy in collision systems.

In a system of two sliding blocks with a wall where collisions are perfectly elastic and frictionless, the total number of collisions between the blocks equals the digits of pi when the mass ratio of the blocks is a power of 100; this surprising connection between pi and discrete collision counting was discovered by Gregory Galperin in 1995.

When the ratio between the radii of two rolling circles is Pi, the resulting epicycloid curve never closes because Pi is irrational; however, the 22/7 approximation produces a nearly closed curve, and after 113 revolutions, the curve generates 355 intersection points with the original circle, visually demonstrating that Pi is approximately 355/113.

The worst-case complexity of chaotic iterations is O(n⁵), derived from: n² outer loop iterations (each set can change by adding one element), n inner loop iterations (one per instruction), O(n) set union operations, and O(n) predecessors per node. However, this is a loose upper bound. In practice, with proper node ordering, the algorithm converges in 2-3 iterations. Under reasonable assumptions (sparse control flow graphs with at most 2 predecessors/successors per node), the practical complexity becomes linear O(n). Empirical evidence from points-to analysis confirms this linear behavior.

In 1995, David Bailey, Peter Borwein, and Simon Plouffe discovered the BBP formula: π = Σ [1/(16^k) × (4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6))] for k=0 to ∞. This allows direct calculation of hexadecimal or binary digits without computing preceding digits. As of January 2020, Timothy Mullican set the world record with over 50 trillion decimal places using Google Cloud. Only 39 digits are needed for physics calculations involving the observable universe. The open question of decimal BBP formulas remains unsolved. The Chudnovsky brothers calculated 2 billion digits using their series. Emma Haruka Iwao calculated 31 trillion digits in 2019 using Google cloud virtual machines.

The study of complex systems is indissolubly correlated with computational complexity. Models of both artificial and natural phenomena are based on algebraic and differential equations requiring numerical resolution. Artificial phenomena include search engines and economic simulations; natural phenomena include meteorology and fluid dynamics. Resolution uses iterative methods providing increasingly precise approximations. Problems defined on the continuum are discretized through numerical linear algebra, graph theory, and matrix calculus.

Exact computation yields undecidable problems (Julia set membership), but approximation makes them tractable. Pseudo-orbits with ε-error shadow real orbits in hyperbolic systems, but verifying pseudo-orbits relates to P vs NP. This connects dynamical systems theory with fundamental computational complexity questions, showing how approximation transforms theoretical limits into practical computability.

When integrating chaotic dynamical systems, propagating uncertainty requires integrating bundles of nearby trajectories simultaneously; however, in non-compiled languages like Python and MATLAB, using nested for loops to integrate each trajectory individually is extremely inefficient (potentially thousands of times slower), whereas vectorized approaches that leverage linear algebra operations and compiled Fortran routines can achieve orders of magnitude speedup while revealing how sensitive dependence on initial conditions causes nearby trajectories to diverge and smear across the attractor over time.