Math in motion
See the moment math turns strange
11 wonders
How can a chain of spinning circles draw any curve?
Any periodic motion splits into circles turning at steady speeds. More circles get closer, but the ≈ 9% spike at a jump never goes away.
How does one line, z² + c, grow endless detail?
A one-line rule with endless complexity: new structure all along the edge, and small copies of the whole hidden everywhere.
If the equations are exact, why can’t we predict the outcome?
Chaos is not randomness: every step follows the equations exactly, but any error at the start grows exponentially, so you can predict the shape, never the position.
Why do sunflowers favour an angle of 137.5°?
The golden angle is the "most irrational" turn, the hardest to approximate by fractions, so seeds never line up in rows and space is used best; the Fibonacci spiral counts come from the same fact.
Why do fifteen unconnected pendulums snake, split into rows and line up again?
Each pendulum does nothing but the simplest periodic swing; every pattern comes from the phase gap between neighbours growing steadily: at p/q of the cycle there are q groups, and the second half replays in reverse because a wave too dense for fifteen points is seen as a slow one. That is aliasing.
Plot the primes in polar coordinates: why do spirals and rays appear?
The spirals belong to the whole numbers, not the primes: 6, 44 and 710 radians are each almost whole turns (44/7 and 710/113 are the best fractions for 2π). The primes only avoid the arms that share a factor with those numbers, and split evenly among the rest (Dirichlet’s theorem).
Solve z³ = 1 by Newton’s method: why do fractals grow between the roots’ territories?
Near a point where the derivative is 0 the tangent is almost flat, so one step can fling a start anywhere and neighbouring starts head for different roots. The result: every point on the border touches all three colours at once, however far you zoom in.
Balls bounce left or right at random: why do they always pile into a bell?
Which slot a ball lands in depends only on how many times it turned right, and the number of paths with k rights is a binomial coefficient: many in the middle, few at the edges. Add up many small independent nudges and the result is always close to the normal curve. That is the central limit theorem.
Drop needles on a floor ruled with parallel lines: why does that give you π?
The average number of crossings depends only on a needle’s length, not its shape; a ring as wide as the gap always crosses exactly twice yet is π times its width around, which is how π gets into the ratio. Random throws can estimate π, but the error only shrinks like 1/√N: one more correct digit costs a hundred times more needles.
Halve it if even, triple it and add 1 if odd: why does every number end at 1?
Each tripling is followed on average by two halvings, a net factor of about 3/4, so numbers "usually" shrink; but usually is not always. It has been checked up to 2⁶⁸, and in 2019 Terence Tao proved that "almost all" numbers fall very low, yet a full proof is still missing.
Two lines of formula and four numbers: how do they draw pictures like these?
Every step is fully determined, yet the point never repeats and never wanders off, roaming for ever over a shape with fractal detail: a strange attractor. A positive Lyapunov exponent means chaos; a negative one means the orbit settles into a cycle of a few points.
More wonders are on the way.
Learning by the syllabus?
The labs follow the syllabus: functions, conics, geometry, probability, motion and waves, each chapter with a board to drag, a guided tour and a quiz.