PrismaX

The Galton board: from chance to the bell curve

1 chapter

A ball drops through rows of pegs and bounces left or right at random at every peg: nobody can tell where it will land. Yet after a few hundred balls, the bins always build the same bell shape.

The butterfly effect: the Lorenz system

1 chapter

Weather forecasts are computed from equations with nothing random in them, yet they lose reliability day after day. The reason already shows in a model of three equations:

Flocking: three rules, one collective behaviour

1 chapter

A flock of starlings turns all at once, yet it has no leader: each bird reacts to the few neighbours around it.

The wave packet: where a particle is

1 chapter

In quantum mechanics an electron does not have a precise position, as a small ball does:

Antifragility: convexity and decisions under uncertainty

1 chapter

Faced with an uncertain future, some choices lose value and others gain it.

The double pendulum: chaos inside a formula

1 chapter

A simple pendulum is the very image of regularity. Hang a second one below the first and the motion becomes unpredictable, even though the formula that governs it is exact and contains nothing random.

Gravity assist: borrowing speed from a planet

1 chapter

A ball thrown at an oncoming train bounces back faster than it arrived. A probe that flies close to a moving planet does the same without touching it:

Quadratic functions: vertex, roots and the best price

1 chapter

A quadratic function $f(x) = ax^2 + bx + c$ draws a parabola, and at its vertex the function takes its largest or smallest value.

Exponential growth, logarithms and "default alive"

1 chapter

A startup whose revenue grows by the same percentage every month does not climb a straight line: it multiplies.

Gradient descent: how a model learns

1 chapter

Training a model means adjusting its parameters until its errors are as small as possible. Gradient descent does this by always stepping downhill on the error surface.

Reinforcement learning: learning by trial and error

1 chapter

Nobody tells the agent in the figure where the goal is or how to get there. It moves, collects rewards and penalties, and from them alone it learns which move is best in every cell.

Physics in code: an orbit in a few lines

1 chapter

A computer does not solve the equations of motion in one go: it moves the planet forward in many small time steps. Four lines of code, repeated, are enough to trace an orbit.

A/B tests: how many users you need (and why not to peek)

1 chapter

You show version A of a page to half of your visitors and version B to the other half, and B converts better. Is B really better, or did chance do it?

Ready-made lessons to start from.

The Galton board: from chance to the bell curve

The butterfly effect: the Lorenz system

Flocking: three rules, one collective behaviour

The wave packet: where a particle is

Antifragility: convexity and decisions under uncertainty

The double pendulum: chaos inside a formula

Gravity assist: borrowing speed from a planet

Quadratic functions: vertex, roots and the best price

Exponential growth, logarithms and "default alive"

Gradient descent: how a model learns

Reinforcement learning: learning by trial and error

Physics in code: an orbit in a few lines

A/B tests: how many users you need (and why not to peek)