Fun contents

Hidden order

The world often looks messy. River lengths begin with unexpected digits. A tiny equation draws an endless fractal. A population becomes chaotic. Thousands of random bounces make a smooth bell. A language model turns next-token guesses into sentences.

Each example looks different, but they share one idea: a simple rule can create a large and surprising pattern.

This series explores that idea one step at a time. Each article begins with a concrete example, keeps the important mathematics, and explains every new symbol along the way.

1. A digit with a favourite

In many real-world lists, numbers begin with 1 much more often than they begin with 9. Benford’s law describes this surprising pattern:

P(d)=\log_{10}\!\left(1+\frac{1}{d}\right)

The article explains logarithms through examples and calculates why about 17.6% of suitable numbers begin with 2.

Explore Benford’s law →

2. A picture made by escaping numbers

The Mandelbrot set begins with one repeated rule:

z_{n+1}=z_n^2+c

Every point in the picture supplies a different value of $c$ . Some number orbits remain bounded; others escape. Colouring their escape times reveals a fractal.

Explore the Mandelbrot set →

3. A population that becomes chaotic

The logistic map models growth with another short rule:

x_{n+1}=r\,x_n(1-x_n)

As the growth rate $r$ changes, the population settles, alternates, splits again and again, and finally becomes chaotic. A video helps make the journey visible.

Explore the logistic map →

4. Enough noise becomes a bell

One random bounce is unpredictable. Thousands of independent bounces create a smooth pattern. The central limit theorem explains why many different kinds of random sums approach the same bell curve.

\frac{1}{\sqrt{2\pi}}e^{-z^2/2}

Explore the central limit theorem →

5. A sentence from next-token guesses

A language model learns to place probabilities on the next piece of text. Repeating that small task across an enormous collection of writing produces sentences that can carry structure, style, and meaning.

P(w_1,\ldots,w_n)=\prod_{t=1}^{n}P(w_t\mid w_1,\ldots,w_{t-1})

Explore language models →

The pattern behind the series

Example	What looks messy	The hidden order
Benford’s law	measurements across many sizes	a probability for the first digit
Mandelbrot set	number orbits that may escape	an escape-time fractal
Logistic map	a population that will not settle	repeated splitting and a long-term distribution
Bell curve	many independent little changes	one curve after centring and resizing
Language model	a huge collection of human writing	probabilities for the next token

None of these rules predicts every detail. Instead, each reveals something steady inside the mess. That is hidden order: a pattern large enough to surprise us, produced by a rule small enough to write down.

Start with Benford’s law, or choose whichever mystery interests you most.