STADIA / NUMBERS & THE UNIVERSE
Simple rules.
Surprising forms.
Explore Fibonacci, repeating patterns and fractals. Learn to understand mathematics by looking, experimenting and explaining one step at a time.
Start with the method ↓Try the first experiment →MATHEMATICS FOR READING THE WORLD
Begin with a question.
Give the symbols meaning.
A formula expresses a relationship. Understanding it means knowing what its symbols stand for, how its parts work together and when the relationship applies.
This first chapter starts with patterns you can build yourself. Experiments help you form a conjecture; explanations and proofs help you decide whether it is true.
THE METHOD / FIVE SMALL STEPS
How to understand mathematics
Observe
Describe what changes and what stays the same. Start with words and a drawing.
Try a small case
Use a few numbers or shapes. Write down each step so you can check it.
Find a rule
Ask which operation connects the examples. A pattern suggests a rule; it does not prove one.
Use symbols
Give each quantity a name. Read the formula aloud and connect it to your example.
Explain and test
Explain why the rule works. Try to find a counterexample and state its conditions.
FIBONACCI / BUILDING A SEQUENCE
Two numbers make the next.
Start with 0 and 1. Add the last two numbers to obtain the next: 0, 1, 1, 2, 3, 5, 8…
Read the rule
Fₙ = Fₙ₋₁ + Fₙ₋₂ (n ≥ 2)
The small index below F tells you the position in the sequence. Fₙ means “the Fibonacci number at position n”, not F multiplied by n.
Why do the ratios approach the golden ratio?
For positive consecutive terms, divide the recurrence by the previous term. If their ratios approach a positive limit r, then r = 1 + 1/r. This gives r² = r + 1, whose positive solution is (1 + √5)/2. A full proof must also establish that the ratios converge.
These ratios approach the golden ratio; a finite ratio of integer Fibonacci numbers is not exactly equal to it. Not every spiral in nature follows Fibonacci.
Before moving the slider: what will the next term be? Explain how you found it.
REPETITION / SAME RULE, NEW POSITION
Build a pattern from one tile.
A periodic pattern repeats after a fixed shift. Move the same tile horizontally and vertically to create a grid.
Separate the tile from the rule.
The tile is the object being repeated. In geometry, a translation moves every point by the same displacement. The tile keeps its orientation and size.
Here a and b are the horizontal and vertical shifts. Rotating alternate tiles adds a second rule and can change which symmetries the pattern has.
Try it with paper
Draw an asymmetric tile, such as a diagonal with a dot at one end. Copy it into four squares. Rotate alternate copies by 180°. Which movements make the whole pattern match itself?
A repeating grid and a fractal use different kinds of repetition: a grid repeats across positions; an exactly self-similar fractal repeats across scales.
Total tiles: 36
Does doubling the number of tiles per side double the total, or multiply it by four? Explain using rows and columns.
FRACTALS / REPEAT AT A SMALLER SCALE
One triangle becomes three.
Start with a filled equilateral triangle. Join the midpoints of its sides, remove the central triangle, and repeat on the three remaining triangles.
The Sierpiński construction
Each step leaves three copies with half the previous side length. After n steps, there are 3ⁿ small triangles, and the remaining area is (3/4)ⁿ of the starting area.
Aₙ / A₀ = (3/4)ⁿ
The screen shows a finite stage. The mathematical fractal is the limit of this construction as the number of steps grows without bound.
Why does 75% of the area remain after each step?
The midpoints divide a triangle into four congruent triangles. Removing one leaves three of the four equal areas. Repeating multiplies the remaining area by 3/4 again at each step.
Natural branching and irregular surfaces can resemble fractals over limited scales. They do not contain infinitely small copies of themselves.
Predict the number of triangles at the next step. Why can their count grow while their total area decreases?
PRACTISE / EXPLAIN THE IDEA YOURSELF
Understanding grows through practice.
Change representation
Write a rule as a sentence, a table, a drawing and a formula. What does each representation make easier to see?
Look for a counterexample
“All increasing sequences double at each step.” Fibonacci already gives a counterexample: 2 is followed by 3. One counterexample disproves a universal claim.
Distinguish seeing from proving
A thousand matching examples can support a conjecture. A proof explains why a statement holds for every case covered by its assumptions.
A short exercise with an explanation
At step 2, the Sierpiński construction has 9 triangles. At step 3 it has 27. The remaining area changes from 9/16 to 27/64 of the original. Each surviving triangle is replaced by three smaller ones, while one quarter of its area is removed.
