Math Wonders · WonderLab

Math wonders: a Sierpiński triangle grown from random jumps, the Koch snowflake's infinite perimeter, a fractal tree swaying in the wind, an ant, paint and scissors on a 3D Möbius strip, a golden-ratio game with an endlessly zooming spiral, and the sunflower's golden angle.

Subject: Mathematics · Level: Middle school, High school · Topics: Fractals, Möbius strip, Golden ratio, Phyllotaxis

Chapters

Sierpiński triangle

Play a game: start anywhere in a triangle and repeat—pick a random vertex, jump halfway towards it, and plot a dot.

Every step is random, yet after tens of thousands of dots an intricate, orderly pattern appears: the Sierpiński triangle. Each small piece looks exactly like the whole; this self-similarity is the mark of a fractal.

Koch snowflake

Start with an equilateral triangle. Split each side into thirds and grow a small equilateral triangle outwards on the middle third. Repeat on every new side.

The edge gets ever more crinkled: the perimeter has no limit, yet the area never exceeds a fixed value.

Fractal tree

A trunk sprouts two shorter branches, each of which sprouts two more… one rule grows a whole tree.

Trees, rivers, blood vessels and lightning all share this branching, ever-shrinking fractal structure.

Möbius strip

Give a long strip of paper a half-twist and glue the ends to get a Möbius strip (August Möbius, 1858).

It has one side and one edge. Drag to turn it, then let the ant walk, paint it, or cut it down the middle and see what happens.

Golden ratio

Split segment AB so that AG : AB = GB : AG. The ratio is about 0.618, the golden ratio; its reciprocal is φ = (1 + √5)/2 ≈ 1.618.

First find the golden point by eye, then see how it relates to the golden spiral and the Fibonacci numbers.

Sunflower seeds

On a sunflower head, each new seed is turned by a fixed angle from the previous one.

Which angle packs the seeds most tightly? Drag the slider: a fraction of a degree changes everything.

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