Fibonacci, the golden ratio and fractals, in 3D

What the simulation shows

• Squares of Fibonacci sizes (1, 1, 2, 3, 5, 8, 13) appear one by one, each fitting against the last.
• A golden spiral is drawn through the squares, a quarter turn in each.
• A sunflower head fills with seeds, each turned 137.5° from the last (the golden angle): perfect spirals.
• The angle changes to 137.0°: the seeds break into spokes with gaps.
• The Mandelbrot set computed live, the camera diving in thousands of times: the pattern never ends.
• A wheel one unit across rolls one full turn: it travels 3.14159… (π).
• A Galton board: balls bounce left or right at random off pegs and pile into a bell curve.
• The flower of life drawn circle by circle: 19 equal circles, each centred on another's edge.

The real numbers it runs on

The simulation is built on these facts, so the narration and the picture always agree:

• The Fibonacci sequence: each number is the sum of the two before (1, 1, 2, 3, 5, 8, 13, 21…). The ratio of neighbours approaches the golden ratio, φ ≈ 1.618.
• Squares with Fibonacci sides tile together, and quarter circles through them make a golden spiral.
• Sunflowers place each new seed 137.5° (the golden angle) from the last: it packs seeds with no gaps and makes spirals in Fibonacci numbers (often 34 and 55). A slightly different angle leaves gaps.
• The Mandelbrot set comes from repeating one rule, z → z² + c; its edge shows new detail however far you zoom.
• π (3.14159…) is a circle's circumference divided by its diameter: roll a wheel one turn and it travels π times its width. Its digits never end or repeat.
• Random 50/50 choices add up to predictable patterns: a Galton board's balls pile into a bell curve (the normal distribution).
• The flower of life is 19 equal circles, each centred on the edge of others, a pattern found in ancient art across many cultures.

The script this video was made from

“Nature's secret number”: the narration of the video above. Each highlighted moment in the simulation happens on the word that names it.

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