Fibonacci numbers do not have to stay in a list. We can turn them into geometry.
Build a Fibonacci rectangle
Start with two 1×1 squares. Add a 2×2 square, then a 3×3 square, then 5×5, then 8×8. Each new square has side length equal to the next Fibonacci number.
Because each new side length is the sum of the previous two, the pieces fit together into larger and larger rectangles.
Draw the spiral
Inside each square, draw a quarter-circle joining two opposite corners along the growing edge. The arcs join to make a spiral-like curve.
The familiar spiral made from quarter-circles is an approximation. As the rectangles get larger, its overall shape becomes closer to a logarithmic spiral associated with the golden ratio.
You have built squares of side 1, 1, 2, 3, 5, 8 and 13. What should the side length of the next square be, and what are the dimensions of the new outer rectangle?
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Reveal answer
The next side length is 21. A 13×21 Fibonacci rectangle becomes 21×34 after the new square is added.
Do spirals in nature prove anything?
You may see claims that sunflowers, pinecones, shells and galaxies are “made from Fibonacci spirals”. Some natural growth patterns do produce counts related to Fibonacci numbers, especially in plant arrangements. But many pictures online are exaggerated or fitted afterwards.
A good mathematical habit is to ask: what exactly was measured? and how close does it really have to be?
Calculate the long-side ÷ short-side ratio for the Fibonacci rectangles 2×3, 3×5, 5×8, 8×13 and 13×21. What seems to be happening?
Reveal answer
The ratios are about 1.5, 1.667, 1.6, 1.625 and 1.615. They wobble around while settling near a number close to 1.618.
That number is the subject of the next lesson.
What have we learned?
- Fibonacci squares fit together because of the addition rule.
- The famous quarter-circle construction gives a Fibonacci spiral.
- The rectangle ratios seem to approach about 1.618.
That mysterious 1.618… is the golden ratio.