What it shows
Squares of side 1, 1, 2, 3, 5, 8 and 13 cm fit together into a rectangle of 13 cm by 21 cm, and a quarter circle in each square joins into a spiral. The ratio of the sides, 21 divided by 13, is about 1.615, close to the golden ratio 1.618, and the ratio gets closer as more squares are added.
The science
The golden ratio phi is the number such that a rectangle of sides phi and 1 can be cut into a square and a smaller rectangle of the same proportions. Phi equals (1 + root 5) divided by 2, about 1.618. The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21 have the property that each ratio of neighbours (2, 1.5, 1.667, 1.6, 1.625, 1.615) gets closer and closer to phi. Stacking Fibonacci squares therefore makes rectangles that look more and more golden, and the quarter circles form an approximation to a logarithmic spiral.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- Chart paper and coloured paper
- A compass, ruler and set square
- Scissors and glue
- A calculator
- A board 30 cm by 40 cm
Safety
For this project: Keep the compass point capped when not in use.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
Step by step
7 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Cut squares from coloured paper with sides 1, 1, 2, 3, 5, 8 and 13 cm.
Glue the two 1 cm squares side by side, then the 2 cm square below them, the 3 cm square to the left, the 5 cm square above, the 8 cm square to the right and the 13 cm square below, turning the same way each time.
Check the whole rectangle measures 13 cm by 21 cm.
With the compass, draw a quarter circle in each square from corner to opposite corner, radius equal to the side, so the arcs join smoothly.
Write the ratio of neighbouring Fibonacci numbers beside each square: 1, 2, 1.5, 1.667, 1.6, 1.625, 1.615.
Compute phi on the calculator as (1 + square root of 5) divided by 2 and write 1.618 at the top.
Draw a second rectangle 21 cm by 34 cm beside it and show that cutting off a 21 cm square leaves a 13 by 21 rectangle with almost the same proportions: 34 divided by 21 is 1.619 and 21 divided by 13 is 1.615.
What you should see
A 13 by 21 cm Fibonacci rectangle with a smooth spiral through its squares, and a table of ratios approaching 1.618 as the numbers grow.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01What is the golden ratio to three decimal places?
1.618.
02What is special about a golden rectangle?
Cutting off a square leaves a smaller rectangle with the same proportions.
03Why do Fibonacci ratios approach phi?
Each Fibonacci number is the sum of the two before, and the ratio of neighbours settles to the value x where x = 1 + 1/x, which is phi.
04Is the spiral drawn with quarter circles a true golden spiral?
No, it is an approximation; the true spiral is a logarithmic spiral that grows by phi every quarter turn.
Take it further
Measure the length-to-width ratio of a playing card, a credit card and a sheet of A4 paper, and your height divided by your navel height, and decide from your own data which of them really come near 1.618.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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