The Golden Ratio
Math is full of surprises, One is the amazing link between Fibonacci numbers and a remarkable number know as the golden ratio.
This painting by Salvador Dali, which includes hints of a dodecahedron in the background, incorporates elements suggestive of the golden ratio.
Below is a special rectangle of length A and width B. The length of side A divided by side B is about 1.618. If you tried to write out the exact decimal value, you would find that the string of digits never comes to an end. The number 1.61803398875… is called the golden ratio.
Any rectangle with sides that correspond to the golden ratio is known as a golden rectangle.
If you cut off a square (B x B) from the original golden rectangle, that leaves a new, smaller rectangle. The length of this new rectangle divided by its width is also 1.61803398875…, making the remnant itself a golden rectangle.
If you cut off a square from the smaller rectangle, it would leave another rectangle with the sides in the same "golden" ratio. You can continue this cutting forever, creating smaller and smaller rectangles and squares, with the same result each time.
When you draw a curve connecting the corners of these nested rectangles,you form a spiral!
The diagonally opposite corners of a set of squares nested within a golden rectangle can be joined by curves to create a spiral.
You can find similar spirals in seashells, snails, spiderwebs, and even stars clustered in galaxies. Many of these spirals, however, don't fit exactly with the specs for a spiral based on the golden ratio, but some come close.
The golden ratio also shows up in regular geometric shapes such as the pentagon (five-sided figure) and the pentagram (five-pointed star).
A pentagram (dashed lines) inscribed within a regular pentagon (solid lines).
If you work out the lengths of lines LN and LM, for example, and calculate the ratio LN/LM, you would get the golden ratio. What other pairs of lines in the diagram would also have lengths that correspond to the golden ratio?
TRY IT!
Look for a link between Fibonacci numbers and the golden ratio.
Write out the first 10 or so Fibonacci numbers. Taking the numbers in pairs and using a calculator, work out the ratio of consecutive pairs of numbers, dividing the larger number by the smaller.
2/1 = ____
3/2 = ____
5/3 = ____
8/5 = ____
13/8 = ____
21/13 = ____
34/21 = ____
What do you observe about the values that you get?
Answers:
As the numbers in the sequence get larger, the values of the ratios of consecutive numbers get closer and closer to the golden ratio, 1.61803398875…:
2, 1.5, 1.666…, 1.6, 1.625, 1.61538…, 1.619…
Notice that the first ratio overshoots the value of the golden ratio and the second ratio goes below. This pattern continues, with one value over, then the next under, getting nearer to the value of the golden ratio.
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