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*Fibonacci Numbers and the

Golden Ratio

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*The Fibonacci Numbers start with 1 and 1.

*The next number represents the sum of the previous two numbers.

*1 1 2 3 5 8 13 21 34 55 89 144 etc.

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*1 1 2 3 5 8 13 21 34 55 89 etc.

If you take any two consecutive Fibonacci numbers and divide them, then the value of that ratio is approximately 1.6

This is often called the golden ratio

Small Number

Large Number

Ratio

3 5 1.666666

5 8 1.6

8 13 1.625

13 21 1.615

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*1 1 2 3 5 8 13 21 34 55 89 etc.

Rectangles (called golden rectangles) can be drawn using consecutive Fibonacci numbers as the dimensions.

This golden rectangle (55 wide by 34 high) can be subdivided into smaller golden rectangles.

A nice spiral can be drawn in each golden (“Fibonacci”( rectangle – no matter what the size.

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*The Golden Ratio in Nature

Spirals of a pineapple:

Three arms of 5, 8, and 13

Spirals of a pinecone:

5 arms and 8 arms (small ones)

8 arms and 13 arms (large ones)

Scientists speculate that plants grow in spiral formation (with Fibonacci numbers) because this arrangement makes for the perfect spacing for maximum growth and survival.

Bananas have 3 or 5 flat sides.

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*The Golden Ratio in Nature

* Lilies and irises = 3 petals

Black-eyed Susan’s = 21 petalsCorn marigolds = 13 petals

Buttercups and wild roses = 5 petals

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~ Fibonacci spiral found in both snail and sea shells

*The Golden Ratio in Nature

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*The Golden Ratio in Music

A scale has 8 notes

There are 13 notes through its octave

5 black notes, 8 white notes

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*The Golden Ratio in The Human BodyBody parts:

1 nose

2 eyes

3 segments in each limb

5 fingers on each hand

DNA molecules:

21 angstroms wide

34 angstroms long

(in each full cycle of the double helix)

The ratio of consecutive parts of the fingers is equal to 1.6

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*The Golden Ratio in Architecture

The Parthenon ( an ancient temple in Athens) fits almost precisely into a golden rectangle.

In the United Nations building, the width of the building compared with the height of every ten floors is a golden rectangle

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*The Golden Ratio in Art

The Parthenon ( an ancient temple in Athens) fits almost precisely into a golden rectangle.

Mona Lisa’s beautiful smile appears to change and her eyes seem to follow you as you move about her. Her proportions have been examined extensively, and Leonardo seems to have made extensive use of the golden rectangle in its creation.

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*Fibonacci Numbers in Everyday Life

The “Rule of Thirds” in photography

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*Fibonacci Numbers in Everyday Life

• Cigarette packs (89 x 55 mm)

• Matchboxes (55 x 34 mm)

• Business cards (89 x 55 mm)

• Index cards

• Some calculators and cell phones

• Some computer and TV screens

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*Links on YouTube

*Fibonacci: Fingerprint of God

*Golden Ratio in the Human Body

*1.618 Phi