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**– its origin and usefulness in engineering**

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**Fibonacci Series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,… ( add the last two to get the next ) What is the next number?**

**Ratio between numbers**

Leonardo Fibonacci c1175-1250.

Plant branches could be modeled to grow such that they can branch into two every month once they are two months old.

This leads to a Fibonacci series for branch counts

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**Fibonacci’s rabbits**

Rabbits could be modeled to conceive at 1 month of age and have two offspring every month thereafter.

This leads to a Fibonacci series for rabbit counts for each subsequent month

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**3 petals (or sepals) : lily, iris Lilies often have 6 petals formed from two sets of 3 4 petals Very few plants show 4 e.g. fuchsia 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia), orchid 8 petals: delphiniums **

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**Petals on flowers 13 petals: ragwort, corn marigold, cineraria, some daisies 21 petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, Asteraceae family**

**Ratio of Fibonacci numbers**

Divide each number by the number before it, we will find the following series of numbers:

**1 / 1 = 1, 2 / 1 = 2, 3 / 2 = 1·5, 5 / 3 = 1·666..., 8 / 5 = 1·6, 13 / 8 = 1·625, 21 / 13 = 1·61538...**

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**These values converge to a constant value, 1.61803 39887……, the golden section,**

**Dividing a number by the number behind it: 0·61803 39887..... 1/**

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**The golden section in geometry**

**The occurrence of the ratio, **

**The meaning of the ratio **

**The use of **

**in engineering**

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**Constructing the golden section**

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**Geometric ratios involving **

**: **

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**Geometric ratios involving **

**: **

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**Golden Spiral Construction Start with a golden rectangle Construct a square inside Construct squares in the remaining rectangles in a rotational sequence Construct a spiral through the corners of the squares**

**Golden Spiral Shortcut**

http://powerretouche.com/Divine_ proportion_tutorial.htm

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**Golden Triangle and Spirals**

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**1**

1/

**1 1**

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**Golden proportions in humans**

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**Echinacea – the Midwest Coneflower**

Note the spirals originating from the center. These can be seen moving out both clockwise and anti-clockwise.

These spirals are no mirror images and have different curvatures. These can be shown to be square spirals based on series of golden rectangle constructions.

**Cauliflower and Romanesque (or Romanesco) BrocolliXCauliflower**

Note the spiral formation in the florets as well as in the total layout

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The spirals are, once again, golden section based

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**Pine cone spiral arrangements**

The arrangement here can once more be shown to be spirals based on golden section ratios.

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**Pine cone spirals**

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**Fibonacci Rectangles and Shell Spirals**

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**Construction: Brick patterns**

The number of patterns possible in brickwork Increases in a Fibonacci series as the width increases

**Phi in Ancient Architecture**

A number of lengths can be shown to be related in ratio to each other by Phi

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**Golden Ratio in Architecture**

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The Dome of St. Paul, London.

**Windsor Castle**

**Golden Ratio in Architecture**

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**Baghdad City Gate The Great Wall of China**

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**Modern Architecture: Eden project The Eden Project's new Education Building **

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Modern Architecture: California Polytechnic Engineering Plaza

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**More examples of golden sections**

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**Mathematical Relationships for Phi**

The Number Phive

**5 0.5**

**x .5 + .5 = 1.61803399 = phi phive to the power of point phive times point phive plus point phive = phi 1.61803399 2 = 2.61803399 = phi +1 1 / 1.61803399 (the reciprocal) = 0.61803399 = phi - 1 .61803399**

**2 + .61803399 = 1**

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**Golden Ratio in the Arts**

Aztec Ornament

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**Golden Ratio in the Arts**

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**Golden Ratio in the Arts Piet Mondrian’s Rectangles**

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**Design Applications of Phi**

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**Design Applications of Phi**

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**Phi in Advertising**

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**Three-dimensional symmetry: the Platonic solids**

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**Octahedron 8-sided solid**

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**This 12-sided regular solid is the 4 th Platonian figure**

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**Icosahedron 20-sided solid Note the three mutually orthogonal golden rectangles that could be constructed **

**Three-dimensional near-symmetry**

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http://www.mathconsult.ch/showroom/unipoly/list-graph.html