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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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