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BioE 202: AestheticsBioE 202: AestheticsBioE 202: AestheticsBioE 202: Aesthetics

The Golden Section – its origin and usefulness in engineering

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The Fibonacci SeriesThe Fibonacci SeriesThe Fibonacci SeriesThe Fibonacci Series

Fibonacci Series0, 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.

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Fibonacci and plant Fibonacci and plant growthgrowthFibonacci and plant Fibonacci and plant growthgrowth

Plant branches couldbe modeled to grow such that they can branchinto two every month once they are two months old.This leads to a Fibonacciseries for branch counts

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Fibonacci’s rabbitsFibonacci’s rabbitsFibonacci’s rabbitsFibonacci’s rabbits

Rabbits could be modeled to conceive at 1 month of ageand have two offspring every month thereafter.This leads to a Fibonacciseries for rabbit counts for each subsequent month

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Petals on flowersPetals on flowersPetals on flowersPetals on flowers

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 flowersPetals on flowersPetals on flowersPetals 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

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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...

                                                                           

Ratio of Fibonacci numbers

These values convergeto 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 The golden section in geometrygeometryThe golden section in The golden section in geometrygeometry

The occurrence of the ratio, The meaning of the ratio The use of in engineering

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Constructing the golden Constructing the golden sectionsectionConstructing the golden Constructing the golden sectionsection

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Geometric ratios Geometric ratios

involving involving : : PentagonPentagonGeometric ratios Geometric ratios

involving involving : : PentagonPentagon

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Geometric ratios Geometric ratios

involving involving : : DecagonDecagonGeometric ratios Geometric ratios

involving involving : : DecagonDecagon

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Golden Spiral Golden Spiral ConstructionConstructionGolden Spiral Golden Spiral ConstructionConstruction

Start with a golden rectangleConstruct a square insideConstruct squares in the remaining rectangles in a rotational sequence

Construct a spiral through the corners of the squares

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Golden Spiral ShortcutGolden Spiral ShortcutGolden Spiral ShortcutGolden Spiral Shortcut

http://powerretouche.com/Divine_proportion_tutorial.htm

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Golden Triangle and Golden Triangle and SpiralsSpiralsGolden Triangle and Golden Triangle and SpiralsSpirals

1

1

1

1/

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Golden proportions in Golden proportions in humanshumansGolden proportions in Golden proportions in humanshumans

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Echinacea – the Midwest Echinacea – the Midwest ConeflowerConeflowerEchinacea – the Midwest Echinacea – the Midwest ConeflowerConeflower

Note the spirals originatingfrom the center. These can be seen moving out bothclockwise and anti-clockwise.These spirals are no mirrorimages and have differentcurvatures. These can be shown to be square spiralsbased on series of golden rectangle constructions.

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Cauliflower and Romanesque Cauliflower and Romanesque (or Romanesco) (or Romanesco) BrocolliXCauliflowerBrocolliXCauliflower

Cauliflower and Romanesque Cauliflower and Romanesque (or Romanesco) (or Romanesco) BrocolliXCauliflowerBrocolliXCauliflowerNote the spiral formation in the florets as well as in the total layout

The spirals are, once again, golden section based

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Pine cone spiral Pine cone spiral arrangementsarrangementsPine cone spiral Pine cone spiral arrangementsarrangements

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

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Pine cone spiralsPine cone spiralsPine cone spiralsPine cone spirals

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Fibonacci Rectangles Fibonacci Rectangles and and Shell SpiralsShell Spirals

Fibonacci Rectangles Fibonacci Rectangles and and Shell SpiralsShell Spirals

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Construction: Brick Construction: Brick patternspatternsConstruction: Brick Construction: Brick patternspatterns

The number of patternspossible in brickwork Increases in a Fibonacciseries as the width increases

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Phi in Ancient Phi in Ancient ArchitectureArchitecturePhi in Ancient Phi in Ancient ArchitectureArchitecture

A number of lengthscan be shown to berelated in ratio to eachother by Phi

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Golden Ratio in Golden Ratio in ArchitectureArchitectureGolden Ratio in Golden Ratio in ArchitectureArchitecture

The Dome of St. Paul, London. Windsor Castle

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Golden Ratio in Golden Ratio in ArchitectureArchitectureGolden Ratio in Golden Ratio in ArchitectureArchitecture

Baghdad City Gate The Great Wall of China

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Modern Architecture: Eden Modern Architecture: Eden projectprojectModern Architecture: Eden Modern Architecture: Eden projectproject

The Eden Project's new Education

Building

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Modern Architecture: California Polytechnic Engineering Plaza Modern Architecture: California Polytechnic Engineering Plaza

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More examples of golden More examples of golden sectionssectionsMore examples of golden More examples of golden sectionssections

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Mathematical Relationships Mathematical Relationships for Phifor PhiMathematical Relationships Mathematical Relationships for Phifor Phi

The Number Phive

 50.5 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 

 .618033992 + .61803399 = 1

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Golden Ratio in the ArtsGolden Ratio in the Arts Aztec Ornament

Golden Ratio in the ArtsGolden Ratio in the Arts Aztec Ornament

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Golden Ratio in the ArtsGolden Ratio in the ArtsGolden Ratio in the ArtsGolden Ratio in the Arts

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Golden Ratio in the ArtsGolden Ratio in the ArtsGolden Ratio in the ArtsGolden Ratio in the Arts

Piet Mondrian’s Rectangles

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Design Applications of PhiDesign Applications of PhiDesign Applications of PhiDesign Applications of Phi

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Design Applications of PhiDesign Applications of PhiDesign Applications of PhiDesign Applications of Phi

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Phi in DesignPhi in DesignPhi in DesignPhi in Design

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Phi in AdvertisingPhi in AdvertisingPhi in AdvertisingPhi in Advertising

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

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

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Octahedron Octahedron 8-sided solid8-sided solidOctahedron Octahedron 8-sided solid8-sided solid

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DodecahedronDodecahedronDodecahedronDodecahedron

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

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Note the three mutuallyorthogonal goldenrectangles that could be constructed

Icosahedron Icosahedron 20-sided solid20-sided solidIcosahedron Icosahedron 20-sided solid20-sided solid

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Three-dimensional near-Three-dimensional near-symmetrysymmetryThree-dimensional near-Three-dimensional near-symmetrysymmetry

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