Constructing The Universe Activity Books

Volume 3

Fibonacci Numbers And The Golden Mean

Explore Harmony in Mathematics, Nature and Art


Michael S. Schneider, M. Ed.

The Fibonacci Numbers form a sequence starting with 0 and 1, nothing and something. The rule is that each two numbers add to make the next, resulting in the famous Fibonacci Series:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

Active involvement, not just reading, is the key phrase for this highly illustrated book about the Fibonacci numbers and Golden Mean. It's for teachers of children and adults who are looking for hands-on activities to explore these fascinating topics. It also has great value for artists, architects, craftspeople and designers who wish to see how the Fibonacci numbers and Golden Mean have been applied to art since pre-dynastic Egypt.

The main idea behind the Fibonacci numbers and Golden Mean, besides building on the recent past, is harmony. These numbers and shapes grow in patterns which also appear in the harmonious designs of nature and in the proportions of great art. The book's underlying theme is interconnected unity, wholeness, balance and stability during growth and change. The idea of harmony weaves through every page.

"Geometry has two great treasures:
one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio.
The first we may compare to a measure of gold; the second we may name a precious jewel."
-- Johannes Kepler

This book will guide some people to become interested in "number theory", the wonderful connections 'behind' and among numbers. Artists, craftspeople, architects and designers will see how the Fibonacci numbers and Golden Mean have been applied by creative people across the centuries and cultures with most masterful excellence. Teachers will appreciate that the activities are clearly described and thoroughly illustrated.

This book has four parts:

* The Fibonacci Numbers, demonstrating the underlying principles of this harmony through "Figurate" arithmetic, representing numbers as piles of beans.

* Fibonacci Numbers In Nature, showing where, how and why these principles are expressed in living growth, with plenty of activities suggested.

* Geometry of the Golden Mean, teaching geometric skills for exploring patterns in nature and art. Get your compass!

* The Golden Mean In Art, seeing how the Golden Mean was applied to art, and with many hands-on examples to explore.

164 pages

Table Of Contents

Introduction 6

The Fibonacci Numbers 7
Weaving Fibonacci Number Patterns 10
Dividing Fibonacci Numbers 13
Adding Consecutive Fibonacci Numbers 14
Adding Every Other Fibonacci Number (Odd Subscripts) 15
Adding Every Other Fibonacci Number (Even Subscripts) 16
Adding Fibonacci Number Squares 17
A Fibonacci Rhythm 18
The Math Behind The Checkerboard Puzzle 18
Fibonacci Rectangles And Squares 21
Subtracting From Fibonacci Rectangles 22
Creating Fibonacci Rectangles 23
Creating Fibonacci Squares 24
Square Fibonacci Symmetry 25
Triple Fibonacci Numbers 26
Ten Fibonacci Piles Will Split Into Eleven Equal Parts 27
Adding Fibonacci Squares Makes Fibonacci Rectangles 28
Fibonacci Numbers In Three Dimensions 29

Fibonacci Numbers In Nature 30
The Whirlpool Structure 32
Draw The Spirals 33
Fibonacci Spiral Flower Software 35
Painting Pine Cones 36
A Spiraling Leaf 38
Painting Cauliflowers 39
Fibonacci Potatoes 40
Pineapple Patterns 41
Pineapple Stars 42
Weave a Pineapple Basket 43
Fibonacci Branching 44
A Spiraling Plant Model 46
Pentagonal Growth Pattern 48
Fibonacci Numbers In The Solar System 50
Audible Fibonacci Rhythms 51
Fibonacci Numbers In The Stock Market 52
Fibonacci Patterns In Art 53
Phi -- The Golden Mean 55
Phi Body Balance 59

The Geometry Of The Golden Mean 64
Divide A Line At The Golden Mean 65
Byrne's Euclid 66
Finding The Golden Mean In Double-Squares 68
Phi In The Vesica Piscis 69
Phi In A Triangle 70
Phi In The Pentagon 71
Unfurling Powers Of Phi 72
Whirling Golden Triangles 73
Flowing Vines 75
The Golden Rectangle 76
The "Most Pleasing Rectangle" Survey 77
A Three-Dimensional Golden Frame 78
Visual Points Of Interest 79
Whirling Squares 80
Another Turn At The Golden Spiral 81
Balance Of The Golden Spiral 82
Enlarging A Golden Rectangle 83
Rabatment 84
Subdivide A Square 87
The Root-Five Rectangle 92
Phi Rectangles and the Pentagram 97
Build Golden Mean Calipers 98
Golden Mean Calipers And The Human Body 99

The Golden Mean In Art 101 (The Golden Mean of this book)
Gebel el-Arak Knife 105
Dual Aker Lions 106
Tutankhamun's Ring With Solar Barque 106
Tutankhamun's Ushabti Box 106
Stela of Bai (Hearing Ear Stela) 107
Mortuary Temple of Neb-Hepet-Ra Mentuhotep (Mentuhotep II) 108
Lintel of Sesostris III 109
Household Stele of Akhenaton and Family 110
Eyes Painted On A Sarcophagus 111
Stela of Ahmose 112
Tutankhamen's Alabaster Boat 113
Stela of Djed-Amon-Iu-Ankh 114
Karnak Temple Floorplan 115
Stela of King Djet 116
The Great Pyramid and Squared Circle 118
“Ebony Fan” of Tutankhamen 120
Tablet of Shamash 121
Meditation And Serpent 122
Narada's Great Ancient Tale 122
Ragmala Paintings 123
The Forbidden City 124
Bonsai Trees 125
The Parthenon 126
Greek Pottery 128
European Painting 135
Jason and the Quest for the Golden Fleece 135
The Ascension 135
Très Belles Heures of the Duc the Berry 136
The Baptism Of Jesus 137
The Virgin of the Rocks 138
The Haywain (Hay Cart) 139
The Birth Of Venus 140
Saint Gregory the Great 141
The Seventh Trumpet 142
Alexander being lowered from a ship in a glass barrel to view the wonders of the sea 142
Lindisfarne Gospels 143
The Annunciation 144
Philosophy Consoling Boethius and Fortune Turning the Wheel 145
Man: Body and Soul 145
Trinity 146
Fall of the Rebel Angels 146
Abbot Vivian and the Monks of Tours Present the Bible to Charles the Bald 147
Philosophy Instructing Boethius On The Role of God 147
Saint Foy Abbey-Church 148
Ka'aba Door 149
Salaman and Absal on the Heavenly Isle 150
Woman 151
Harmony in Red/La desserte 151
The Comtesse d'Haussonville 152
Anne In White 152
Guernica 153
Herter Brothers Furniture 154
Everyday Household Objects 156
Modern Soup Ladle 156
Eames Chair 156
Classic Coffee Pot 156
Desktop Radio 157
Early Transistor Radio 157
Credit Cards 158
Candy Wrappers 158

Hambidge's Phi-related rectangles 159

Web Pages and Books 163

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Overall description of the Activity Books

Thank You and Enjoy!

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© 2008 Michael S. Schneider