Constructing The Universe Activity Books

Volume 4

Dynamic Rectangles

Explore Harmony in Mathematics and Art


Michael S. Schneider


Dynamic Rectangles, also called Root Rectangles, are the series of rectangles which grow from the diagonal of a square. They're harmoniously linked to each other and can have a powerful effect on viewers. That's why they're specifically chosen as the shape for many designs. You'll be surprised at the many places these particular rectangles appear around us from diplomas, writing paper and household objects to worldwide sacred art, crafts and architecture from the many cultures who discovered their remarkable properties. This highly illustrated book is its own geometry course and will teach you about the fascinating geometry of these rectangles through hands-on activities with a compass and straightedge. After learning the simple geometry, you're invited to do constructions on images of art and objects which use these proportions and discover for yourself how they were intelligently designed. If you're an artist, architect or designer, this information can be quite valuable. Learn from the masters how they used these remarkable rectangles to create the most beautiful and compelling works. You're probably familiar with many of them without knowing the secrets of their proportions.

Constructing a Root Two Rectangle

The entrance to Chartes Cathedral

Dynamic Rectangles

141 pages

Table Of Contents

Introduction 6

The Mathematics of Root Rectangles 8
A Square Foundation 8
Construct a Square 10
The Square’s Diagonal 11
Find The Missing Sides 12
The Scarecrow’s Misstatement 13
The Square Root 14
Spiral Of Doubling Areas 15
The Expansive Power Of The Diagonal 16
Construct a Root Two Rectangle 16
The Root Rectangle Ruler 17
Construct a Root Rectangle Ruler 17
Calculate Square Roots 18
Multiples Of Root Rectangles 19
A Sequence of Squares 20
Root Rectangle Ripples 21
Draw a Root Rectangle Platform 22
Square Roots In Three Dimensions 23
A Spiral Of Square Roots 24
Explore The Root Ruler’s Extensions 25
Root Rectangle Extension Patterns 26
Constructing Root Rectangles Within a Square 27
Construct A Root Ruler Within A Square 28
Relative Areas In A Root Rectangle 29
Divide Any Rectangle Into Equal Parts 30
Serlio’s Door 32
Divide Root Rectangles Into Equal Parts 33
Spirals Within Root Rectangles 35
Root Rectangle Rabatment 39
Rabat Squares In Root Rectangles 40
Hotspots: Points, Lines and Areas Of Visual Interest 41
Locate More Visual Points Of Interest 43

Root Two Rectangles 45
The International Paper Standard 46
Divide A Root Two Rectangle In Half 47
Another Way To Halve A Root Two Rectangle 48
Subdivide A Root Two Rectangle By Rabatment 49
Another Way To Subdivide The Root Two Rectangle 50
Replicate These Subdivisions of the Root Two Rectangle 51
A Bouquet Of Harmonious Rectangles 52
Another Bouquet Of Harmonious Rectangles 53
Patterns With Circles 54
Division Into Only Root Two Rectangles 55
Replicate These Patterns 56
Construct Root Two Rectangles Within A Square 57
Root Two Rectangles In The Octagon 58
Construct Overlapping Root Two Rectangles 59

Root Two Rectangle Art 60
Offering Tablet 61
Box Lid From Tutankhamun’s Tomb 62
Tutankhamun’s Cartouche Jewelry Box 63
Tutankhamun’s Double-Cartouche Perfume Box 64
Baboons Adoring Sunrise 65
Metternich Magical Stela 66
The Pyramid Complex Of Userkaf 68
Menorah From The Jewish Temple In Jerusalem 69
A Greek Oinochoe 70
A Greek Kylix 71
A Red Figure Kylix 72
Shakyamuni Buddha Thangka 74
Temple of Venus At Baalbeck 75
Kiva at Kawaika-A 76
Omayyad Mosque of Damascus 77
Taj Majal And Garden 78
Fatehpuri Mosque Gate 79
The Funeral Of Isfandiyar 80
Coptic Christian Funerary Stela 81
The Divine Geometer 82
Christ With Apostles 84
Page From An Ethiopian Illuminated Gospel 86
The Birth Of Adam 88
Coronation Of The Virgin 89
Laurentian Medici Library Entrance 90
Laurentian Medici Library Floor Tile 91
Silver Wine Goblet 92
West Entrance To Chartres Cathedral 93
Cathedral Of Notre-Dame of Amiens 94
Ely Cathedral 97
The Old Basilica Of Saint Peter, Rome 98
St. Patrick’s Cathedral, New York City 99
Palladio’s Villa Foscari 100
The United States Capitol Building 101
Herter Brothers Cabinet 102
Slave Market With The Disappearing Bust Of Voltaire 103
Classic Wrist Watch 104
Entrance To City Hall (San Francisco) 105
Paper Money Geometry 106

Root Three Rectangles 107
Construct A Root Three Rectangle 108
Rabatment In A Root Three Rectangle 109
Construct A Root Three Rectangle In A Square 111
Rotating Root Three Rectangles 112
Root Three In The Triangle 113
Construct A Double Root Three Rectangle 114
Overlapping Hexagons 114
Subdivide Hexagrams For Root Three Rectangles 115
Overlap Root Three Rectangles 155
A Different Subdivision Of Hexagrams 116
Draw Diminishing Root Three Rectangles 117
Replicate These Root Three Rectangle Constructions 118

Root Three Rectangle Art 120
Pectoral Of Princess Mereret 121
Another Pectoral Of Mereret 122
Stela Of Amenhotep’s Purification 123
The Sphinx Dream Stela 124
Tutankhamun’s Scarab Pectoral 125
Stela Of Ha-hat 126
Chinese Imperial Throne 127
Mourning Athena 128
Greek Cup 129
Greek Skyphos 130
Greek Hydria 131
Arch Of Constantine 132
Mayan Stela of the Maize (Foliated) Cross 133
Polyptych Of Saint Anthony 134
Gunzo’s Dream 135
Gunzo Directing The Monks 136
The Throne Of Grace 137
Strasbourg Cathedral Altar 138
A Young Woman Standing At A Virginal 139
Ivory Comb From Sri Lanka 140

Final Word 141

Bibliography 141 (= 100 x Square Root of Two)

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© 2008 Michael S. Schneider