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<br /> <br /> First Edition<br /> FORM GEOMETRY<br /> STRUCTURE<br /> FROM NATURE<br /> TO DESIGN<br /> Author<br /> Daniela Bertol<br /> Editorial<br /> Hyde Park<br /> Design<br /> Quad/Graphics<br /> Exton, Pennsylvania USA<br /> Form Geometry Structure<br /> From Nature to Design<br /> First Edition<br /> Cover Image Credits:<br /> Front cover rendering by Daniela Bertol; Back cover background rendering by<br /> Daniela Bertol. Photographs by Z Corporation and Daniela Bertol of a physical<br /> model designed by the author and realized with the ZPrinter model 650 3D printer<br /> (courtesy of Z Corporation)<br /> Copyright © 2011 by Bentley Institute Press<br /> Bentley Systems, Incorporated<br /> 685 Stockton Drive<br /> Exton, Pennsylvania 19341<br /> www.bentley.com/books<br /> Printed in the United States of America<br /> All rights reserved. No part of this publication may be reproduced, stored in a retrieval<br /> system, or transmitted, in any<a title="Free Preview page 1" href="http://viewer.zmags.com/publication/7aacda28?page=1"> </a> <a title="Free Preview page 2" href="http://viewer.zmags.com/publication/7aacda28?page=2"> First Edition FORM GEOMETRY STRUCTURE FROM</a> <a title="Free Preview page 3" href="http://viewer.zmags.com/publication/7aacda28?page=3"> Form Geometry Structure From Nature to Design </a> <a title="Free Preview page 4" href="http://viewer.zmags.com/publication/7aacda28?page=4"> To the Memory of My Father MARCELLO BERTOL</a> <a title="Free Preview page 5" href="http://viewer.zmags.com/publication/7aacda28?page=5"> </a> <a title="Free Preview page 6" href="http://viewer.zmags.com/publication/7aacda28?page=6"> v C O N T E N T S T</a> <a title="Free Preview page 7" href="http://viewer.zmags.com/publication/7aacda28?page=7"> II DESIGN, MODELS, DIGITAL MODELS AND GENER</a> <a title="Free Preview page 8" href="http://viewer.zmags.com/publication/7aacda28?page=8"> vii IV SYMMETRY IN TWO-DIMENSIONAL SPACE ....</a> <a title="Free Preview page 9" href="http://viewer.zmags.com/publication/7aacda28?page=9"> V SYMMETRY IN THREE-DIMENSIONAL SPACE: PO</a> <a title="Free Preview page 10" href="http://viewer.zmags.com/publication/7aacda28?page=10"> ix VI T R I A N G L E S A N D H E X A G O N S</a> <a title="Free Preview page 11" href="http://viewer.zmags.com/publication/7aacda28?page=11"> VIII SPIRALS, NUMBERS AND PHYLLOTAXIS ...........</a> <a title="Free Preview page 12" href="http://viewer.zmags.com/publication/7aacda28?page=12"> xi Ta b l e o f Contents</a> <a title="Free Preview page 13" href="http://viewer.zmags.com/publication/7aacda28?page=13"> FOREWORD While Daniela was working on this book </a> <a title="Free Preview page 14" href="http://viewer.zmags.com/publication/7aacda28?page=14"> xiii An overuse</a> <a title="Free Preview page 15" href="http://viewer.zmags.com/publication/7aacda28?page=15"> INTRODUCTION TOWARD AN INTERDISCIPLINARY DES</a> <a title="Free Preview page 16" href="http://viewer.zmags.com/publication/7aacda28?page=16"> xv mensuration employing 60-degree vectorial c</a> <a title="Free Preview page 17" href="http://viewer.zmags.com/publication/7aacda28?page=17"> Thompson was a biologist, classicist and mathemat</a> <a title="Free Preview page 18" href="http://viewer.zmags.com/publication/7aacda28?page=18"> xvii Form Geome</a> <a title="Free Preview page 19" href="http://viewer.zmags.com/publication/7aacda28?page=19"> N A T U R E , G E O M E T R Y , FORM AND </a> <a title="Free Preview page 20" href="http://viewer.zmags.com/publication/7aacda28?page=20"> Matter presents itself as geometric forms at any </a> <a title="Free Preview page 21" href="http://viewer.zmags.com/publication/7aacda28?page=21"> Nature, G e o m e t r y, Form and S t r u c t u r</a> <a title="Free Preview page 22" href="http://viewer.zmags.com/publication/7aacda28?page=22"> 5 The study of</a> <a title="Free Preview page 23" href="http://viewer.zmags.com/publication/7aacda28?page=23"> I.A Form is a Diagram of Forces The definition o</a> <a title="Free Preview page 24" href="http://viewer.zmags.com/publication/7aacda28?page=24"> 7 nature, from inanimate matter to living org</a> <a title="Free Preview page 25" href="http://viewer.zmags.com/publication/7aacda28?page=25"> Nature, G e o m e t r y, a. Form and Structure </a> <a title="Free Preview page 26" href="http://viewer.zmags.com/publication/7aacda28?page=26"> 9 I.3 Forces and fields: </a> <a title="Free Preview page 27" href="http://viewer.zmags.com/publication/7aacda28?page=27"> Any state of matter—solid, fluid, gas—is</a> <a title="Free Preview page 28" href="http://viewer.zmags.com/publication/7aacda28?page=28"> 11 Form is a Diagram of Forces </a> <a title="Free Preview page 29" href="http://viewer.zmags.com/publication/7aacda28?page=29"> I.B Historical Highlights </a> <a title="Free Preview page 30" href="http://viewer.zmags.com/publication/7aacda28?page=30"> 13 Historical Highlights </a> <a title="Free Preview page 31" href="http://viewer.zmags.com/publication/7aacda28?page=31"> Nature, </a> <a title="Free Preview page 32" href="http://viewer.zmags.com/publication/7aacda28?page=32"> 15 Historical Highlights </a> <a title="Free Preview page 33" href="http://viewer.zmags.com/publication/7aacda28?page=33"> I.6d Albrecht Dürer: human proportion Nat</a> <a title="Free Preview page 34" href="http://viewer.zmags.com/publication/7aacda28?page=34"> 17 Perceptions and Scale in Scientific E</a> <a title="Free Preview page 35" href="http://viewer.zmags.com/publication/7aacda28?page=35"> I.9a–d Patterns in nature from the seventeeth ce</a> <a title="Free Preview page 36" href="http://viewer.zmags.com/publication/7aacda28?page=36"> 19 Perceptions and Scale in Scientific Experim</a> <a title="Free Preview page 37" href="http://viewer.zmags.com/publication/7aacda28?page=37"> c. Nature, G e o m e t r y, Form and Struc</a> <a title="Free Preview page 38" href="http://viewer.zmags.com/publication/7aacda28?page=38"> 21 Perceptions and Scale in Scientific Experim</a> <a title="Free Preview page 39" href="http://viewer.zmags.com/publication/7aacda28?page=39"> The science of botany as developed by mi</a> <a title="Free Preview page 40" href="http://viewer.zmags.com/publication/7aacda28?page=40"> 23 Perceptions and Scale in Scientific Experim</a> <a title="Free Preview page 41" href="http://viewer.zmags.com/publication/7aacda28?page=41"> I.11 Illustration of the Moon as seen by telesco</a> <a title="Free Preview page 42" href="http://viewer.zmags.com/publication/7aacda28?page=42"> 25 The advent o</a> <a title="Free Preview page 43" href="http://viewer.zmags.com/publication/7aacda28?page=43"> In the last century, tools of scientific</a> <a title="Free Preview page 44" href="http://viewer.zmags.com/publication/7aacda28?page=44"> 27 The electron microscope uses a particle bea</a> <a title="Free Preview page 45" href="http://viewer.zmags.com/publication/7aacda28?page=45"> I.D Forms in Remote Space From time immemorial, </a> <a title="Free Preview page 46" href="http://viewer.zmags.com/publication/7aacda28?page=46"> 29 constellations is an example of the human d</a> <a title="Free Preview page 47" href="http://viewer.zmags.com/publication/7aacda28?page=47"> After the discovery of the telescope, as</a> <a title="Free Preview page 48" href="http://viewer.zmags.com/publication/7aacda28?page=48"> 31 Forms in Remote Space </a> <a title="Free Preview page 49" href="http://viewer.zmags.com/publication/7aacda28?page=49"> I.18 Time lapse photography showing the Sun path</a> <a title="Free Preview page 50" href="http://viewer.zmags.com/publication/7aacda28?page=50"> 33 Forms in Remote Space </a> <a title="Free Preview page 51" href="http://viewer.zmags.com/publication/7aacda28?page=51"> I.20 Diagrams of star trails at different latitu</a> <a title="Free Preview page 52" href="http://viewer.zmags.com/publication/7aacda28?page=52"> 35 Forms in Remote Space</a> <a title="Free Preview page 53" href="http://viewer.zmags.com/publication/7aacda28?page=53"> I.E Taxonomies of Morphology The categorization </a> <a title="Free Preview page 54" href="http://viewer.zmags.com/publication/7aacda28?page=54"> 37 Hexagons and Hexagonal Symmetry </a> <a title="Free Preview page 55" href="http://viewer.zmags.com/publication/7aacda28?page=55"> Nature, G e o m e t r y, Form and S t r u c t u r</a> <a title="Free Preview page 56" href="http://viewer.zmags.com/publication/7aacda28?page=56"> 39 H e x a g o n s a n d H e x a g o n a l S y</a> <a title="Free Preview page 57" href="http://viewer.zmags.com/publication/7aacda28?page=57"> SOAP BUBBLES The Belgian physicist Joseph Platea</a> <a title="Free Preview page 58" href="http://viewer.zmags.com/publication/7aacda28?page=58"> 41 Hexagons and Hexagonal Symmetry </a> <a title="Free Preview page 59" href="http://viewer.zmags.com/publication/7aacda28?page=59"> HONEYCOMBS Bees' honeycombs (figure I.24a) are an</a> <a title="Free Preview page 60" href="http://viewer.zmags.com/publication/7aacda28?page=60"> 43 Hexagons and Hexagonal Symmetry </a> <a title="Free Preview page 61" href="http://viewer.zmags.com/publication/7aacda28?page=61"> The shared end of the honeycomb (figure </a> <a title="Free Preview page 62" href="http://viewer.zmags.com/publication/7aacda28?page=62"> 45 S N O W F L A K E S </a> <a title="Free Preview page 63" href="http://viewer.zmags.com/publication/7aacda28?page=63"> d. Snowflakes from Wilson Bentley, Snow Crystals</a> <a title="Free Preview page 64" href="http://viewer.zmags.com/publication/7aacda28?page=64"> 47 A N D A F E W O T H E R H E X A G O N S</a> <a title="Free Preview page 65" href="http://viewer.zmags.com/publication/7aacda28?page=65"> Another geological formation, Giant's Ca</a> <a title="Free Preview page 66" href="http://viewer.zmags.com/publication/7aacda28?page=66"> 49 Hexagons and Hexagonal Symmetry </a> <a title="Free Preview page 67" href="http://viewer.zmags.com/publication/7aacda28?page=67"> I.G Radial and Spherical Patterns The s</a> <a title="Free Preview page 68" href="http://viewer.zmags.com/publication/7aacda28?page=68"> 51 Radial and Spherical Patterns I.30 Radi</a> <a title="Free Preview page 69" href="http://viewer.zmags.com/publication/7aacda28?page=69"> Nature, G e o m e t r y, I.31 </a> <a title="Free Preview page 70" href="http://viewer.zmags.com/publication/7aacda28?page=70"> 53 Radial and Spherical Patterns </a> <a title="Free Preview page 71" href="http://viewer.zmags.com/publication/7aacda28?page=71"> Nature, G e o m e t r y, I.33 </a> <a title="Free Preview page 72" href="http://viewer.zmags.com/publication/7aacda28?page=72"> 55 Radial and Spherical Patterns </a> <a title="Free Preview page 73" href="http://viewer.zmags.com/publication/7aacda28?page=73"> Nature, G e o m e t r y, I.35 </a> <a title="Free Preview page 74" href="http://viewer.zmags.com/publication/7aacda28?page=74"> 57 Diatoms are </a> <a title="Free Preview page 75" href="http://viewer.zmags.com/publication/7aacda28?page=75"> Nature, G e o m e t r y, b. F</a> <a title="Free Preview page 76" href="http://viewer.zmags.com/publication/7aacda28?page=76"> 59 Radial and Spherical Patterns </a> <a title="Free Preview page 77" href="http://viewer.zmags.com/publication/7aacda28?page=77"> I.38 Pentagonal Symmetry in flowers:Verbascum (V</a> <a title="Free Preview page 78" href="http://viewer.zmags.com/publication/7aacda28?page=78"> 61 Radial and Spherical Patterns </a> <a title="Free Preview page 79" href="http://viewer.zmags.com/publication/7aacda28?page=79"> I.H Polyhedra and Radiolaria Many forms in natur</a> <a title="Free Preview page 80" href="http://viewer.zmags.com/publication/7aacda28?page=80"> 63 Polyhedra and Radiolaria </a> <a title="Free Preview page 81" href="http://viewer.zmags.com/publication/7aacda28?page=81"> As opposed to the skeletons of sponges, </a> <a title="Free Preview page 82" href="http://viewer.zmags.com/publication/7aacda28?page=82"> 65 Polyhedra and Radiolaria </a> <a title="Free Preview page 83" href="http://viewer.zmags.com/publication/7aacda28?page=83"> I.42 The mesh of the silica skeleton of Radiolar</a> <a title="Free Preview page 84" href="http://viewer.zmags.com/publication/7aacda28?page=84"> 67 Polyhedra and Radiolaria </a> <a title="Free Preview page 85" href="http://viewer.zmags.com/publication/7aacda28?page=85"> b. Nature, G e o m e t r y, Form and Structure </a> <a title="Free Preview page 86" href="http://viewer.zmags.com/publication/7aacda28?page=86"> 69 Polyhedra and Radiolaria </a> <a title="Free Preview page 87" href="http://viewer.zmags.com/publication/7aacda28?page=87"> N a t u r e , I.44 Tetrahed</a> <a title="Free Preview page 88" href="http://viewer.zmags.com/publication/7aacda28?page=88"> 71 I.I Partitioning Space: Lattices and Cryst</a> <a title="Free Preview page 89" href="http://viewer.zmags.com/publication/7aacda28?page=89"> The most common methods for observing an</a> <a title="Free Preview page 90" href="http://viewer.zmags.com/publication/7aacda28?page=90"> 73 Partitioning Space: Lattices and Crystals </a> <a title="Free Preview page 91" href="http://viewer.zmags.com/publication/7aacda28?page=91"> I.48 Crystal structure: computer model of the sod</a> <a title="Free Preview page 92" href="http://viewer.zmags.com/publication/7aacda28?page=92"> 75 Neural Networks </a> <a title="Free Preview page 93" href="http://viewer.zmags.com/publication/7aacda28?page=93"> a. </a> <a title="Free Preview page 94" href="http://viewer.zmags.com/publication/7aacda28?page=94"> 77 Neural Networks </a> <a title="Free Preview page 95" href="http://viewer.zmags.com/publication/7aacda28?page=95"> I.51 Images of astrocytes from electron microsco</a> <a title="Free Preview page 96" href="http://viewer.zmags.com/publication/7aacda28?page=96"> 79 I.K Spirals and Phyllotaxis </a> <a title="Free Preview page 97" href="http://viewer.zmags.com/publication/7aacda28?page=97"> The shells of Nautilus (figure I.54), sn</a> <a title="Free Preview page 98" href="http://viewer.zmags.com/publication/7aacda28?page=98"> 81 Spirals and Phyllotaxis </a> <a title="Free Preview page 99" href="http://viewer.zmags.com/publication/7aacda28?page=99"> Many forms in botany also follow spiral </a> <a title="Free Preview page 100" href="http://viewer.zmags.com/publication/7aacda28?page=100"> 83 Spirals and Phyllotaxis </a> <a title="Free Preview page 101" href="http://viewer.zmags.com/publication/7aacda28?page=101"> I.58 Spiral pattern in phyllotaxis at a Norway s</a> <a title="Free Preview page 102" href="http://viewer.zmags.com/publication/7aacda28?page=102"> 85 I.L Self Similarities and Fractals </a> <a title="Free Preview page 103" href="http://viewer.zmags.com/publication/7aacda28?page=103"> Fractual geometry describes natural phen</a> <a title="Free Preview page 104" href="http://viewer.zmags.com/publication/7aacda28?page=104"> 87 Self Similarities and Fractals </a> <a title="Free Preview page 105" href="http://viewer.zmags.com/publication/7aacda28?page=105"> D E S I G N, M O D E L S, D I G I T A L ODEL M </a> <a title="Free Preview page 106" href="http://viewer.zmags.com/publication/7aacda28?page=106"> The recognition of the presence of geometry in na</a> <a title="Free Preview page 107" href="http://viewer.zmags.com/publication/7aacda28?page=107"> Previous Page Digital model of the structur</a> <a title="Free Preview page 108" href="http://viewer.zmags.com/publication/7aacda28?page=108"> 91 solution, or at least to initiate the parti</a> <a title="Free Preview page 109" href="http://viewer.zmags.com/publication/7aacda28?page=109"> II.B Models and Simulation Design can be greatly</a> <a title="Free Preview page 110" href="http://viewer.zmags.com/publication/7aacda28?page=110"> 93 Models and Simulation </a> <a title="Free Preview page 111" href="http://viewer.zmags.com/publication/7aacda28?page=111"> II.C Diagrams, Graphs and Networks The informati</a> <a title="Free Preview page 112" href="http://viewer.zmags.com/publication/7aacda28?page=112"> 95 Diagrams, Graphs and Networks </a> <a title="Free Preview page 113" href="http://viewer.zmags.com/publication/7aacda28?page=113"> replaced by the variables in the model. The varia</a> <a title="Free Preview page 114" href="http://viewer.zmags.com/publication/7aacda28?page=114"> a. c. </a> <a title="Free Preview page 115" href="http://viewer.zmags.com/publication/7aacda28?page=115"> Graphs are diagrams based on abstract re</a> <a title="Free Preview page 116" href="http://viewer.zmags.com/publication/7aacda28?page=116"> 99 Arch</a> <a title="Free Preview page 117" href="http://viewer.zmags.com/publication/7aacda28?page=117"> II.12 Two views of the Panorama, the physical mo</a> <a title="Free Preview page 118" href="http://viewer.zmags.com/publication/7aacda28?page=118"> 101 Digital Models in Architecture </a> <a title="Free Preview page 119" href="http://viewer.zmags.com/publication/7aacda28?page=119"> II.13 A section detail in a construction documen</a> <a title="Free Preview page 120" href="http://viewer.zmags.com/publication/7aacda28?page=120"> 103 Digital Models in Architecture </a> <a title="Free Preview page 121" href="http://viewer.zmags.com/publication/7aacda28?page=121"> BUILDING INFORMATION MODELS Building Informatio</a> <a title="Free Preview page 122" href="http://viewer.zmags.com/publication/7aacda28?page=122"> 105 GenerativeComponents(GC) </a> <a title="Free Preview page 123" href="http://viewer.zmags.com/publication/7aacda28?page=123"> Design, Models, Digital Models and Generative C o</a> <a title="Free Preview page 124" href="http://viewer.zmags.com/publication/7aacda28?page=124"> 107 Generative- Components (GC) </a> <a title="Free Preview page 125" href="http://viewer.zmags.com/publication/7aacda28?page=125"> II.16b An excerpt of GenerativeComponents script </a> <a title="Free Preview page 126" href="http://viewer.zmags.com/publication/7aacda28?page=126"> 109 G e n e r a t i v e - C o m p o n e n t s </a> <a title="Free Preview page 127" href="http://viewer.zmags.com/publication/7aacda28?page=127"> II.G Digital Fabrication In the last three decad</a> <a title="Free Preview page 128" href="http://viewer.zmags.com/publication/7aacda28?page=128"> 111 Digital Fabrication a1. </a>