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Ad Quadratum Construction and Study of the Regular Polyhedra Ad Quadratum Construction and Study of the Regular Polyhedra by Jean Le Mée The Louis and Jeanette Brooks Engineering Design Center Department of Mechanical Engineering Studies in the Design Process Albert Nerken School of Engineering and Design Education The Cooper Union for the Advancement of Science & Art Series A, Number 8 New York, NY 10003 November 2001 Ad Quadratum Construction and Study of the Regular Polyhedra by Jean Le Mée The Louis and Jeanette Brooks Engineering Design Center Department of Mechanical Engineering Studies in the Design Process Albert Nerken School of Engineering and Design Education The Cooper Union for the Advancement of Science & Art Series A, Number 8 New York, NY 10003 November 2001 Copyright ¤ 2001 by Jean Le Mée iii Table of Contents Ad Quadratum Construction and Study of the Regular Polyhedra Acknowledgements ..................................................................................................vi Preface.......................................................................................................................vii Introduction..............................................................................................................1 Generation of the Platonic Forms ..........................................................................3 1. A view from the Center .............................................................................3 2. Complementary views ...............................................................................5 Platonic Forms and Circumsphere.........................................................................5 Cube...............................................................................................................7 Tetrahedron...................................................................................................9 Octahedron....................................................................................................9 Dodecahedron ...............................................................................................11 Icosahedron...................................................................................................11 Construction of the Polyhedra – Example of the Cube........................................13 The Ad Quadratum Construction..........................................................................15 The Challenge of Abul Wefa...................................................................................21 Ratio of Insphere to Circumsphere Radii..............................................................23 Tetrahedron...................................................................................................25 Cube...............................................................................................................25 Octahedron....................................................................................................25 Dodecahedron ...............................................................................................27 Icosahedron...................................................................................................29 Ratio of Intersphere to Circumsphere Radii.........................................................35 Tetrahedron...................................................................................................35 Cube...............................................................................................................35 Octahedron....................................................................................................35 Dodecahedron ...............................................................................................36 Icosahedron...................................................................................................36 Ratio of Insphere to Intersphere Radii ..................................................................36 Tetrahedron...................................................................................................36 Cube...............................................................................................................37 Octahedron....................................................................................................37 iv Dodecahedron ...............................................................................................37 Icosahedron...................................................................................................38 Comparison with Orthoscheme Approach...................................................39 Polyhedra and Regularity .................................................................................41 Stellated Polyhedra ............................................................................................43 1. Stella Octangula.....................................................................................45 2. The Process of Stellation .......................................................................47 (a) Face Stellation...................................................................................49 (i) Case of the dodecahedron ............................................................51 (ii) Case of the icosahedron...............................................................51 (b) The three-dimensional approach......................................................51 3. The Four regular stellated polyhedra ....................................................53 (a) The small stellated dodecahedra (SSD)............................................57 (b) The great dodecahedra (GD) ............................................................65 (c) The great stellated dodecahedra (GSD) ...........................................71 (d) The great icosahedron (GI) ..............................................................79 4. Construction of the Golden Rectangles Determining the Structure of Dodecahedra and Icosahedra......................................................................89 (a) Case of Convex Polyhedra................................................................89 (b) Case of Stellated Polyhedra ..............................................................93 x Case of SSD..................................................................................93 x Case of GD...................................................................................95 x Case of GSD.................................................................................95 x Case of GI ....................................................................................96 Alternative Methods of Generating the Platonic Forms ............................97 1. Historical Perspective ..............................................................................97 2. Method of Internal Angle Vertices Location on the Sphere ................101 (a) with the golden rectangles.................................................................101 (b) by individual polyhedra.....................................................................105 3. Method of Interference............................................................................108 x Tetrahedron........................................................................................110 x Cube ...................................................................................................111 x Octahedron ........................................................................................111 x Icosahedron........................................................................................112 x Dodecahedron....................................................................................112 v 4. Interference Method through Ad Quadratum......................................114 5. Method of the Six Directions of Space ...................................................114 Ad Quadratum Method and the Generation of Spirals .......................................119 Ad Quadratum and the Pythagorean Triples .......................................................123 Ad Quadratum and Music ......................................................................................128 Music and World View................................................................................128 Music and the Platonic Forms ....................................................................129 Pythagorean Scale.........................................................................................131 Just Intonation ..............................................................................................137 Ad Quadratum Construction of the Pythagorean consonances ..............139 Ad Quadratum and the Just Intonation scale ...........................................143 The Aural and the Visual/Esthetic of Proportion .................................................145 Ad Quadratum, the Millennium Sphere, and Astronomy ...................................153 The Millennium Sphere and the Liberal Arts Curriculum .................................157 Appendices: ..............................................................................................................164 x Table 1: Trigonometric Properties of Internal Angles ............................165 x Table 2: Trigonometric Properties of Half Internal Angles ...................166 x Table 3: Regular convex Polyhedra Metrics ...........................................167 x Table 4: Stellation Growth Factors...........................................................168
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