<div id="LayerE10" style="visibility: visible; overflow: visible; z-index:0; left:10px; top:5px; position:absolute; "><?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> <head> <!-- ======================================================= --> <!-- Created by AbiWord, a free, Open Source wordprocessor. --> <!-- For more information visit http://www.abisource.com. --> <!-- ======================================================= --> <meta content="text/html;charset=UTF-8" http-equiv="content-type" /> <title>Harmonics & Astronomy - Kepler's Platonic Solids.html</title> <style type="text/css"><!-- #toc, .toc, .mw-warning { border: 1px solid #aaa; background-color: #f9f9f9; padding: 5px; font-size: 95%; } #toc h2, .toc h2 { display: inline; border: none; padding: 0; font-size: 100%; font-weight: bold; } #toc #toctitle, .toc #toctitle, #toc .toctitle, .toc .toctitle { text-align: center; } #toc ul, .toc ul { list-style-type: none; list-style-image: none; margin-left: 0; padding-left: 0; text-align: left; } #toc ul ul, .toc ul ul { margin: 0 0 0 2em; } #toc .toctoggle, .toc .toctoggle { font-size: 94%; }@media print, projection, embossed { body { padding-top:1.0000in; padding-bottom:1.0000in; padding-left:1.2500in; padding-right:1.2500in; } } body { text-indent:0in; text-align:left; font-weight:normal; text-decoration:none; font-variant:normal; color:#000000; font-size:12pt; font-style:normal; widows:2; font-family:'Times New Roman'; } table { } td { border-collapse:collapse; text-align:left; vertical-align:top; } p, h1, h2, h3, li { color:#000000; font-family:'Times New Roman'; font-size:12pt; text-align:left; vertical-align:normal; } --></style> </head> <body> <div> <div style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"></span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> <table style="border-collapse: collapse;" width="600" cellspacing="0" cellpadding="3" bordercolor="#000000" border="0"><tbody><tr><td width="100%" valign="top"> <div align="center"></div> <div> <div align="center"> <span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Image 2</span></div> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Platonic Solids as a Model for the Solar System</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">from: </span><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">Mysterium Cosmographicum</span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US"> by Johannes Kepler (1571-1630)</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Format 5 : 9 (Minor Seventh)</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Johannes Kepler was convinced from his youth onwards that the harmonic order of the universe could be proved empirically. His investigation of this was groundbreaking for the modern approach to natural science: according to his Laws of Celestial Mechanics, still applied today, a planet's distance from the sun can be calculated from its observed orbital period-but this does not explain why it has exactly this distance for this orbital period. This was precisely the question that interested Kepler, and this type of observation is typical for the harmonic way of thinking: it is a matter of the identification of morphological correspondences, which comprise a “physiognomic” statement for the observer. His first approach to this subject was geometric:</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> When one attempts to assemble regular three-dimensional solids from equilateral polygons, it becomes apparent that only five such solids are possible. From equilateral triangles, three solids can be built: the tetrahedron from four, the octahedron from eight, and the icosahedron from twenty triangles. From squares, only the cube can be built, and finally the pentagon-dodecahedron can be built from twelve pentagons. These five solids were first described by Plato, and are therefore called the “Platonic solids”-they can be seen in Image 3. Now each solid can be placed inside a sphere, so that all the corners of the solid touch the surface of the sphere on the inside; and another sphere can be placed inside each solid, so that the sphere's surface is tangent to the center of each surface of the solid from the inside.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Kepler then discovered that when these solids are placed inside each other so that the outer sphere around one solid is also the inner sphere inside the next, the radii of the spheres accurately correspond to the distances between the planets. This is illustrated as a model in Image 2, the lower figure being a magnification of the central part of the upper figure.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Admittedly, the outer planets (the first of which, Uranus, was discovered two hundred years after Kepler's time) have no place in this model, since only five such solids are geometrically possible; and Kepler himself was not completely satisfied with the precision of the ratios. But this discovery encouraged him to look further wherever he could for clues to the harmonic structure of the universe.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">Peter Neubäcker</span></p> <p style="text-align: left;" dir="ltr"><img src="../kayser2/imagelibrary/Astronomy/0.jpg" alt="" /><span lang="en-US" xml:lang="en-US"><br /> </span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Image 3</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">The Platonic Solids and their Relationships</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">from: </span><span style="font-weight: bold; font-style: italic;" lang="en-US" xml:lang="en-US">Harmonices Mundi </span><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">by Johannes Kepler (1571-1630)</span></p> <p style="text-align: center;" dir="ltr"><span style="font-weight: bold;" lang="en-US" xml:lang="en-US">Format 8 : 9 (Whole-Tone)</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US">In 1619, Johannes Kepler published </span><span style="font-style: italic;" lang="en-US" xml:lang="en-US">Harmonices Mundi - the Five Books of Universal Harmonics</span><span lang="en-US" xml:lang="en-US">, which he considered his own most significant work, since it described the musical harmony of the solar system which he had spent his life seeking and had finally found.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In the first two books of this work, he derives qualities of numbers from geometric investigations of the ways in which the regular solids can be arranged. Image 3 comes from the second book; here he puts the regular figures together into three-dimensional bodies. The Platonic solids appear again, here connected to the elements, fire, water, air, and earth, and to the cosmos. A few figures derived from these solids also appear, such as the star-solids first described by Kepler.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In the third book, he transfers the number qualities found in the first two books into the audible domain, using the monochord, and develops an all-embracing musical theory. In the fourth book he applies the insights gained in geometry and music to astrological observations.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> In the fifth book, the astronomical part, he presents his great harmonic discovery: the realization of musical harmonies through the movements of the planets. After first musically examining the distances and orbital periods of the planets from various viewpoints, and obtaining no satisfying results, he then examines the angular velocities of the planets, as an observer on the sun would perceive them, and establishes that these velocities, at the perihelion and aphelion of each planet, when positioned on the monochord, yield the most beautiful musical harmonies-for each individual planet as well as for the various planets compared with one another. This result also applies to the outer planets discovered later, as calculated in modern times; it appears to be a universally valid principle of the structure of our solar system.</span></p> <p style="text-align: left;" dir="ltr"><span lang="en-US" xml:lang="en-US"> Astonished at his discovery, Kepler writes: “...Whether people read this book now or later does not matter. It can wait a hundred years for the readers, if God Himself waited six thousand years for His work to be seen...”</span></p> <p style="text-align: center;" dir="ltr"><img src="../kayser2/imagelibrary/Astronomy/1.jpg" alt="" /></p></div></td></tr></tbody></table> </span></div> <p style="text-align: left;" dir="ltr"></p> </div> </body> </html></div>
History of Harmonics
Ancient Civilizations
Haase - Hermann Bahr & Harmonic Thought
Haase - Leibniz & Kayser
Summary of History
Harmonics & Mathematics
Magic Squares
Number Distribution
Number Symbolism
Fractal Harmonies
Harmonics & Geometry
Musical Ratios
Parabola, Hyperbola, Ellipse
The Proportions
Tone Spirals & Tone Curves
Harmonics & Astronomy
Kepler
Haase on Plato's Harmony of The Spheres
Harmony of The Spheres
Kepler`s Platonic Solids
Planetary Harmonics
Cosmology & Cosmogony
Harmonic Cosmogony
I Ching
Pythagoreanism
Robert Fludd
Robert Fludd Images
Octaves & Spirals
Harmonics & Architecture
Harmonics & Architecture
Architectural Examples
Harmonics & Language
Harmonics & Language
Harmonics & Symbolism
Cosmic Harmony in Symbolism
Doctrine of Correspondences
Harmonics as Symbolism
Hermann Hesse & The Glass Bead Game
Gann & Financial Markets
Partial Tone Diagram & Square of 9
Polar Diagrams & Spiral Charts
Harmonics in Nature
Human Canon
Animals
Plants
Snow Crystals
Harmonics & Music Theory
Harmonics as a Science
Interference
Harmonic Theories & Diagrams
Thought as a Sense
Color & Partial-Tone Coordinates
The Lambdoma
Lambdoma Applications
The Monochord
Library & Resources
Hans Kayser Biography
Hans Kayser's Horoscope
Rudolf Haase Biography
Bibliography of Kayser's Works
Bookstore
Law of the Cosmos
Natural Architecture
The Textbook of Harmonics
News - May 2010
February 2008 Newsletter
Links
