CONTACT 2006

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CONTACT 2006

• Music of the Spheres in More Than 3
  Dimensions

Carlo H. Séquin EECS Computer Science
Division University of California, Berkeley
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The world is a mysterious place !
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Astrology ? Astronomy ? Astrophysics ?
Cosmology
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Pythagoras of Samos (569-475 BC)

• Harmony of the Spheres

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World Model of the Pythagoreans

• Earth is at the center.
• It is surrounded by 5 crystalline
  spheres,spanned and held up by the 5 Platonic
  solids.
• The planets and the stars are attached to these.
• As they rotate, they created musical harmonies.

? Music of the Spheres
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Claudius Ptolemy (85-165)
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Johannes Kepler (1571-1630)
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Kepler the Scientist

• Planetary orbits
• 1. ellipses sun in one focal point.
• 2. equal areas swept out in equal time.
• 3. (revolution times)2 (long orbit axes)3

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Kepler the Geometrician

• tilings, polyhedra

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Kepler the Mystic
The meaning of the five Platonic solids

• Octahedron Tetrahedron Dodecahedron
  Cube Icosahedron
• Air Fire
  the Universe Earth Water

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Johannes KeplerMusic of the Worlds

• Diagrams from Keplers De Harmonices Mundi
  (1618), showing the melody sung by each
  heavenly body, and the way in which they join
  in six-part counterpoint.

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Kepler the Mystic

• Trying to relate the sizes of the planetary
  orbits

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Keplers Mysterium Cosmographicum
(1596)

• relating the sizes of the planetary orbitsvia
  the fivePlatonic solids.

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Diameters of Inter-Planetery Spheres from the
Book of Copernicus

• Jup./Sat. .635 Cube .577 gt
  -9
• Mars/Jup. .333 Tetra .333 gt
  0
• Earth/Mars .757 Dodeca .795 gt 5
• Venus/Earth .794 Icosa .795 gt
  0
• Merc./Venus .723 Octa .577 gt -20

mid-edge radius of Octa .707 gt -2
J. V. Field "Kepler's Geometrical
Cosmology" Univ. of Chicago Press, 1988, page 65.
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A Later Table Expressed in Earth Radiiwith
corrections by Aiton (1981)

• Saturn aph 9.727 --gt 10.588 gt 9
  peri 8.602 --gt 9.364
• Jupiter aph 5.492 --gt 5.403 gt -2
  peri 4.999 --gt 4.918
• Mars aph 1.648 --gt 1.639 gt -1
  peri 1.393 --gt 1.386
• Earth aph 1.042 --gt 1.102 gt 0 by
  def. peri 0.958 --gt 0.898
• Venus aph 0.721 --gt 0.714 gt -1
  peri 0.717 --gt 0.710
• Mercury aph 0.481 --gt 0.502 gt 4
  peri 0.233 --gt 0.242

Adding the orbit of the moon to make a thicker
shell for the earth
Explanation of errors Saturn "too far away,
Mercury "too close to sun"
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A Problem More than Six Planets !

• There are only 5 Platonic solids,
• but there are more than 5 orbit intervals!
• Universe has more than 3 dimensions
• Look into higher dimensions for additional
  Platonic solids.
• Higher dimensions ... ? ...

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Simplest Regular Objects in Any Dimension ?
Simplex Series

• Connect all the dots among D1 equally spaced
  vertices(Find next one above centroid). 1D
  2D 3D

. . .
This series goes on indefinitely!
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Another Infinite Seriesthe Hypercube Series

• Also called Measure Polytope Series
• Consecutive perpendicular sweeps

. . .
1D 2D 3D
4D
This series also extends to arbitrary dimensions!
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The 6 Regular Polytopes in 4D
Projections to 3D Space
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The Regular 4D 120-Cell (projected to 3D)

• 600 vertices, 1200 edges.

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The Regular 4D 600-Cell (projected to 3D)
120 vertices, 720 edges.

• David Richter

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Advantage of Using 4D Polytopes

• Four different sphere radii on each polytope
• Through its vertices Rv
• Through its edge-midpoints Re
• Through its face centers Rf
• Through its cell centers Rc

For Hypercube 2.000 1.732 1.414 1.000
Thus we can form 6 different radius ratios !
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Ratios of Sphere Radii of 4D Polytopes
Rc/Rv Rc/Re Rc/Rf Rf/Rv Rf/Re Re/Rv
Simplex .250 .408 .408 .612 .667 .612 Tesseract .5
00 .577 .707 .707 .816 .866 Crosspoly .500 .707 .5
77 .866 .817 .707 24-Cell .707 .816 .817 .866 .943
.866 120-Cell .926 .934 .973 .951 .982 .991 600-C
ell .925 .973 .934 .991 .982 .951
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How Well Do the New Numbers Fit ?
Planet Orbit Ratio Best Fit
Error
Mercury 0.39 Venus 0.72 Earth 1.00 Mars
1.53 Asteroids 2.22 Jupiter 5.22 Saturn
9.58 Uranus 19.28 Neptune 30.21 Pluto 39.63 Sedna
70.47
0.537 0.577 7.4 0.725 0.707 -2.5 0.654 0.6
67 2.1 0.689 0.707 2.6 0.425 0.408 -4.1
0.545 0.577 5.9 0.497 0.5
0.6 0.638 0.612 -4.1 0.762 0.816
7.1 0.562 0.577 2.6
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Johann Daniel Titius (Tietz) (1729-96)

• Prussian astronomer, physicist, and biologist
  whose law (1766) expressing the distances between
  the planets and the Sun was confirmed by J.E.
  Bode in 1772.

"Titius, Johann Daniel." Encyclopædia Britannica.
2006. Encyclopædia Britannica Premium
Service. 12 Mar. 2006 http//www.britannica.com/e
b/article-9072653
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Table by Johann Titius (1766)

• PLANET ORBIT 10R-4
• Mercury 0.39 0
• Venus 0.72 3
• Earth 1.00 6
• Mars 1.53 12
• Jupiter 5.22 48
• Saturn 9.58 96

• Selene ? 2.80 24 (missing planet)

• Georgian Pl. 19.18 192 (1781 Uranus)

• Neptune 30.21 298 (1846)

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Table by Johann Titius (revisited)

• PLANET ORBIT 10R-4
• Mercury 0.39 0
• Venus 0.72 3
• Earth 1.00 6
• Mars 1.53 11
• (asteroids) - - -
• Jupiter 5.22 48
• Saturn 9.58 92
• Uranus 19.18 188
• Neptun 30.06 296

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Is the Universea Dodecahedral Poincaré Space?

• Oct. 2003

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Evidence for Dodecahedral Universe ?

• Power spectrum of the cosmic microwave background
  (CMB) radiation. Data from WMAP have extended the
  accuracy of the spectrum far beyond what was
  known from earlier measurements. This plot
  reflects the small differences in the temperature
  of the CMB across the sky. There are a series of
  peaks in the spectrum at small angular
  separations, but at large scales that structure
  disappears. Standard cosmological models cannot
  explain this, but Luminet and colleagues
  topological model for a finite universe can
  (image and text credit Nature 425 566).

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String Concert in 10 Dimensions ?

• String theory, the current favorite ...
• 1200 scientists, mathematicians work on it.
• Subatomic particles are resonances of very small
  (10-35m) loopy strings.
• Need to introduce 7 extra dimensions to make
  numbers work out sort of ...
• These strings are as invisible as Platos
  crystalline spheres.

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The Great Pyramidhttp//www.infinitetechnologies.
co.za/articles/thegreatpyramid.html

• Mean Distance to the Sun The height of the
  pyramid times 109 represents the mean radius of
  the Earth's orbit around the sun.
• Mean Distance to the Moon The length of the
  Jubilee passage times 7107 is the mean
  distance to the moon. ( Dont ask ! )
• Tropical Year The length of the Antechamber
  used as the diameter of a circle produces a
  circumference of 365.242 (accurate to 6 digits).
• Many more ...

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Adventures in Scienceand Cyclosophy

• Cornelis De Jager (astrophysicist),
• Skeptical Inquirer,Vol 16, No 2, Winter 1992, pp
  167 - 172. 

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Dutch Bicycle
B
L
W
P

• W Wheel diameter (defines direction of
  path)P Pedal diameter (gives power, forward
  dynamics) L Lamp diameter (enlightens the
  search path) B Bell diameter (means of
  communication...)

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Amazing Results
Mass of Proton Mass of Electron

• P2 ( L B )1/2 1823
• P4 W2 137.0 Fine Structure Constant
• P-5 ( L / WB )1/3 6.6710-8 Gravitation
  Constant
• P1/2 B1/3 / L 1.496 Distance to Sun
  (108 km)
• W? P2 L1/3 B5 2.999105 Speed of
  Light (km/s)

2.998error of measurement ?
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Computerized Search

• ? Aa Bb Cc Dd
• a, b, c, d can assumeall integer values from
  5 to 5,and also the values 1/2, 1/3,
  ?.
• A, B, C, D, are arbitrary assumed constants.
• Compare ? (83521 combinations) with databaseof
  natural constants or simple ratios thereof.

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Matching Your Measurements to Your Favorite
Theory ...

• You can always find good matches, if you look
  hard enough and ignore measurement
  uncertainties.
• So this seems like a pretty silly game ...
• Millions of people are doing it !!

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Golden Ratio is Everywhere ...length to width of
rectangle 1.61803 39887 49894 84820
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Statistics on Random Rectangles
Golden Ratio
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• In range of rectangle ratios from 1.0 to 2.0
• 1/3 of all rectangles fit within 10 (1.45-1.78)
• 1/30 fit within 1 (1.602-1.634) of golden ratio.

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Key Message !

• The number-matching game is too easy to play.
• Most of the found results are meaningless !

MUSIC as Art ...
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Music of the SpheresIs it still playing ??

• Lets look on the Web ...

Acknowledgements Thanks to the Internet and to
the Google search engine !
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Music of the Spheres www.spectrummuse.com

• The Science of Harmonic Energy and Spirit
• unification of the harmonic languages of color,
  music, numbers and waves 

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Sand Mandela by Rosalind Gittings
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Music of the Spheresby Lisa
shukti_at_techren.com
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Music of the Spheres by Isabel Rooney
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A Novel by Elizabeth Redfern

• London, 1795
• Spy story
• French astronomersin exile,
• sending secret informationhidden in tables of
  astronomical data.
• Describes numbers gameby Johannes Titius ...

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Music of the Spheres by Bernard Xolotl
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Yorkshire Building Society Band
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Deutsche Bläserphilharmonie
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Wind Chimes
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Music of the Spheres - John Robinson
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Music of the Spheres by Paul Katrich
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Music of the Spheres Kinetic Sculpture by
Susan Pascal Beran
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Music of the Spheres - Nancy Mooslin
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Music of the Spheres - Nancy Mooslin
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Music of the Spheres by Brent Collins
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Music of the Spheres