CONTACT 2006CONTACT 2006

Music of the Spheres in More Than 3 Dimensions

Carlo H. Séquin

EECS Computer Science DivisionUniversity of California, Berkeley

The world is a mysterious place !The world is a mysterious place !

Astrology Astrology

Astronomy Astronomy

Astrophysics Astrophysics

Cosmology Cosmology

Pythagoras of Samos (Pythagoras of Samos (569-475 BC569-475 BC))

“Harmony of the Spheres”

World Model of the PythagoreansWorld 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

Claudius Ptolemy (Claudius Ptolemy (85-16585-165))

Johannes Kepler (1571-1630)Johannes Kepler (1571-1630)

Kepler – the ScientistKepler – 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

Kepler – the GeometricianKepler – the Geometrician

tilings, polyhedra

Kepler – the MysticKepler – the Mystic

Octahedron: Tetrahedron: Dodecahedron: Cube: Icosahedron:

Air Fire the Universe Earth Water

The “meaning” of the five Platonic solids

Johannes Kepler:Johannes Kepler:“Music of the Worlds”“Music of the Worlds”

Diagrams from Kepler’s

De Harmonices Mundi (1618),

showing the melody “sung”

by each heavenly body,

and the way in which they

join in six-part counterpoint.

Kepler – the MysticKepler – the Mystic

Trying to relate the sizes of the planetary orbits

Kepler’s Mysterium CosmographicumKepler’s Mysterium Cosmographicum

relating the sizes of the planetary orbitsvia the fivePlatonic solids.

(1596)

Diameters of Inter-Planetery Spheres Diameters of Inter-Planetery Spheres from the Book of Copernicusfrom the Book of Copernicus

Jup./Sat. = .635 Cube: .577 => -9%

Mars/Jup. = .333 Tetra: .333 => 0%

Earth/Mars = .757 Dodeca: .795 => +5%

Venus/Earth = .794 Icosa: .795 => 0%

Merc./Venus = .723 Octa: .577 => -20%

J. V. Field: "Kepler's Geometrical Cosmology"

Univ. of Chicago Press, 1988, page 65.

mid-edge radius of Octa: .707 => -2%

A Later Table Expressed in Earth RadiiA Later Table Expressed in Earth Radiiwith corrections by Aiton (1981)with corrections by Aiton (1981)

Saturn aph 9.727 --> 10.588 => +9% peri 8.602 --> 9.364

Jupiter aph 5.492 --> 5.403 => -2% peri 4.999 --> 4.918

Mars aph 1.648 --> 1.639 => -1% peri 1.393 --> 1.386

Earth aph 1.042 --> 1.102 => 0% by def. peri 0.958 --> 0.898

Venus aph 0.721 --> 0.714 => -1% peri 0.717 --> 0.710

Mercury aph 0.481 --> 0.502 => +4% peri 0.233 --> 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"

A Problem – More than Six Planets !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 ... ? ...

Simplest Regular Objects in Any Dimension:Simplest Regular Objects in Any Dimension: Simplex Series Simplex Series

Connect all the dots among D+1 equally spaced vertices:(Find next one above centroid).

1D 2D 3D

This series goes on indefinitely!

. . .

Another Infinite Series:Another Infinite Series:the Hypercube Seriesthe Hypercube Series

Also called “Measure Polytope” Series

Consecutive perpendicular sweeps:

1D 2D 3D 4D

This series also extends to arbitrary dimensions!

. . .

The 6 Regular Polytopes in 4DThe 6 Regular Polytopes in 4D

Projections to 3D Space

The Regular 4D 120-Cell (The Regular 4D 120-Cell (projected to 3Dprojected to 3D))

600 vertices, 1200 edges.

The Regular 4D 600-Cell (The Regular 4D 600-Cell (projected to 3Dprojected to 3D))

David Richter

120 vertices,

720 edges.

Advantage of Using 4D PolytopesAdvantage 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 !

Ratios of Sphere Radii of 4D PolytopesRatios of Sphere Radii of 4D Polytopes

Simplex .250 .408 .408 .612 .667 .612Tesseract .500 .577 .707 .707 .816 .866Crosspoly .500 .707 .577 .866 .817 .70724-Cell .707 .816 .817 .866 .943 .866120-Cell .926 .934 .973 .951 .982 .991600-Cell .925 .973 .934 .991 .982 .951

Rc/Rv Rc/Re Rc/Rf Rf/Rv Rf/Re Re/Rv

How Well Do the New Numbers Fit ?How Well Do the New Numbers Fit ?

Mercury 0.39Venus 0.72Earth 1.00Mars 1.53Asteroids 2.22Jupiter 5.22Saturn 9.58Uranus 19.28Neptune 30.21Pluto 39.63Sedna 70.47

0.537 0.577 7.40.725 0.707 -2.50.654 0.667 2.10.689 0.707 2.60.425 0.408 -4.10.545 0.577 5.90.497 0.5 0.60.638 0.612 -4.10.762 0.816 7.10.562 0.577 2.6

Planet Orbit Ratio Best Fit %Error

Johann Daniel Titius (Tietz) (Johann Daniel Titius (Tietz) (1729-961729-96))

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/eb/article-9072653

Table by Johann Titius (Table by Johann Titius (17661766))

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

Neptune 30.21 298 (1846)

Georgian Pl. 19.18 192 (1781: Uranus)

“Selene” ? 2.80 24 (missing planet)

Table by Johann Titius (Table by Johann Titius (revisitedrevisited)) 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

Is the UniverseIs the Universea Dodecahedral a Dodecahedral PoincarPoincaréé Space? Space?

Oct. 2003

Evidence for Dodecahedral Universe ?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).

String Concert in 10 Dimensions ?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 Plato’s crystalline spheres.

The Great PyramidThe Great Pyramidhttp://www.infinitetechnologies.co.za/articles/thegreatpyramid.htmlhttp://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 7*107 is the mean distance to the moon. ( * Don’t 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 ...

““Adventures in ScienceAdventures in Scienceand Cyclosophy” and Cyclosophy”

Cornelis De Jager (astrophysicist),

Skeptical Inquirer,Vol 16, No 2, Winter 1992, pp 167 - 172. 

Dutch BicycleDutch Bicycle

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...”)

B

P

L

W

Amazing ResultsAmazing Results P2 * ( L B )1/2 = 1823 =

P4 * W2 = 137.0 = Fine Structure Constant

P-5 * ( L / WB )1/3 = 6.67*10-8 = Gravitation Constant

P1/2 * B1/3 / L = 1.496 = Distance to Sun (108 km)

W * P2 * L1/3 * B5 = 2.999*105 ~ Speed of Light (km/s)

Mass of Proton Mass of Electron

2.998error of measurement ?

Computerized SearchComputerized Search

= Aa * Bb * Cc * Dd

a, b, c, d can assume:all 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.

Matching Your Measurements Matching Your Measurements to Your Favorite Theory ...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 !!

Golden Ratio is Everywhere ...Golden Ratio is Everywhere ...length to width of rectangle = 1.61803 39887 49894 84820length to width of rectangle = 1.61803 39887 49894 84820

Statistics on Random RectanglesStatistics on Random Rectangles

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.

Golden Ratio

1:1 1:2

Key Message !Key Message !

The number-matching game is too easy to play.

Most of the found results are meaningless !

MUSIC as Art ...

Music of the SpheresMusic of the Spheres

Is it still playing ?? playing ??

Let’s look on the Web ...

Acknowledgements

Thanks to the Internet and to the Google search engine !

The Science of Harmonic Energy and Spirit

unification of the harmonic languages of color,

music, numbers and waves 

““Music of the Spheres” Music of the Spheres” www.spectrummuse.comwww.spectrummuse.com

Sand Mandela by Rosalind GittingsSand Mandela by Rosalind Gittings

““Music of the Spheres”Music of the Spheres”by Lisa by Lisa shukti@techren.comshukti@techren.com

““Music of the Spheres” by Isabel RooneyMusic of the Spheres” by Isabel Rooney

A Novel by Elizabeth RedfernA Novel by Elizabeth Redfern London, 1795

Spy story

French astronomersin exile,

sending secret informationhidden in tables of astronomical data.

Describes numbers gameby Johannes Titius ...

““Music of the Spheres” by Bernard XolotlMusic of the Spheres” by Bernard Xolotl

Yorkshire Building Society BandYorkshire Building Society Band

Deutsche BlDeutsche Blääserphilharmonieserphilharmonie

Wind ChimesWind Chimes

““Music of the Spheres” - John RobinsonMusic of the Spheres” - John Robinson

““Music of the Spheres” by Paul KatrichMusic of the Spheres” by Paul Katrich

““Music of the Spheres” Music of the Spheres” Kinetic Sculpture by Susan Pascal BeranKinetic Sculpture by Susan Pascal Beran

““Music of the Spheres” - Nancy MooslinMusic of the Spheres” - Nancy Mooslin

““Music of the Spheres” - Nancy MooslinMusic of the Spheres” - Nancy Mooslin

Music of the Spheres by Brent CollinsMusic of the Spheres by Brent Collins

Music of the SpheresMusic of the Spheres