Transcript PPT - Berkeley EECS - University of California, Berkeley

CONTACT 2006
Music of the Spheres
in More Than 3 Dimensions
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
The world is a mysterious place !
Astrology
 Astronomy
 Astrophysics
 Cosmology
Pythagoras of Samos (569-475 BC)
“Harmony of the Spheres”
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
 As
planets and the stars are attached to these.
they rotate, they created musical harmonies.
 Music of the Spheres
Claudius Ptolemy (85-165)
Johannes Kepler (1571-1630)
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
Kepler – the Geometrician

tilings, polyhedra
Kepler – the Mystic
The “meaning” of the five Platonic solids
Octahedron:
Air
Tetrahedron: Dodecahedron:
Fire
the Universe
Cube:
Icosahedron:
Earth
Water
Johannes Kepler:
“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 Mystic
Trying to relate the sizes of the planetary orbits
Kepler’s Mysterium Cosmographicum
(1596)
 relating
the sizes of the
planetary orbits
via the five
Platonic solids.
Diameters of Inter-Planetery Spheres
from the Book of Copernicus

Jup./Sat.
= .635
Cube:
.577 => -9%

Mars/Jup.
= .333
Tetra:
.333 =>

Earth/Mars = .757

Venus/Earth = .794

Merc./Venus = .723 Octa:
mid-edge radius of Octa:
0%
Dodeca: .795 => +5%
Icosa:
.795 =>
0%
.577 => -20%
.707 => -2%
J. V. Field: "Kepler's Geometrical Cosmology"
Univ. of Chicago Press, 1988, page 65.
A Later Table Expressed in Earth Radii
with corrections by Aiton (1981)

Saturn
aph 9.727 --> 10.588
peri 8.602 --> 9.364

Jupiter aph 5.492 --> 5.403
peri 4.999 --> 4.918
=> -2%

Mars
aph 1.648 --> 1.639
peri 1.393 --> 1.386
=> -1%

Earth
aph 1.042 --> 1.102
peri 0.958 --> 0.898
=> 0% by def.

Venus
aph 0.721 --> 0.714
peri 0.717 --> 0.710
=> -1%

Mercury aph 0.481 -->
peri 0.233 -->
0.502
0.242
=> +9%
=> +4%
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 !

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:

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:
the 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 4D
Projections to 3D Space
The Regular 4D 120-Cell (projected to 3D)

600 vertices, 1200 edges.
The Regular 4D 600-Cell (projected to 3D)
120 vertices,
720 edges.
 David
Richter
Advantage of Using 4D Polytopes

Four different sphere radii on each polytope:
For Hypercube:

Through its vertices = Rv
2.000

Through its edge-midpoints = Re
1.732

Through its face centers = Rf
1.414

Through its cell centers = Rc
1.000
Thus we can form 6 different radius ratios !
Ratios of Sphere Radii of 4D Polytopes
Rc/Rv Rc/Re Rc/Rf Rf/Rv Rf/Re Re/Rv
Simplex
Tesseract
Crosspoly
24-Cell
120-Cell
600-Cell
.250
.500
.500
.707
.926
.925
.408
.577
.707
.816
.934
.973
.408
.707
.577
.817
.973
.934
.612
.707
.866
.866
.951
.991
.667
.816
.817
.943
.982
.982
.612
.866
.707
.866
.991
.951
How Well Do the New Numbers Fit ?
Planet
Orbit
Ratio
Best Fit
%Error
Mercury
Venus
Earth
Mars
Asteroids
Jupiter
Saturn
Uranus
Neptune
Pluto
Sedna
0.39
0.72
1.00
1.53
2.22
5.22
9.58
19.28
30.21
39.63
70.47
0.537
0.725
0.654
0.689
0.425
0.545
0.497
0.638
0.762
0.562
0.577
0.707
0.667
0.707
0.408
0.577
0.5
0.612
0.816
0.577
7.4
-2.5
2.1
2.6
-4.1
5.9
0.6
-4.1
7.1
2.6
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/eb/article-9072653
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

“Selene” ?
2.80
24

Jupiter
5.22
48

Saturn
9.58
96

Georgian Pl.
19.18
192
(1781: Uranus)

Neptune
30.21
298
(1846)
(missing planet)
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
---
Is the Universe
a Dodecahedral
Poincaré Space?
Oct. 2003
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 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 Pyramid
http://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 Science
and Cyclosophy”
Cornelis De Jager (astrophysicist),
Skeptical Inquirer,
Vol 16, No 2, Winter 1992, pp 167 - 172.
Dutch Bicycle
B
L
W
P
= 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...”)
W
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 = 6.67*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)
2.998
error of measurement ?
Computerized 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 database
of natural constants or simple ratios thereof.
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 !!
Golden Ratio is Everywhere ...
length to width of rectangle = 1.61803 39887 49894 84820
Statistics on Random Rectangles
Golden Ratio
1:1
1:2
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.
Key Message !
The number-matching game is too easy to play.
Most of the found results are meaningless !
MUSIC as Art ...
Music of the Spheres
Is it still playing ??
 Let’s
look on the Web ...
Acknowledgements
Thanks to the Internet and to
the Google search engine !
“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
Sand Mandela by Rosalind Gittings
“Music of the Spheres”
by Lisa
[email protected]
“Music of the Spheres” by Isabel Rooney
A Novel by Elizabeth Redfern
 London,
 Spy
1795
story
 French
astronomers
in exile,
 sending
secret information
hidden in tables of
astronomical data.
 Describes
numbers game
by Johannes Titius ...
“Music of the Spheres” by Bernard Xolotl
Yorkshire Building Society Band
Deutsche Bläserphilharmonie
Wind Chimes
“Music of the Spheres” - John Robinson
“Music of the Spheres” by Paul Katrich
“Music of the Spheres”
Kinetic Sculpture by Susan Pascal Beran
“Music of the Spheres” - Nancy Mooslin
“Music of the Spheres” - Nancy Mooslin
Music of the Spheres by Brent Collins
Music of the Spheres