1
1. The TitiusBode
Law
Sir Isaac Newton (1642–1727) showed in his Principia Mathematica
that a gravitational force varying as the inverse square of their distance from the Sun binds
the planets in orbits that are ellipses (Fig. 1). These curves are characterised by their eccentricities:
e = (1 – b^{2}/a^{2})^{½},
where a is the length of the semimajor axis and b is the length of the
semiminor axis. A circle, which has a = b, has zero eccentricity. All the planets have nearly
circular orbits, except Mercury with e = 0.21 and Pluto with e = 0.25. Planets revolve in an
anticlockwise sense around the Sun situated at one of the two foci of each ellipse.
The distance of each focus from the centre of the planet’s ellipse is ae.
Its distance at its closest point A to the Sun is therefore (a – ae) and its distance at its
farthest point B is (a + ae). Their sum is 2a, so that the average of these distances is a, the
length of the semimajor axis. Unless stated otherwise, it is this average of its
largest and shortest distances from the Sun that is meant when the text refers to a planetary
distance.
In 1766, the German mathematician, Johan Daniel Titius (1729–1796) of
Wittenberg, translated into German “Contemplation de la Nature,” by the French natural
philosopher Charles Bonnet. To the paragraph where Bonnet remarks: “We know seventeen planets
that enter into the composition of our solar system;* but we are not sure that there are no more,” Titius added what is now
known as the ‘TitiusBode Law’ (or sometimes ‘Bode’s Law’):
“Take notice of the distances of the planets from one another, and recognize
that almost all are separated from one another in a proportion which matches their bodily
magnitudes. Divide the distance from the Sun to Saturn into 100 parts; then Mercury is
separated by four such parts from the Sun, Venus by 4 + 3 = 7 such parts, the Earth by 4 + 6
= 10, Mars by 4 + 12 = 16. However, notice that from Mars to Jupiter there comes a deviation
from this so exact progression. From Mars there follows a space of 4 + 24 = 28 such parts,
but so far no planet was sighted there. But should the Lord Architect have left that space
empty? Not at all. Let us therefore assume
____________________________
* That is, major planets and their satellites.
2
that this space without doubt belongs to the still undiscovered satellites
of Mars, let us also add that perhaps Jupiter still has around itself some smaller ones,
which have not been sighted yet by any telescope. Next to this for us still unexplored space
there rises Jupiter's sphere of influence at 4 + 48 = 52 parts; and that of Saturn at 4 + 96
= 100 parts. What a wonderful relation!”^{1}
The German astronomer, Johan Elert Bode (1747–1826), was putting the
finishing touches in 1772 to the second edition of his introduction to astronomy “Anleitung zur
Kenntniss des gestimten Himmels,” which he originally published in 1768 at the age of 19, when
he came across the relationship proposed by Titius in a footnote to the second edition of his
translation. Convinced by it, he added it as a footnote in his text, although only
acknowledging Titius as his source in later editions, possibly because of some urging by
him.
Despite this plagiarisation of Titius’s discovery, the relationship came to
be known as Bode’s Law, although he merely popularized it. In fact, it is not a physical law at
all because that status requires a conceptual foundation for what remains merely an empirical
relationship between numbers and average planetary distances. Nevertheless, this article will
follow contemporary practice by referring to it as the ‘TitiusBode Law,’ whilst at the same
time recognising that it is but a rule.
Titius had noticed that, if 0 were assigned to Mercury, 3 to Venus, 6 to
Earth, 12 to Mars, etc, that is, 3 times successive powers of 2, and then 4 added, the
resulting integers when divided by 10 were approximately equal to the average distances of the
planets then known from the Sun in terms of the SunEarth distance:

Mercury

Venus

Earth

Mars

Asteroids

Jupiter

Saturn

Uranus

Neptune

Pluto


0

3

6

12

24

48

96

192

384

768

Add 4: 
4

7

10

16

28

52

100

196

388

772

Divide by 10: 
0.4

0.7

1.0

1.6

2.8

5.2

10.0

19.6

38.8

77.2

The division by 10 enables the distances to be compared with that of the
Earth, whose distance from the Sun is about 93 million miles, or one Astronomical Unit
(AU).*
One way to help visualize the relative sizes in the Solar System
(Fig. 2) is to imagine a model in which it is reduced in size by a factor of a
billion. Then the Earth is about 1.3
____________________________
* The exact figure is 149,597,871 Km.
3
cm in diameter (the size of a grape). The Moon orbits about a foot away. The
Sun is 1.5 metres in diameter (about the height of a man) and 150 metres (about a city block)
from the Earth. Jupiter is 15 cm in diameter (the size of a large grapefruit) and 5 blocks away
from the Sun. Saturn (the size of an orange) is 10 blocks away; Uranus and Neptune (lemons) are
20 and 30 blocks away. A human on this scale is the size of an atom; the nearest star would be
over 40,000 km away.
Table 1 indicates that the distances of the planets from the Sun show good
agreement with those predicted by the TitiusBode Law as far as Uranus but fail for the next
two planets, Neptune and Pluto.
Table 1
Planet 
TitiusBode law

Actual distance (AU)

Mercury 
0.4

0.39

Venus 
0.7

0.72

Earth 
1.0

1.00

Mars 
1.6

1.52

Asteroids 
2.8

2.77

Jupiter 
5.2

5.20

Saturn 
10.0

9.54

Uranus 
19.6

19.19

Neptune 
38.8

30.06

Pluto 
77.2

39.48

It was first tested in 1781 when William Herschel discovered Uranus at a
distance predicted by the relationship. It was accepted by astronomers until the discovery of
Neptune in 1846. It is interesting that some of the larger asteroids between Mars and Jupiter
satisfy the law. This indicates that the Asteroid Belt is likely remnants of the
protoplanetary nebula that failed to form a planet. Ceres, discovered by G. Piazzi on January
1, 1801, is the largest asteroid and the first to be discovered. It comprises over onethird of
the total mass of all the asteroids and has a distance from the Sun of 2.77 AU, which compares
with the predicted value of 2.8 AU. The larger asteroids have distances that spread about this
figure. The asteroid Kleopatra shows the best agreement with the TitiusBode Law with a
distance of 2.793 AU. One object in the Asteroid Belt, Chiron, discovered in 1977, is anomalous
in that its orbital period of 50.7 years is much larger than typical asteroid periods of 3–5
years, whilst its mean distance from the Sun is 13.63 AU, which compares with their typical
values of 2–3 AU. Because it is emitting supervolatiles, it could not have been in its present
orbit for very long. It is thought likely to be an intruder from a much colder region outside
the Solar System — probably a comet from the Kuiper Belt — rather than a remnant of a planet
between Mars and Jupiter that broke up. This is further suggested by its possession of a coma,
which asteroids do not have.
According to the TitiusBode Law, the mean distance in Astronomical Units
from the Sun of the nth planet from Mercury can be written:
d_{n} = 0.4 + 0.3×2^{n1} 
(n = 1, 2, 3, … 9) 
(1) 
4
= 4 + 3×2^{n1}×1. 

(2)

(1+2+3+4)



where 0.4 is Mercury’s mean distance from the Sun. What astronomers have
failed to notice in Equation 1 is that it can be expressed wholly in terms of the set
of four integers 1, 2, 3 & 4, as Equation 2 indicates. These integers are symbolised by the
rows of dots in the tetractys symbolising for the Pythagoreans the perfect number 10:
This is the first clue to what until now has been the complete mystery of
the mathematical regularity observed by the mean distances from the Sun of planets other than
Neptune and Pluto. Indeed, it was the absence of any credible theory underlying the law that
made many astronomers dismiss the excellent agreement between the numbers as a series of lucky
coincidences when they found that it broke down for the two outermost planets. However, their
displays of professional scepticism have been neither convincing nor unified. The TitiusBode
Law has remained an enigma, often mentioned in books on astronomy with a mixture of scientific
reserve and curiosity that conceals a measure of embarrassment about what to make of a simple,
numerical regularity that is suggestive far more of a designing Creator than of what the force
of gravity might have produced if it had acted on a fledgling Solar System subject only to
Newtonian mechanics and the rule of chance!
It is important to point out that the number 10 used as a divisor in the
TitiusBode Law is the number in the sequence of integers starting with 4 that corresponds to
the planet Earth. It is what turns this term in the sequence into 1, making comparison
of planetary mean distances simpler when they are expressed in Astronomical Units. Presumably,
a hypothetical Martian astronomer discovering this empirical relationship would have divided
these integers by 16 in order to make a convenient comparison with the distances of the planets
measured in terms of his Astronomical Unit — the SunMars mean distance. Similarly, a
Venusian astronomer would have divided them by 7 and a Jovian astronomer would have used the
divisor of 52 to make comparison easier. As the correct explanation of the rule cannot, of
course, be expected to favour any particular planet by having one of these numbers as the
divisor in its mathematical formulation, it is clear that the procedure of dividing every
integer by 10 is both parochial in an astronomical sense and unnecessary in a theoretical
sense, because only human astronomers would want to make this division in order to ease
comparison between the actual and predicted numbers. It is only the relative proportions of the
numbers in the sequence that matter, not their absolute values, which only become actual
distances when a particular planet is arbitrarily chosen to set the unit of distance. A true
explanation of the rule must not discriminate between planets and will need to explain only the
ratios of the set of integers: 4, 7, 10, 16. 28, 52, 100, etc, not their absolute magnitudes,
which have been used to express a relationship in a way that favours a particular planet,
namely, Earth. That said, a remarkable connection exists, as already mentioned, between the
terrestrial formulation of the rule and what the author has found to be the universal
mathematical lexicon expressing numbers with cosmic significance, namely, the Pythagorean
mathematical formulation of whole systems in terms of the integers 1, 2, 3 & 4.
This gives unique significance to the mathematical formulation of the TitiusBode law in terms
of the SunEarth mean distance, for these integers do not appear when the rule is expressed in
terms of any other planet’s distance from the Sun.
5
Another point that must be made at this stage is that, by assuming that its
conventional form has to apply to all the planets, astronomers have introduced an
unnecessary complication into the TitiusBode equation:
d_{n} = (4 + 3×2^{n1})/10. 
(3) 
It is counterintuitive to require n = –∞ so that d_{1} = 0.4 for Mercury, when n is a
positive integer for all other planets. As it appears in Equation 3, the value of n signifies
the order of location of the planet from Mercury. What is so special about Mercury that it
should be differentiated in this radical way from the other planets? It should have been
obvious that Equation 3 holds (in theory, that is) for all planets except Mercury
because the first term is already the distance of Mercury from the Sun, whilst n = 1
corresponding to Venus. Equation 3 needs to be modified or at least be understood in a new way
that makes sense (if there is any) of the special mathematical status attributed for Mercury by
requiring n = –∞. People may have been reluctant to create a new mystery by not letting the
equation apply to Mercury, as though this was tantamount to saying that this planet disobeyed
the TitiusBode Law, thereby discrediting its historical status as a genuine regularity
observed by all the planets except Neptune and Pluto. However, this is not the logical
implication of allowing the values of n to start only with 1. Like it or not, it is distances
of planets that are measured from Mercury — not from the Sun — that increase by
successive, integer powers of 2. The canonical formulation of the TitiusBode Law appears to
give to Mercury a special status in making its distance simply an added constant in Equation 3
that is falsely taken into account by bizarrely requiring n = –∞ for this planet. As we shall
see, however, this is an illusion arising from the fact that the planetary average distances
stem from other distances defined by the underlying theory. One cannot let n = 0 for
Mercury and change the added constant in Equation 3 from 0.4 to 0.25 so that d_{0} =
0.4 because this would reduce all ensuing values by 0.15, significantly worsening the agreement
for Venus, Earth, the Asteroid Belt and Jupiter, although marginally improving it for Mars,
Saturn, Uranus, Neptune and Pluto. The natural meaning of n as the number signifying the order
of a planet from the Sun becomes lost if — as it is often written — the power of 2 in Equation
3 is n, not n–1, because Venus is then the case n = 0, so that n denotes the order in the
sequence of planets counting from Venus. This makes even less sense in terms of a fundamental
theory of planetary distances than counting from Mercury because it attributes a false
theoretical significance to what is merely the second planet!
The form of the TitiusBode Law that has to be explained is not its
normalised, canonical form but the equation for the distance (measured in arbitrary units) of
the (n+1)th planet from the Sun:
d_{n+1} = 4 + 3×2^{n1}, 
(n = 1–9)

(4) 
where n = 1 applies to Venus, i.e., the value of n refers to the nth planet beyond Mercury.
Equation 4 can also be written as
d_{n+1} =

4 + 3/2×2^{n} = 1 + 3/2(2^{1} + 2^{n}) = 1
+ (1 + 2^{1})(2^{1} + 2^{n})/2 

=

[1 + (2^{1} + 2^{2})/2] + (2^{n} + 2^{n+1})/2. 
(5)

Mercury’s distance is the first term (shown in square brackets), the second component of which
((2^{1} + 2^{2})/2) has the same form as the second term in Equation 5
representing the distance between Mercury and the nth planet after it. Indeed, for n = 1
(Venus), the latter is merely a repetition of it. This shows that Mercury at least belongs to
the same mathematical pattern as the other planets, which is certainly not what
requiring n = –∞ for this planet suggests! However, (2^{1} + 2^{2})/2 cannot be
treated as the first term in a
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geometric progression involving successive powers of 2 because, as was just
stated, the second term associated with Venus is exactly the same. As (2^{0} +
2^{1})/2 = 3/2>1, the first component, 1, cannot be split up into an analogous
expression without introducing a negative component, ½, which lacks meaning in the context of
distances from the Sun. This may be regarded as an argument against the existence of an unseen
planet between the Sun and Mercury, which would require n to assume negative values if (as
seems reasonable) it, too, obeyed the TitiusBode Law. If, instead, the number ‘1’ denoted its
distance from the Sun, the next two expressions for the mean distance of Mercury and Venus from
the Sun would correctly be (1 + 3×2^{0} = 4) and (1 + 3×2^{1} = 7). However,
the distances of planets beyond Venus would then have to be (1 + 3×2^{n1}) instead of
(4 + 3×2^{n1}), which leads to unacceptably more inaccurate, predicted values.
Therefore, a hypothetical planet between Mercury and the Sun does not restore mathematical
generality to the TitiusBode Law in an acceptable way.
Although it would not persuade astronomers, another powerful argument
against the possibility of the existence of such an unobserved planet is that it would imply
the existence of eleven heliocentric planets,^{1} which would violate the Pythagorean view of the Solar System as a
whole system modelled on the archetypal, tenfold tetractys. As we shall see
shortly, although one principle determines the relative sizes of all planetary average
distances, its mathematical expression takes two forms; the transition from one to the other
corresponds to the changeover from Uranus to Neptune — the first planet to exhibit serious
deviation from the TitiusBode Law. In fact, the analysis predicts that this departure is an
illusion. By so doing, it proves that Pluto is a true planet.
2. The Pythagorean musical
scale
Any musical scale is defined by its starting note — the tonic C, with a tone
ratio of 1 — and its finishing note — the octave C', with a tone ratio of 2^{2}. In the Pythagorean scale, the arithmetic mean of these tone
ratios: (1 + 2)/2 = 3/2 defines the tone ratio of the ‘perfect fifth’ G, socalled because
it is the fifth note in the Pythagorean scale (Fig. 3).
The pitch interval between the perfect fifth and the octave is 2/(3/2) =
4/3, which defines the tone ratio of the fourth note F, the socalled ‘perfect fourth.’
Descending a perfect
____________________________
^{1}This includes the Asteroid Belt, which is the remains of a planet
that failed to form.
^{2} The tone ratio of a musical note is the ratio of its frequency to
that of the tonic.
7
fourth from G creates the second note D, the major second, with a tone ratio
of (3/2)×(3/4) = 9/8. An ascent from D by a perfect fifth then creates A, the major sixth with
a tone ratio of (9/8)×(3/2) = 27/16. Stepping down a perfect fourth generates E, the major
third with a tone ratio of (27/16)×(3/4) = 81/64. Ascending by a perfect fifth from this note
creates the last note B in the scale, the major seventh, with a tone ratio of (81/64)×(3/2) =
243/128. The perfect fifth thus divides C and C' into all other notes.
The first seven notes of the Pythagorean scale:
C

D

E

F

G

A

B

1

9/8

81/64

4/3

3/2

27/16

243/128

are repeated on each higher or lower octave, corresponding notes,
respectively, increasing or decreasing in pitch by a factor of 2. The octave is spanned by five
whole tone intervals of 9/8 and two ‘leimmas’ of 256/243, which correspond to (but are 10%
flatter than) the modern, equaltempered semitone: (9/8)^{5}×(256/243)^{2} =
2.
Table 2 shows the tone ratios of the first eleven octaves of the Pythagorean
musical scale. The last column shows the running total of overtones — notes above the tonic
with tone ratios that are integers. Notice that the first nine overtones in purple
cells: 3, 6, 12, 24, 48, 96, 192, 384, 768,
that are successive octaves of a perfect fifth include the very integers
that Titius noticed denote the distances (when divided by 10) of the (then known) planets from
Mercury. This is the second clue to the Pythagorean basis of the TitiusBode Law.
Table 2. The Tone Ratios of the Pythagorean Musical Scale.
C

D

E

F

G

A

B

Number
of
Overtones

1

9/8

81/64

4/3

3/2

27/16

243/128

0

2

9/4

81/32

8/3

3

27/8

243/64

2

4

9/2

81/16

16/3

6

27/4

243/32

4

8

9

81/8

32/3

12

27/2

243/16

7

16

18

81/4

64/3

24

27

243/8

11

32

36

81/2

128/3

48

54

243/4

15

64

72

81

256/3

96

108

243/2

20

128

144

162

512/3

192

216

243

26

256

288

324

1024/3

384

432

486

32

512

576

648

2048/3

768

864

972

38

1024







39

768, the perfect fifth of the tenth octave, is not, however, the
distance of Pluto from Mercury andso it needs to be explained why, instead, 384, the perfect
fifth of the ninth octave, denotes its distance and how (if at all) Neptune fits into the
sequence of overtones.
3. Perimeter of
ellipse
As pointed out in Section 1, the mean distances of the planets from the Sun
are equal to the lengths of the semimajor axes of their elliptical orbits. Any law of scaling
of the former is therefore also one of the latter. Indeed, being a characteristic of orbits, it
is really this length, not the artificial notion of ‘mean distance,’ that is fundamental. The
area of an ellipse with semimajor axis a and semiminor axis b is πab, that is, a simple,
8
algebraic function of a and b. This is not the case with its perimeter. For
an ellipse with eccentricity e, the perimeter P is
where


(7)

is the complete elliptic integral of the second kind.^{2} This cannot be expressed as a simple algebraic function, so
mathematicians have worked out various approximations. An exact, series expansion for P in
ascending powers of e is^{3}:


(8)

= 1 – (1/4)e^{2} – (3/64)e^{4} – (5/256)e^{6} –
(175/16384)e^{8} – (441/65536)e^{10} – … . 

(9)

Notice that this converges to the correct limit 1 of a circle as e → 0.
An astoundingly accurate formula approximating P was given in 1918 by the
Indian mathematical genius, S. Ramanujan (1887–1920)^{4}:
P ≈ π(a+b)[1 + 3h/(10 + √(4–3h))],


(10)



(11)

where
All the terms match the correct series (9) up to and including the
coefficient of e^{18}! It is amazingly accurate for small e and, even when e ≈ 1, the
absolute size of the relative error is only 7π/22 – 1, or about 4×10^{–4}.
Table 3 shows how accurately — even for Pluto, the planet with the largest
eccentricity — the circumferences of the planetary orbits approximate to the value
2πa for a circle:
Table 3
Planet 
e

P/2πa

Mercury 
0.21

0.988882

Venus 
0.01

0.999975

Earth 
0.02

0.998999992

Mars 
0.09

0.997972

Jupiter 
0.05

0.999375

Saturn 
0.06

0.999099

Uranus 
0.05

0.999375

Neptune 
0.01

0.999975

Pluto 
0.25

0.984187

The level of approximation is far better than that between measured
planetary distances and those predicted by the TitiusBode Law. This means that the
discrepancies cannot
9
be due merely to considering their orbitals as circles instead of as
ellipses, otherwise Mars with the largest eccentricity amongst the planets VenusNeptune might
be expected to show the worst agreement, which it does not.
The very accurate proportionality between a, the mean distance of a planet
from the Sun, and its orbital circumference implies that the latter, starting with Venus,
increases as integer powers of 2 in accordance with the TitiusBode Law, although not exactly.
It raises the possibility that circumferences of planetary orbits may be more relevant to the
understanding of the TitiusBode Law than the artificial notion of an arithmetic average of
their maximum and minimum distances from the Sun. As explained in Section 6, however, this
turns out not to be the case.
4. Undertones, tones &
overtones
In Timaeus, his treatise on cosmology, Plato (428 B.C.E.–347
B.C.E.) described how the Demiurge measured the substance of the World Soul according to the
simple proportions of the first three powers of 2 and 3, which came to be represented by what
is known as ‘Plato’s lambda’ because of its resemblance to the Greek letter Λ (Fig. 4).^{5} This came to be recognised as but two sides of a tetractys array of
ten integers whose ratios determine the tone ratios of the notes of the Pythagorean
scale:
The way in which they generate the spectrum of musical notes is, however,
asymmetrical because the pairing of integers to form octaves, such as 4 and 8 or 6 and 12, and
perfect fifths, such as 8 and 12 or 4 and 6, follows the directions of the sides of the
tetractys, whereas the pairing of integers to form perfect fourths, such as 3 and 4 or 9 and
12, is diagonally across the natural geometry of the array of integers in the tetractys.
Moreover, the creation of tone ratios is incomplete in the context of Pythagorean mathematics
because the number 4 is missing as a generative factor from Plato’s Lambda. By considering the
Lambda tetractys with 1^{3} = 1, 2^{3} = 8 and 3^{3} = 27 at its
corners as but one face of a tetrahedron with 4^{3} = 64 at its fourth corner, it was
found^{6} that a complete symmetry appeared in the pairing of integers forming
the fourth face of the tetrahedron (Fig. 5). All successive octaves lie on red lines, all perfect fourths lie on
green lines and all perfect fifths lie on blue lines, these sets of lines being parallel to
the three sides of the tetractys forming the fourth face of the tetrahedron. (In
10
the first face, only octaves and perfect fifths are linked parallel to these
sides). Its hexagonal symmetry means that, when this fourth tetractys is extended to create
other octaves, every number becomes surrounded by six others that are octaves, perfect fourths
or perfect fifths. All the numbers in this infinite, planar array may be divided by any one of
them to generate the same hexagonal lattice of tone ratios of the Pythagorean scale.
It is, of course, not invariant with respect to division by any integer, because not
all integers are present in the lattice. For example, all prime numbers are
11
absent. However, ratios of any pair of numbers are unchanged by division of
each by the same number. Division of all the numbers in the lattice by any number — whether or
not it belongs to the lattice — therefore leaves the tone ratios formed from the pairs
unchanged. It does not matter which number in the lattice is picked as the tonic, or
fundamental frequency, as the tone ratios created by dividing it by the numbers around it are
the same as those formed by the numbers surrounding the number 1. Where one picks one’s tonal
‘origin’ is arbitrary. This simply reflects the way in which the intervals between notes in one
octave are preserved in a different octave because every tone ratio is changed by the same
factor.
Figure 6 shows the lattice of tone ratios, starting with the tonic, 1.
Overtones are shown in yellow circles. Red lines connect octaves (×2), green lines connect
perfect fourths (×4/3) and blue lines connect perfect fifths (×3/2). The Pythagorean tone
interval 9/8 is
12
13
also indicated by the orange line joining the tonic at the centre of the
tetractys (coloured grey) to one corner of it. The tone ratio 27/16 of the major sixth and the
tone ratio 243/128 of the major seventh are similarly defined by, respectively, indigo and
violet diagonals extending from the number 1 to corners of larger triangles. Successive notes
of the musical scale are joined by dashed lines. They zigzag between the octave, the major
seventh and the perfect fourth, i.e., between the extremities of each octave and its
midpoint.
The perfect fourth of the nth octave has a tone ratio of
2^{n1}(4/3) = 2^{n}(2/3) and the perfect fifth has a tone ratio of
2^{n1}(3/2) = 2^{n}(3/4). The corresponding undertones of the nth octave have
tone ratios of, respectively, 2^{n}(4/3) and 2^{n}(3/2). As
2^{n}(2/3) is the reciprocal of 2^{n}(3/2) and 2^{n}(3/4) is the
reciprocal of 2^{n}(4/3), the tone ratio of the perfect fourth for a given octave of
undertones is the reciprocal of the tone ratio of the perfect fifth of the corresponding octave
of tones, and vice versa. This is illustrated in Figure 7 for five octaves of tones and their undertones. Red arrows link
octaves, green arrows link the perfect fourths of undertones to the perfect fifths of tones
and blue arrows link the perfect fifths of undertones to the perfect fourths of tones. Only
octaves, perfect fourths and fifths share this property of reciprocity.
As frequency and wavelength are inversely related, the wavelength of an
undertone that is a perfect fourth is the same as the tone ratio of the perfect fifth of the
same octave of tones. Table 4 displays the wavelengths of the undertones for eleven octaves. The
last column lists the number of their subharmonics as a running total. We see that only
octaves and perfect fourths have wavelengths that are whole numbers. More important is the
fact that the latter are the very integers that appear in the TitiusBode Law measuring
the distances of the planets from Mercury. This is the third clue to its physical
basis.
Table 5 shows the zigzag pattern of the Pythagorean musical undertones and
their wavelengths. The second column numbers the integer values of the perfect fourths in the
purple cells up to 192. The reason for stopping at this number for the eighth octave
will be given in Section 8.
5. Theories of the Solar
System
The oldest theory of the Solar System is the nebula theory. Originally
proposed in the middle of the eighteenth century by Immanuel Kant (1724–1804), the great German
philosopher, and developed in 1796 by the French astronomer, Pierre Laplace (1749–1827), it
starts with a cloud of interstellar gas and/or dust that was triggered to collapse under its
own gravity by some disturbance (perhaps the shockwave from a nearby supernova). The centre of
the cloud became compressed as it collapsed and heated up until it formed into a protostar.
Viscous drag between the rotating protostar and the gas flowing around it made the latter start
to rotate. Some material fell into the protostar and the rest condensed into an ‘accretion
disk,’ which rotated around the star and cooled off enough for metal, rock and ice to condense
into tiny particles. The metals condensed almost as soon as the accretion disk formed (4.55 to
4.56 billion years ago according to measurements of certain meteors). The rock condensed later
(between 4.4 and 4.55 billion years ago). Particles collided and aggregated into larger
particles until they became the size of small asteroids. Then gravity took over and pulled in
more, smaller particles. They grew to a size that depends on their distance from the star and
the density and composition of the protoplanetary nebula. The accretion of these
‘planetesimals’ is supposed to have taken a few hundred thousand to about twenty million years,
the outermost taking the longest to form because of the lower density of
14
material near the rim of the disk. About one million years after the nebula
cooled, the star’s nuclear reactions expelled its outer layers, this socalled ‘T Tauri Wind’
sweeping away all the gas left in the protoplanetary nebula. Gas giants like Jupiter and Saturn
formed because they were massive enough to hold on a relatively large quantity of nebula gas,
which was swept away from the smaller planets. The planetesimals slowly collided with one
another and became more massive, moonsized bodies that continued to collide until the planets
formed about ten to a hundred million years later.
There were two main problems with the original version of this theory.
First, as angular momentum is conserved, the condensation process should have left the Sun with
99% of the Solar System mass with most of the angular momentum, whereas 99% of it resides in
the planets’ orbital and rotational motions. The central mass could not have transferred this
much momentum to the planets. Second, a hot gaseous ring of the type postulated by Laplace
would disperse into space and would not pull itself together gravitationally to form a planet.
A variation of the theory suggested that the protoplanetary nebula was a
system of rings that were radiated away from the Sun, somewhat like a series of smoke rings
puffed out by someone smoking a cigarette. Apart from the problem whence this chain of rings
came, it would require much more time than the estimated five billion years the Sun has
existed.
Problems with the nebula theory made people think of an alternative. The
French naturalist George Buffon (1707–1788) proposed in 1745 that material ripped off from the
Sun by collision with a comet had condensed into the planets. This encounter theory was
developed by the American geologist Thomas Chamberlin (1843–1928) and the American astronomer
Forest Moulton (1872–1852), who suggested that giant eruptions were pulled off the Sun by the
gravitational attraction of a passing star. Later, another geologistastronomer pair in
England, Sir Harold Jeffreys (1891–1989) and Sir James Jeans (1877–1946), theorized that a
cigarshaped gaseous filament was pulled from the Sun by the sideswiping action of a passing
star. The middle section condensed into the Jovian planets, and the ends condensed into the
smaller planets. This theory accounts for all the planets orbiting in the same direction and in
the sense of the Sun's rotation, as well as for the planets' nearly circular and coplanar
orbits. In either version, however, this theory has serious failings in that solar matter,
whether pulled or ejected, could not have acquired sufficient angular momentum nor could hot
gas have condensed into planets. Besides, the probability of a near encounter in our region of
the Galaxy is vanishingly small, less than one in many millions. Finally, encounter theories
cannot explain why the Earth and other planets display so many elements not found in the
Sun.
Improvements to the nebula hypothesis were made in the mid twentieth
century. A fragment was imagined to first separate from an interstellar cloud of gas and dust.
This was followed by the separation of other fragments. The central region of the cloud was
denser than its outer parts and collapsed more quickly. As the rotating cloud broke up,
rotation was transferred to each fragment, the motion speeding up as the solar nebula
contracted in order to conserve angular momentum. The solar nebula grew by accretion as
material continued to fall inward from its surroundings. Largescale turbulence from
gravitational instabilities ruptured the thin accretion disk into eddies, each containing many
small particles. These particles gradually built up into larger bodies by some combination of
adhesive forces. Repeated encounters among them resulted in the accretion of literally billions
of still larger asteroidsized aggregates (planetesimals), which orbited the centre of the
solar nebula. Mutual gravitational attraction led to further encounters and gradual coalescence
into many roughly Moonsize bodies, or proto
15
planets, which in turn coalesced to form the planets.
The Asteroid Belt likely represents not the remains of a planet that broke
up owing to catastrophic collision with some invading object but one that failed to form
because the mass of all the bodies there (less than a quarter of the mass of the Earth’s Moon)
was insufficient to create massive enough planetesimals to draw them together by the force of
gravity. If it had been the former, one would have expected its orbit to have been perturbed
enough to cause some deviation from the TitiusBode Law, with which the Asteroid Belt agrees
well. Alternatively (and more likely), the formation of large bodies could have been disrupted
by the powerful gravitational pull of neighbouring Jupiter, which would have tugged them
completely out of the belt if they occupied socalled ‘resonant orbits’ that periodically
brought them close to the giant planet.
In January 2002 a strange object called a ‘brown dwarf’ was
reported^{7} orbiting a star nearly as closely as Saturn is to the Sun. Brown
dwarfs are large balls of gas, much heavier than Jupiter but not massive enough to generate
the thermonuclear fusion that powers a star. The nearest, confirmed brown dwarf is about 16
lightyears^{3} from Earth (the nearest star, Proxima Centauri, is 4.2
lightyears away), although an as yet unconfirmed brown dwarf has been found^{8} about 13 lightyears away. Between 55 and 78 times as heavy as
Jupiter, its planetaryscale distance of 14 AU from the star is uncomfortably too close for
its size to be explainable by the nebula theory of planetary systems. The problem
raised by this object is similar to that raised by many of the massive, extrasolar
planets that have been discovered orbiting stars. They are much closer to these stars
than current ideas
of planetary formation allow and generally have large orbital
eccentricities, raising the question whether the Solar System and its formation is actually
atypical of planetary systems. The most likely situation is that there is no unique process by
which planetary systems form. That said, in 1992 the Hubble Space Telescope obtained the first
images of protoplanetary disks in the Orion Nebula.^{9 10} Dr. C. Robert O’Dell, a Rice University astronomer, surveyed with
the Hubble Space Telescope 110 stars in the Orion Nebula 1500 lightyears away and found
disks around 56 of them (Fig. 8). At the centre of each disk was a young star. The images showed that
the objects were pancakeshaped disks of dust. Some of these disks are
____________________________
^{3} A lightyear is the distance travelled by light in one year. It is
about 5.88 trillion miles (9.7 trillion km).
16
visible as silhouettes against a background of hot, bright interstellar gas,
while others are seen to shine brightly. Hubble’s images provide direct evidence that dust
surrounding a newborn star has too much spin to be drawn into the collapsing star. Instead, the
material spreads out into a broad, flattened disk through a combination of centrifugal force
and gravitational attraction between objects on either side of the central plane of rotation.
The disks are roughly on the same scale as the Solar System and lend strong support to the
nebular theory of its origin.
6. Planetary orbitals as musical
perfect fourths
The problem of the nebula theory visàvis the TitiusBode Law is that the
condition for stabilising a future planetary orbit, namely the balancing of the centrifugal
force acting on the orbiting, wouldbe planet with the Sun’s gravitational force, merely
creates the relationship between its period and average distance described by Kepler’s Third
Law. Another dynamical condition that gravity does not seem able to provide is required to
determine the relative sizes of the planetary orbits. The fact that physics could not supply
one made some astronomers question whether the numerical relationship discovered by Titius was
anything other than
coincidence. The fact, however, that both the four largest moons of Jupiter
and some extrasolar planetary systems exhibit spacing rules in their orbitals, albeit not of
the TitiusBode kind, discredits this viewpoint because these rules, too, would then have to be
due to coincidence, which is
17
implausible. If, as the nebula theory asserts, some disturbance, such as the
shockwave from an exploding star, pushed out a clump of gas and dust in the rotating accretion
disk to a point where the centrifugal force exceeded the inward gravitational pull of the
protosun at the centre of the disk, this would have created a tear in it. In fact, many tears
would be created as the radiation and gas blast from the exploding star accelerated different
clumps of matter as it passed through the disk. Travelling faster than material nearby that had
been shielded by the blast, the ejected matter would collide more often with objects in its
path and build up its mass owing to the greater likelihood of their greater speed causing
cohesion with them. It would literally dig a path through the gas and dust, the trail being
made up of bodies that were larger than most of the material in the accretion disk
undisturbed by the exploding star. As the disk rotated, the trail of larger,
faster objects moving with it as a whole would curve round and eventually become elliptical,
their centrifugal force overcoming the inward gravitational pull by the protosun for a while
until they had slowed down enough through collisions and had become sufficiently massive for
the latter force to counterbalance the former. When this happened, the various trails of
aggregated bodies formed rings of material that, being of higher density, attracted matter in
the accretion disk, thus widening the gaps between the rings.
A point on a logarithmic curve has polar coordinates (r, θ) related by:
r(θ) = ae^{bθ} 
(–∞≤θ≤∞)

(12)

(a and b are positive constants). Hence, after n further, complete
revolutions:
r(θ+2πn) 
= e^{2πbn} = (e^{2πb})^{n} = 2^{n}

(13)

r(θ)



if b = ln2/2π ≈ 0.1103. Therefore:
r(θ) = ae^{(ln2/2π)θ}= (2^{θ/2π})a.

(14)

This particular logarithmic curve crosses any straight line passing through the point r = 0
around which it spirals at points r(2πn) = 2^{n}a = a, 2a, 4a, 8a, etc. The distance
between
18
successive crossings is twice the previous one (Fig. 10). The same is true for the length of the curve up to these points,
as now shown. The differential length ds is given by
(ds)^{2} = (dr)^{2} + r^{2}(dθ)^{2}.

(15)

The length of the logarithmic spiral up the point with polar coordinates (r, θ)
is
Every revolution of the spiral increases its length by a factor of 2, just
as the distance between successive crossings of any straight line through its centre does. As
r(2πn) = 2^{n}a and s(2πn) ≈ 9.12a×2^{n}, the length of the curve from its
centre up to any point on it is just over nine times the distance of the point from the
centre. As r(πn) = (√2)^{n}a and s(π(n+1))/s(nπ) = (√2), successive revolutions by 180° increases both
the radius and the length of the spiral by √2 ≈1.414, showing the meaning of this
smallest surd as the factor by which this logarithmic spiral expands in every
halfrevolution.
It is important to point out here that n can take negative values because
the curve winds endlessly in smaller and smaller spirals around its asymptotic centre θ= –∞.
The part of the curve that n defines is arbitrary because the logarithmic spiral is
selfsimilar — every corresponding section whose ends are defined by the same pair of angles
modulo 2πn is similarly shaped, differing only in scale. n = –∞
denotes the asymptotic centre.
Selfsimilar spirals are ubiquitous in nature as the form taken by living
things that do not change in shape as they grow in size. Let us suppose that the inswirling
material of the solar nebula that gave birth to the Solar System followed the inward path of a
logarithmic spiral.
This contraction was the opposite to the kind of expansive, spiral
development that occurs, for example, in seashells and vertebrate embryos. The spiral motion of
the material continued in the accretion disk. It aggregated into spiral bands of
19
denser material that eventually turned into stable, elliptical annuli of
bodies in orbit around the young Sun. Assuming that the decrease in density of accretion
material with distance was uniform, the average distance of each ring was set by the arithmetic
mean of the radii of successive spirals, each halfrevolution of which caused that section of
the spiral inflow of material to break off and to go into its own orbit, its parameters set by
Kepler’s Third Law. Material belonging to the most tightly wind spirals was closest to the Sun
and therefore collapsed into it. As the Sun was forming at the same time, sucking dust and gas
into it, there is no reason why it should have been at the asymptotic centre of the spiral,
which was not an orbital path generated by its gravitational pull on the material of the
accretion disk.* The distance from the asymptotic centre of the nth crossing point of
the spiral with the major axes of the elliptical orbits is 2^{n}a. This centre is
distance R from the common centre of the ellipses. The distance R_{n} of the nth
crossing point from the centre of the orbits is
R_{n} = R + 2^{n}a 
(n = 1, 2, 3, etc)

(19)

The mean distance of the nth ring from the centre (Fig. 11) is
d_{n} = ½(R_{n} + R_{n+1}) = R +
½a(2^{n} + 2^{n+1}) = R + 3a×2^{n1}. 

(20)

The distance between the nth and (n+1)th planet after Mercury is
d_{n+1} – d_{n} 
= ½(R_{n+2} – R_{n}) = ½a(2^{n+2} – 2^{n})
= 3a×2^{n–1}

(21)


= 3a, 6a, 12a, 24a, etc. 
(22)

As the first planet, Mercury was formed from material of the accretion disk
within the distance 0≤r≤R_{1}, the average value of which = (0+R_{1})/2 =
½R_{1}. Its distance from the Sun is d_{M} = ½R_{1} = ½(R + 2a).
According to Equation 20, the distance of Venus from the Sun is
d_{1} = ½(R_{1} + R_{2}) = R + 3a.
The distance of Venus from Mercury is
d_{1} – d_{M} = ½(R + 4a).


(23)

According to Equation 22, the distance between consecutive planets is an
even or odd multiple of a. If one supposes that this also true of the distance between Mercury
and Venus, then
d_{1} – d_{M} = 2Na or (2N–1)a,


(24)

where N = 1, 2, 3, etc. Hence,
½(R + 4a) = 2Na or (2N–1)a.


(25)

R = 4(N–1)a = 4a, 8a, 12a, etc,


(26)

or
R = (4N–6)a = 2a, 6a, 10a, etc.



R = 2a is excluded because it implies d_{M} = 2a as well, i.e., that
Mercury is at the asymptotic centre of the logarithmic spiral, which would not allow its own
annulus (and hence itself) to develop. The minimum value of R is 4a. Substituting in Equation
20,
d_{n} = 4a + 3a×2^{n1}.


(27)

d_{2} = 4a + 6a = 10a,


(28)

and
d_{n}/d_{2} = 0.4 + 0.3×2^{n1}.


(29)

____________________________
* Sir Isaac Newton proved that a logarithmic spiral is the orbit produced
by a force that varies as 1/r^{3}.
20
This is the TitiusBode Law (Equation 1) expressed in terms of the
astronomical unit d_{2}, the mean distance of the second ring from the centre of the
ellipses. It expresses the average distance from the centre of the elliptical orbits of
successive edges of the spiral segments of the accretion disk that eventually form a planet. R
= 4a is not, as astronomers have thought, the distance of Mercury from the Sun. Instead, it is
the distance of the asymptotic centre of the logarithmic spiral, which, as we shall see, turns
out to be close to Mercury. n = 1 corresponds to Venus, n = 2 corresponds to Earth, etc.
As R = 4a, Equation 19 becomes
R_{n} = 4a + 2^{n}a = (4 +
2^{n})d_{2}/10, 

(30)

R_{∞}= 0.4d_{2} = 4a = R. Why n was wrongly thought to have the singular value
of ∞ in the case of Mercury now becomes clear. It signifies the asymptotic point on the Xaxis
to which the logarithmic spiral converges after winding infinitely many times around it.
According to Equation 19, R_{1} = 3R/2, R_{2} = 2R,
R_{3} = 3R, R_{4} = 5R, etc. Therefore,
d_{n} 
= (R_{n} + R_{n+1})/2 = R + 3×2^{n3}R



= [1 + (2^{n2} + 2^{n1})/2]R. 
(31)

Therefore,
d_{n}/R = 1 + (2^{n2} + 2^{n1})/2 

(32)

Following the convention that the tonic of the first octave has a tone ratio of 1, the second
term on the righthand side of Equation 32 is the perfect fifth of the (n–1)th octave. The
relative distance (d_{n}–R)/R of the nth planet from Mercury is simply the proportion
by which the frequency of the perfect fifth of the (n–1)th octave exceeds that
of the tonic of the first octave. Alternatively, Figure 7 shows that, as the tone ratios of perfect
fifths are reciprocals of those of perfect fourths of their counterpart
undertones
21
and as wavelength and frequency are inversely proportional to each other,
the planets’ distances from Mercury increase as the wavelengths of successive octaves of
perfect fourths of the musical undertones. This reflects the fact that, whereas the tone ratios
of perfect fourths are the harmonic mean of those of the tonic and octave, their wavelengths
are the arithmetic means of these notes.
d_{M} = ½R_{1} = 3R/4 =0.3d_{2}. This, and
not R = 0.4d_{2}, would be the predicted distance of Mercury from the Sun if
the centre of its orbit coincided with those of the other planets, all of whom (apart from
Pluto) have almost zero eccentricities (see Table 3). However, the eccentricity of Mercury’s orbit is e = 0.2056, which
means that, unlike other planets to a very good approximation, the Sun is not at the
geometrical centre of its orbit. If d_{M} is its mean distance from the Sun, its
distance at perihelion is d_{M}(1–e) ≈ 0.7944d_{M}. This must be the average
distance from the Sun of the material that formed Mercury. Hence, 0.7944d_{M} ≈
0.3d_{2}, so that d_{M}/d_{2} ≈ 0.3776. This predicted distance
compares well with the actual value of 0.3871 — certainly better than the value of 0.4
appearing in the TitiusBode Law (the discrepancy is –0.095 compared with +0.1129).
7. Octet patterns in
nature
When chemists in the 1860s began to group the known elements according to
similar chemical and physical properties, they found that the latter repeated in cycles.
Arranged in a periodic table, elements with similar properties occurred in the same vertical
column. The chemists discovered eight main groups, or types, of elements (Table 6). Physicists eventually found the reason for this eightfold pattern.
The chemical properties of elements are due to the electrons that their atoms either give up
or share when they bind to other atoms — their socalled ‘valence electrons.’ Electrons
occupy a discrete number of orbitals in a set of quantum shells. Usually the valence
electrons in the outermost shell participate in the bonding together of atoms. As this shell
possesses eight electrons when filled (Fig. 12), atoms strive to attain this most stable electronic configuration
by combining with those elements whose atoms have sufficient number of valence electrons to
fill up the shell. Elements at the extreme right column of the table (group VIII), known as
the inert, or noble, gases like neon and argon, have atoms with full, outermost shells of
eight electrons. They find difficulty in chemically reacting with other elements. When the
outer shell is full, a new row, or ‘period,’ of elements begins again with one electron in
its outer shell (Group I). These give up an electron in chemical reactions. Group II
elements give up two electrons, and so on. Elements whose atoms give up or share the same
number of electrons will occupy the same group. As it is this number that determines how
they bond to other atoms, elements within the same group display similar chemical
properties. Atoms therefore have up to eight stages in the filling of their valence
shell.
Particle physicists found in the 1960s that strongly interacting subatomic
particles called ‘baryons’ and ‘mesons’ could be classified according to what became known as
the ‘eightfold way.’ This highly successful classification scheme placed these particles in
groups of eight, or ‘octets’ (Fig. 13). The Quark Model proposed by GellMann and Zweig in 1964 explained
these patterns by postulating the existence of a more fundamental particle called the
‘quark.’ Three types of these particles combined
22
as quarkantiquark pairs or as groups of three quarks to form just those
baryons and mesons that had been discovered in the highenergy physics laboratory, as well as
one — the omega minus — which was discovered soon after the model was proposed.
Particle physicists describe the forces operating between subatomic
particles in terms of socalled gauge symmetry groups. Many believe that the basic constituents
of matter are extended objects called ‘superstrings.’ The gauge symmetry group
describing the unified superstring force is the rank8 exceptional group
E_{8}. It was shown in Article 15^{11}
23
that a continuous, mathematical link exists between this group and the
algebra of octonions, which is the most general class of division algebra. Article
16^{12} showed a remarkable analogy between the multiplicative
properties of this 8dimensional algebra (Table 12) and the seven musical octave species known to the ancient
Egyptians and Greeks. This correspondence is too detailed and exact to be due to chance. It
suggests that the muchsought ‘Mtheory’ that encompasses the five superstring theories with
supergravity will incorporate the Pythagorean mathematics of the musical scale.
These examples of eightfold patterns at work in Nature determining the
chemical properties of atoms and the interactions of superstrings arise because the cyclic
process that renews and exhausts all possibilities — either physical (electron shells, barons
and mesons) or mathematical (eight octonions, eightdimensional E_{8}) — requires eight
steps. The spacetime predicted by superstring theory has eight dimensions perpendicular to any
given direction that superstrings may move along. These comprise two largescale dimensions of
the space that is familiar to us and six dimensions of a hidden, curledup space. This 2:6
division corresponds in music to the beginning (tonic) and end (octave) of the eightnote
diatonic scale and the six notes spread between them. In the atom it corresponds to the two
electrons occupying the S orbital and the six electrons in the three P orbitals of the valence
shell that together determine the chemistry of elements other than the rare earths.
In view of these examples, it should not come as a surprise that an
eightfold pattern exists in the planets of the Solar System. In fact, it was encountered in
Section 1, where Table 1 shows that the TitiusBode Law is obeyed by the first eight
planets (including the Asteroid Belt, which is the remnant of a planet that failed to form)
but breaks down for the next two planets. As next explained, this is because, like musical
tones repeated on higher octaves, Neptune and Pluto belong to another octet governed by the
same principle forming the planets up to Uranus. They obey a rescaled version of the
Titius Bode Law appropriate to this new octet, Uranus acting as the tonic of a new
octave.
8. The octet structure of the Solar
System
Logarithmic spiral geometry for the spiralling of matter in the solar
accretion disk before it aggregated into planets was shown in Section 6 to result in planetary
average distances that obey the TitiusBode Law. Average distances from the centre of the
spiral are simply the wavelengths of the perfect fourths of the undertones of the Pythagorean
musical scale:
d_{n} – 4a = (3a/2)×2^{n} 
(n = 1, 2, 3, etc)

(33)

3a/2 is the wavelength of the perfect fourth of the first octave of undertones whose tonic has
the wavelength a. Venus (n = 1) corresponds therefore to the perfect fourth of the second
octave. In general, the (n+1)th planet from the Sun (the nth planet from Mercury) corresponds
to the perfect fourth of the (n+1)th octave. Uranus, the eighth planet from
Mercury,* has a distance from it equal to the wavelength of the perfect fourth of
the eighth octave, namely, (3a/2)×2^{7} = 192a. Let us suppose that, just as the
eighth note of the Pythagorean musical scale is both the last note of one octave and the
tonic of the next higher octave, so Uranus both ends the first octet of planets and
commences the next one. Then, just as Mercury itself does not obey the TitiusBode Law in
the same way as other planets do because it is not a term in the geometrical progression and
is therefore undetermined by it, so Uranus does not obey the law of geometric progression
that corresponds to the new octet. According to this view,
____________________________
* As always, the Asteroid Belt is counted as a planet because it is the
remnant of one that failed to form.
24
Neptune, the first member (n' = 1) of this octet after Uranus, corresponds
to Venus, the first planet (n = 1) after Mercury in the first octet of planets (Table 13).
Table 13
Octave

Planet

n

n'

Distance from 1st asymptotic
centre = (3/2)×2^{n}a

Distance from 2nd asymptotic centre = (3/2)×2^{n}a =
48×2^{n}a

R_{n}

R_{n'}

1

Mercury 


(a)

–

–

–

2

Venus 
1


(3/2)×2^{1} = 3a

–

6a

–

3

Earth 
2


(3/2)×2^{2} = 6a

–

8a

–

4

Mars 
3


(3/2)×2^{3} = 12a

–

12a

–

5

(Asteroids) 
4


(3/2)×2^{4} = 24a

–

20a

–

6

Jupiter 
5


(3/2)×2^{5} = 48a

–

36a

–

7

Saturn 
6


(3/2)×2^{6} = 96a

–

68a

–

8

Uranus 
7


(3/2)×2^{7} = 192a

–

132a

–

9

Neptune 

1

–

48×2^{1} = 96a

260a

260a

10

Pluto 

2

–

48×2^{2} = 192a

–

324a

But now, instead of the distance (3a/2)×2n = 3a×2^{n1} of the nth
planet from the first asymptotic centre, the distance of the n'th planet beyond Uranus will be
given by 96a×2^{n'1}. The reason for this is as follows: The distance from the Sun of
the n'th crossing point of the logarithmic spiral in the second octet is
R_{n'} = R' + 2^{n'}a', 
(n' = 1–8)

(34)

where R' is the distance of the new centre and a' is the distance of the first crossing point
from the centre. The rescaled spiral becomes centred on the orbit of Uranus, whose distance
from the first centre is 192a (Fig. 14). Therefore, R' = 4a + 192a = 196a.
25
The distance from the Sun of the n'th planet beyond Uranus in the second
octet is
d_{n'} = ½(R_{n'} + R_{n'+1}) = R' +
3a'×2^{n'1}


(35)

= 196a + 3a'×2^{n'1}. 

(36)

This compares with
d_{n} = 4a + 3a×2^{n1}


(37)

for planets in the first octet. The orbit of Venus is determined by the second and third
crossing points of the spiral at distances from its centre of, respectively, 2a and 4a, the
first crossing point being at a distance of a. The latter plays the role of the wavelength of
tonic of the first octave of undertones.
Table 14
Planet 
Predicted distance (AU)

Actual distance (AU)

Mercury 
0.38

0.39

Venus 
0.70

0.72

Earth 
1.00

1.00

Mars 
1.60

1.52

Asteroids 
2.80

2.77

Jupiter 
5.20

5.20

Saturn 
10.00

9.54

Uranus 
19.60

19.19

Neptune 
29.20

30.07

Pluto 
38.80

39.48

The distance 3a of Venus from the spiral’s centre is the perfect fourth of
the second octave. Uranus, the first member of the second octet, corresponds to Mercury, which
is the first member of the first octet, and Neptune (n' = 1), the second member of the second
octet, corresponds to Venus (n = 1), the second member of the first octet. The distance between
the two spirals associated with Uranus is 2^{8}a – 2^{7}a = 128a, the outer one
being 64a units from the planet’s orbit. The inner spiral for Venus corresponds to the inner
spiral for Neptune, which is at the same position as the outer spiral for Uranus. Venus’s inner
spiral is distance 2a from the asymptotic centre. Hence, as the spiral for the second octet is
a logarithmic spiral as well,* Neptune’s inner spiral is distance 2a' from its centre, which is at the
crossing point of Uranus’s orbit. Therefore, 2a' = 64a, a' = 32a and Equation 36 becomes
d_{n'} = 196a + 96a×2^{n'1} 

(38)

As 7 + n' = n, the distance of the (n+1)th planet from the Sun is
d_{n} = 4a + 3a×2^{n1}

(n = 1–7)

(39)

= 196a + 3a×2^{n3}

(n = 8–15)

(40)

____________________________
* As a' = 2^{5}a, i.e., larger than a by an integer power of 2,
the spirals for the two octets are identical.
26
Dividing by 10a (d_{2}) to convert d_{n} into Astronomical
Units, the TitiusBode Law for the second octet of planets is
d_{n} = 19.6 + 0.3×2^{n3}. 

(41)

For Neptune (n = 8), d_{8} = 29.2 AU, comparing well with the actual value of 30.06 AU.
For Pluto (n = 9), d_{9} = 38.8 AU, also agreeing well with the actual value of 39.48
AU.
By making just one reasonable assumption that the planets have the same
octet pattern as that found in music, the quark makeup of baryons and mesons and the group
mathematics of superstrings, it has been shown that, far from being anomalous, Neptune and
Pluto actually fit the same musical pattern of perfect fourths underlying the TitiusBode Law
as the other planets do (Table 14). The discrepancies are 2.9% under for Neptune, comparing with the
old value of 29.0% over, and 1.7% under for Pluto, comparing with the huge discrepancy of
95% over that caused doubt among some astronomers that Pluto is a true planet. Mercury plays
the role of the tonic of the musical scale and Uranus its octave, which is the tonic of the
next higher octave of notes. The unit octonions comprise the real number 1 and seven
imaginary octonions. The two planets correspond to the real unit octonion. The meson octets
each comprises a socalled ‘isospin singlet state’ as well as seven other quarkantiquark
bound states (e.g., the η meson in the spin0 octet shown in Fig. 13). This 1:7 differentiation corresponds to the distinction between
Mercury and the seven other members of the Solar System up to Uranus forming the first
octet.
The predicted distance of the next hypothetical planet beyond Pluto is
d_{10} = 19.6 + 0.3×2^{7} = 58.0 AU, not 77.2 AU, as predicted by the
unmodified TitiusBode Law. This value agrees precisely with the current distance of a
large Kuiper Belt Object called 2004 XR 190 (nicknamed “Buffy”) whose discovery^{13} by
astronomer Lynne Allen with the Canada France Hawai Telescope during the operation of the
CanadaFrance Ecliptic Plane Survey was announced on December 15, 2005. The large inclination
of 47° of its orbit to the ecliptic makes astronomers think it is a Kuiper Belt object, some of
which have large inclinations. However, Pluto’s orbit has an inclination of over 17° and so, if
Pluto is a real planet (some astronomers do not think it is, an issue discussed in Section 10),
the even larger inclination of Buffy is no reason to discount it as a true planet because it
could have arisen from a cause similar to what made Pluto’s inclination large. Furthermore,
unlike observed Kuiper Belt objects, it has an almost circular orbit, which is consistent with
its being a real planet, although complex gravitational interactions in the early history of
the solar system may also account for this. The additional fact that the measured distance
agrees exactly with prediction makes one optimistic that it is not just coincidental,
although caution is necessary in deciding whether this does, indeed, amount to a spectacular
confirmation of the explanation of the TitiusBode Law given in this article.
Beauty is a quality of eternal truth and mathematical beauty shines brightly
in the Pythagorean character of the Solar System, as will be evident shortly. Another criterion
is the set of ten Hebrew Divine Names assigned in Kabbalah to the ten Sephiroth of the Tree of
Life, for a large body of evidence both reported by the author^{14} and as yet unpublished^{15} has shown that they mathematically prescribe the archetypal
nature of Pythagorean whole systems through their gematria values.* Examples of this prescription will be discussed next.
____________________________
* By assigning integers to the letters of the Hebrew alphabet, a Hebrew
word can be converted into a number that is the sum of its letter values. This was the basis
of the ancient practice of gematria.
27
Table 1 indicates that the ratio of the average distance from the Sun of
Uranus, the last member of the first octet, and that of the asymptotic centre of the
logarithmic spiral is predicted to be 19.6/0.4 = 49 (actual value ≈49.205). This number is
highly significant because it is the number value of the Divine Name El Chai (“God
Almighty”) assigned to Yesod, the ninth Sephirah. Here is a remarkable illustration of how a
Godname prescribes aspects of a divine archetype, for the Solar System is an arena for
evolution (at least on the third planet from the Sun), and its structure has therefore to
conform to the nature of God, Who has ten aspects or qualities embodied by the Sephiroth.
This is not to suggest, of course, that God created the Solar System in the biblical sense.
Instead, it is to assert that any holistic system — whether a superstring, living cell or
human being must conform to the pattern of the Tree of Life. Just as the Pythagorean,
musical octave is whole and complete, so the first eight planets up to Uranus form a whole
that — because it is a whole — must be prescribed by the Divine Names. El Chai also
determines the crossing point of the inner spiral of Uranus with distance 128a from the
centre because, as the wavelength of the seventh octave of undertones (Table 5), 128 is the 49th note above the tonic 1 of the first octave. It is
also the 50th note, showing how the Godname Elohim assigned to the third Sephirah, Binah,
with number value 50 prescribes the octet of planets.
The number 192, the wavelength of the perfect fourth of the eighth octave,
is the 15th whole integer in Table 5. 15 is the number value of Yah, one of the two Divine Names assigned
to Chokmah, the second Sephirah, which therefore prescribes the distance of Uranus from the
asymptotic centre. Yah prescribes the distance 4a of the latter from the Sun because 4 is
the wavelength of the 15th undertone. According to Equation 30, the distance from the Sun of
the outer spiral associated with Uranus is R_{8} = 4a + 256a = 260a = 26×10a.
Table 5 indicates that 256, the distance of the outer spiral of this planet
from the centre, is the wavelength of the 15th subharmonic, showing how Yah prescribes the
size of the octet of planets. The mean EarthSun distance is 10a (see Equation 28).
Therefore, this point on the outer spiral is exactly 26 AU from the Sun. This shows how the
Divine Name Yahweh with number value 26 prescribes the section of the logarithmic spiral
that generates the full octet of planets.
It is of profound, religious significance that the most commonly known
ancient Hebrew Godname measures through its number value the size of the part of the accretion
disk that forms the octet of planets in terms of the Earth’s average distance from the Sun!
R_{7} = 132a and R_{8} = 260a, so the average distance of Uranus from the Sun =
d_{7}/d_{2} = ½(R_{7} + R_{8})/d_{2} = ½(132a +
260a)/10a = ½(13.2 + 26) = 19.6 AU, comparing well with the actual value of 19.19 AU. This
shows explicitly how the number value 26 of Yahweh determines the distance of Uranus. The
Pythagorean Tetrad (4) determines this distance because the distance (in terms of a) of the
outer spiral for Uranus from the asymptotic centre = 2^{8} = 256 = 4^{4}, a
beautiful, mathematical property of the octet of planets. The distance
28
between Mercury (distance 0.3d_{2} = 3a from the Sun) and Uranus
(distance 196a from the Sun) is 193a. 193 is the 44th prime number! Once again, the Tetrad
appears in this numerical prescription of the distance between the first and last planets of
the first octet. This will be commented upon shortly.
That the octet of planets does, indeed, constitutes a whole or a Tree of
Life pattern is indicated by the fact that the inner form of the Tree of Life — the seven
enfolded, regular polygons shown in Figure 15 — is shaped by 47 tetractyses with 260 yods outside their shared
edge, that is, the yods in 26 separate tetractyses (47 is the 15th prime number and is thus
prescribed by the Godname Yah, whose number value is 15). In the planetary manifestation of
this blueprint, each yod corresponds to the distance a and a tetractys corresponds to 10a —
the mean EarthSun distance.
The Godname Eloha assigned to Geburah, the sixth Sephirah, with number value
36 prescribes the distance a' = 32a of the first crossing point of the second spiral from its
asymptotic centre because, according to Table 5, 32 is the wavelength of the 36th note, counting from the tonic
1.
The distance of the n'th planet from the asymptotic centre of the second
logarithmic spiral is 96a×2^{n'1} = 48a×2^{n'}. Table 4 indicates that 48 is the wavelength of the fortieth undertone,
counting from the tonic of the first octave with a wavelength equal to 1. 40 = 4×10 =
4(1+2+3+4) = 4 + 8 + 12 + 16, i.e., this number is the sum of four integers spaced four
units, starting with 4. This and the earlier results expressed by the number
29
4 are examples of the Tetrad Principle formulated by the author^{16} whereby the fourth member of a class of mathematical object or
the sum of the first four members of a sequence of mathematical objects is always a
parameter of the universe. In this case the Tetrad naturally determines via the musical
scale the parameter 48 setting the scale of the logarithmic spiral associated with the
second octet of planets. The TitiusBode Law is, indeed, a universal law, not merely a rule
that applies only to one of the many planetary systems in the universe. 48 is also the tone
ratio of the fortieth tone, that is, the thirtyninth note after the tonic of the first
octave. There are (39 + 38 = 77) notes and intervals beyond the tonic up to this note. 77 is
the 76th integer after 1. This shows how Yahweh Elohim, Godname of Tiphareth, with number
value 76 prescribes the distance 48a between successive crossing points of the second
logarithmic spiral.
The number 384 measuring the distance of Pluto from the asymptotic centre of
the logarithmic spiral is the 16th subharmonic (16 = 15th integer after 1), showing how the
Divine Name Yah with number value 15 prescribes the distance of the tenth planet from the first
planet. Table 4 shows that 384 is the 61st smallest wavelength, starting from 1, the
tonic of the first octave. 61 is the 31st odd integer, where 31 is the number value of El
(“God”), the Hebrew Godname assigned to Chesed, the fourth Sephirah of the Tree of Life.
Table 2 shows that 384 is the 61st tone and the 30th overtone, that is, there
are 30 tones with fractional tone ratios up to the tone ratio 384. This has the following
remarkable geometrical representation: when the decagon is divided into its triangular
sectors and the latter are each converted into the ten dots of a tetractys, 61 dots are
created, of which 30 lie on its boundary and 31 are in its interior. All overtones up to 384
can therefore be assigned to dots on the sides of the decagon and all fractional tone ratios
between 1 and 384 can be assigned to dots in its interior, with 1, the tonic of the first
octave, appropriately at its centre (Fig. 16). The Decad (10) is the perfect number of the Pythagoreans.
Measuring the fullness of Divine Unity (the Monad), it is symbolised by the tetractys. As
well as measuring the number of planets in the Solar System, it also determines the maximum
distance the logarithmic spiral extends to create them.
The Pythagorean Tetrad defines the number 384 in the following way:
384 =

4! 4! 4! 4!
4! 4! 4! 4!
4! 4! 4! 4!
4! 4! 4! 4! 
(4! = 1×2×3×4)

This serves as another example of the Tetrad Principle at work. The facts
that the first term in the TitiusBode Law for the first octet of planets is 4 and that
planetary distances are determined by the wavelengths of perfect fourths of the
musical undertones further illustrate this potent principle.
The average distance of Pluto from the Sun is 388a and the average distance
of Mercury from the Sun is 3a. The average distance between the first and tenth planets is
385a, where
385 = 
1^{2}
2^{2} 3^{2}
4^{2} 5^{2} 6^{2}
7^{2} 8^{2} 9^{2} 10^{2}.

This is a stunningly beautiful property! Is it a curious accident? No. Who
can deny that it bears the hallmark of a Designer? In Astronomical Units this distance is
385a/10a = 385/10 = 38.5, that is, the arithmetic mean of the squares of the first ten natural
integers. The actual average distance of Pluto from Mercury is 39.09 AU, a discrepancy
30
31
of only 1.5% from the theoretical value of 38.5 AU! These results are very
convincing evidence in favour of Pluto being a true planet, not merely an escaped moon
of another planet.
Table 13 indicates that R_{7} = 132a. The theoretical distance
between the first planet and the inner spiral of the last planet of the first octet = 132a –
3a = 129a. The number value of Yahweh Sabaoth, Godname of Netzach, which is the seventh
Sephirah, is 129, showing how it prescribes this parameter of the first octet of
planets.
Figure 17 depicts how successive division of circles into pairs of similar
circles reproduces the powersof2 scaling of the logarithmic spiral that determines the
average distances of the planets from the Sun.
The distance of Venus from the asymptotic centre is 3a, the distance of
Earth from Venus is 3a and the distance of Pluto from Earth is 378a. These distances are
encoded in the inner form of the Tree of Life (Fig. 18) in the following way: the two sets of the first six polygons are
prescribed by the Godname Elohim because its number value 50 is the number of their corners.
They are made up of 378 coloured yods other than the three white yods (one corner of the
triangle and two corners of the hexagon) that coincide with Chokmah, Chesed and Netzach on
the Pillar of Mercy, the three analogous white yods coinciding with Binah, Geburah and Hod
on the Pillar of Severity and the two endpoints of the root edge shared by each set of
polygons. The three white yods on one side of the central pillar of Equilibrium denote (in
units of 1/10 AU) the distance of Earth from Venus and the three white yods on the other
side denote the distance of Mercury from Venus.
We saw earlier that the octet of planets MercuryUranus constitutes a Tree
of Life pattern because the size of the outer spiral of Uranus — 260a — is the number of yods
creating the shape of this pattern. The above result indicates that the first six polygons also
constitute a Tree of Life pattern. This is confirmed by the following extraordinary correlation
between their yod population and planetary distances: associated with each set of six polygons
are 25 corners and 168 other yods (Fig. 19). Compare this with the prediction* that the mean distance (in terms of 1/10 AU) of the Asteroid Belt
from
____________________________
* This does not take into account the relatively large eccentricity of
Mercury’s orbit, which is probably a feature acquired since the formation of the
planets.
32
Mercury is 25 (actually 23.8) and the average distance between the Asteroid
Belt and Uranus is 168 (actually, 164.2). The 25 corners associated with the six enfolded
polygons define the distance from Mercury of the first four planets after it (counting the
Asteroid Belt formally as a planet), whilst the remaining yods define the distance between the
Asteroids and Uranus — the last member of the octet.
This is truly remarkable, for it has the profound implication that the octet
of planets in the Solar System conforms to the Divine blueprint of the Tree of Life! The four
() yods coinciding with the positions of the Sephiroth in the Tree
of Life symbolise the distance 4a between the Venus and Mercury and the 189 other yods
symbolise the distance 189a between Venus and Uranus.
That this correlation is not a coincidence is indicated by the remarkable
fact that there are 168 yods on the boundaries of both sets of six polygons outside their
shared edge (Fig. 20).
The number 168 determines their shape because it is the number value of
Cholem Yesodeth (מלח
יסודות), the Kabbalistic title of the Mundane Chakra of Malkuth.
Its physical significance as the basic structural parameter of superstrings has been discussed
in most of the author’s earlier articles.
33
In analogy to the notes of the Pythagorean scale, Mercury represents the
tonic, whilst the seven planets beyond Mercury up to Uranus represent the seven notes above the
tonic, Uranus completing the planetary octet and musical octave and playing the same role for
the next octet containing Neptune and Pluto as Mercury does for the first octet, i.e., its
first member or musical tonic.
9. Planetary distances and superstring
theory
The symmetries displayed by the forces of nature are expressed by physicists
in the language of a branch of mathematics called ‘group theory.’ The mathematical fields that
represent the particles mediating a given type of force are said to be the gauge
fields of the gauge symmetry group expressing the symmetry of this force. The
mathematical transformations belonging to a gauge group are defined by its set of
generators. The dimension N of a symmetry group is the number of independent
generators defining its transformations. Each generator is associated with its own gauge field.
N gauge fields are therefore associated with a symmetry group of dimension N. The complete set
of N generators of a group obeys the rules of an abstract algebra called Lie algebra.
This gives rise to a certain algebraic equation, solutions of which are specified by a set of N
points in an ldimensional Euclidean space, where l is the rank of the group. Each
point defines a root of the group. l of the N roots are said to be zero roots
because they denote points at the centre of the diagram representing these roots, and (N–l)
roots are called nonzero roots because they denote points a nonzero distance from
this centre.
Physicists Gary Schwarz and Michael Green made the important discovery in
1984 that the gauge symmetry group describing the symmetries of the forces other than gravity
acting between 10dimensional superstrings had to have the dimension N = 496 in order that the
theory be free of quantum anomalies. They pointed out that two groups:
E_{8}×E_{8} and SO(32)* have this dimension. The former is the one that first became favoured
by string theorists. The group E_{8} is called the exceptional group of rank
8. It has dimension 248 (half of 496). Its 248 roots consist of 8 zero roots and 240
nonzero roots. The 496 roots of E_{8}×E_{8} thus consist of (8+8=16) zero
roots and (240+240=480) nonzero roots. The 8 zero roots of E_{8} comprise a zero
root of one kind (which need not be specified here) and 7 of another kind. Similarly, the
240 nonzero roots of E_{8} consist of 128 nonzero roots of one kind and 112 of
another kind. The root composition of E_{8}×E_{8} is laid out below:
Let us now compare the root composition of E_{8} with the distances
between the planets. Inspection of Table 13 shows that (in terms of a) the outer spiral of Uranus is at a
distance R_{8} = 260 from the Sun and the outer spiral of Earth is at a distance R
_{3} = 12. The distance between the outer spirals of the third and eighth planets =
R_{8} – R_{3} = 260 –
____________________________
* A third group contained in the other two groups has since been found to
be free of anomalies.
34
12 = 248 (Fig. 20). Amazingly, this is the dimension of E_{8}! The outer
spiral of Mars is at a distance of R_{4} = 20. The distance between the outer
spirals of the fourth and eighth planets = R_{8} – R_{4} = 260 – 20 = 240.
This is the number of nonzero roots of E_{8}. The distance between the outer
spirals of Earth and Mars = R_{4} – R_{3} = 20 – 12 = 8. This is the number
of zero roots of E_{8}. The Earth occupies a unique position in the Solar System in
that the distance between its outer spiral and the edge of the octet is 248 units — the very
number of roots of the superstring gauge symmetry group. Its neighbour Mars defines the
number of nonzero roots and the distance between them measures the number of zero roots.
Moreover, the distance of the inner spiral of Uranus is R_{7} = 132, so that the
distance between its inner and outer spirals = R_{8} – R_{7} = 260 – 132 =
128.
As stated earlier, this is the number of nonzero roots of a certain kind.
The distance between the outer spiral of Mars and the inner spiral of Uranus = R_{7} –
R_{4} = 132 – 20 = 112. This is the number of nonzero roots of another kind (see
above). The distance 240 between the innermost and outermost spirals of the outer four planets
of the octet splits into the pair of integers:
measuring, respectively, the width of these spirals for the first three
planets and the width for the fourth one. Remarkably, this division is the same as those
defining the number of the two types of nonzero roots of E_{8}. The pattern of
distances for the octet of planets mirrors the root structure of the mathematical symmetry
group describing superstrings! Hard though this may seem to believe, it has the following
simple but profound reason: as proved in earlier work,^{17} E_{8} belongs to the Tree of Life description of the
forces of nature. The mathematical explanation of the TitiusBode Law presented in this
article is also part of the Tree of Life blueprint, for it applies to any planetary system.
The same pattern of numbers must ipso facto manifest in both the Solar System and
the superstring because both are wholes in the Pythagorean sense, conceived
according to the Divine blueprint of the Tree of Life. Figure 15 shows the geometrical form of this blueprint and how it embodies
the number 260 measuring the
35
size of the octet.*
Once again, these distances are an example of the Tetrad Principle
because
The sum of the first three powers of 2 is 112, which is the distance spanned
by the Asteroid Belt, Jupiter and Saturn. The fourth power of 2, i.e., 2^{7}, is the
distance between the two spirals of Uranus. The 3:1 division in the number of powers reflects
the same pattern in the last four planets of the octet.
The distance between the inner spiral R_{4} of the Asteroid Belt and
the outer spiral R_{8} of Uranus is 240, whilst the distance between the Asteroids and
Uranus is 168. The remaining distance is 72. These numbers are expressed by the tetractys
representation of the number 240:
The sum of the numbers 24 at the corners of the tetractys is 72 and the sum
of the numbers at the corners and centre of the hexagon is 168. The central integer 24 is the
predicted distance between the Asteroid Belt and Jupiter (the actual distance is 24.3). In
terms of superstring theory, 240 is the number of nonzero roots of E_{8}, 72 is the
number of nonzero roots of E_{6}, an exceptional subgroup of E_{8}, and 168 is
the number of nonzero roots of E_{8} that do not belong to E_{6}.
Ignoring the fact that the centre of its eccentric orbit is not located at
the Sun (as in the case of other planets to a good approximation), the distance of Mercury from
the Sun is predicted to be 3 units. The distance of the outer spiral of the eighth planet
completing the octet is R_{8} = 260. The distance between the first planet and the edge
of the octet = 260 – 3 = 257. This is the 55th prime number, where
55 =

1
2 3
4 5 6
7 8 9 10

is the tenth triangular number. This demonstrates for the second time how
the Pythagorean Decad defines properties of the Solar System. It is evidence that the distance
of Mercury from the Sun is correctly given by ½R_{1} for, had this not been the case,
neither this result nor the spectacular property of the distance 385 between Mercury and Pluto
being the sum of the squares of the first ten integers would have
____________________________
* It is amusing that the eccentricity of Pluto, the tenth planet,
is 0.248, whilst its orbital period is 247.92 years, that is, almost 248. Two orbital
parameters of Pluto are approximately the dimension 248 of the superstring symmetry group
E_{8}!
36
been true. We encountered earlier a similar property in finding that the
distance between Mercury and Uranus is 193, the 44th prime number. The extra distance between
Uranus and its outer spiral = 257 – 193 = 64 = 4^{3}, which is another beautiful
illustration of the Tetrad Principle. The Solar System is measured out with the Pythagorean
yardstick of the number 4. No wonder that one of the ancient titles assigned by the early
Pythagoreans to this number was “holding the key of nature”!
10. Is Pluto a
planet?
Although the International Astronomical Union still declares Pluto to be a
planet,^{18} some astronomers believe that it was not formed at the time of
the other planets but is a satellite of Neptune that was knocked out of orbit. The
reasons^{19} they give for this belief are:

Inclination of its orbit compared to the ecliptic is 17.148°;

Large orbital eccentricity: 0.248 (Earth's eccentricity: 0.0167);

Composition: Pluto is composed of:

core of hydrated rock (70% of mass);

mantle of water ice;

atmosphere containing methane ice (and possibly: N_{2}, CO,
CO_{2}). This composition is very different from the other outer
planets because they are mainly composed of gas. Therefore, Pluto’s density is
larger than the other outer planets.

Pluto has a high albido: ±0.5. Remarkably, it is irregular; Pluto has the largest
globalscale contrast in the solar system. This
indicates that the planet is active;

Charon (Pluto’s satellite) is extraordinary large compared to Pluto: Pluto’s
radius:Charon’s radius = 1:0.5, in comparison with:
Earth’s radius:Moon’s radius = 1:0.3 and Mars’s radius:Phobos’s radius = 1:0.003. This
makes some astronomers believe that
Pluto and Charon may be a double planet.
The fact, however, that Pluto’s composition is more like that of the
asteroids in the Kuiper’s Belt is hardly unambiguous evidence that it, too, was once an
asteroid. The relative tenuity and coldness of the material on the edges of the accretion disk
that condensed into the Solar System would lead one to expect any bodies to have formed there
later than those nearer the Sun and to display a composition different from that of the gaseous
giant planets, especially if the outer rim of the gaseous accretion disk was mixed with
material from the Kuiper Belt.. Nor is its large orbital eccentricity evidence that it is not a
planet, for Mercury’s eccentricity is almost as large, yet no one disputes that it is a genuine
planet. Pluto differs from objects in the Kuiper Belt by having an orbital inclination to the
ecliptic of about 17° — much larger than the several degrees of most Kuiper Belt objects. This
difference needs to be explained if Pluto is such an object that left the Kuiper Belt some time
in the past. There have been a number of models proposed. None has worked. It has been
theorized that Pluto was a natural satellite of Neptune and that Triton, now one of its
satellites, was originally in a heliocentric orbit. Their orbits were changed by a collision,
which also created Charon, Pluto’s satellite, through tidal forces. However, computer
simulations of the orbits and dynamics of Neptune and Pluto have made this scenario very
unlikely. It also seems improbable that an object far enough away from the Sun to belong to the
Kuiper Belt could have been captured by Neptune. Charon could have been a Kuiper Belt object
that was captured by Pluto. This would be consistent with its relatively large size, for one
object — Quaoar — has been found in the Kuiper Bet with about onehalf the size of Pluto, that
is, it is about as large as Charon.^{20} Finally, Mercury has an orbital inclination of about 7°, which
is not much smaller than the average of 10±1 degrees^{21}
37
reported by March, 1999 for Kuiper Belt objects. Yet no one would argue that
Mercury is such an object! The relatively large inclination of Pluto’s orbit compared with
other heliocentric planets is not unambiguous evidence that it is an asteroid, for it is
possible that this light body could have been formed from the planetary nebula but have been
subsequently knocked into an orbit of a different inclination through one or more collisions
with other objects. This is consistent with the relatively large eccentricity of its orbit,
which suggests that the orbital plane changed from what it was during the formation of the
planets. It is surely not coincidental that the two planets Mercury and Pluto that have
relatively large eccentricities also have relatively large orbital inclinations. If one planet
can have a large orbital inclination without having to be considered a Kuiper Belt object, why
should the same not apply to another planet?
Contrary to what its protagonists believe, the idea that Pluto is not a
natural planet has no strong argument or evidence to support it. Even if objects in the Kuiper
Belt are found that are as large as Pluto (or even larger), this does not cast doubt on whether
it is a planet, for why should there not be such objects in the primordial debris field? But
for the fact that Pluto did not seem to obey the TitiusBode Law at all, arguably there would
have been little incentive to doubt that Pluto is a planet. However, as we have seen, Pluto
does in fact obey the theory underlying the law to a very good degree of accuracy. This is
inexplicable if it had been merely a satellite of Neptune that was struck by Triton and flew
away into its current, heliocentric orbit. For the TitusBode Law governs those bodies that
condensed out of the primordial gas cloud, not objects that later assumed stable orbits around
the Sun through collisions with other detritus that never managed to aggregate into planets.
Protagonists of the satellite model might be on firmer ground if the theory presented here had
restored Neptune to the fold of planets obeying the TitiusBode law but still excluded Pluto.
However, in the theory presented here restoring Neptune automatically reinstates Pluto. Unless
the agreement between its predicted distance and its actual distance is a highly implausible
coincidence (but, then, what about the beautiful Pythagorean property of the number 385
revealed on page 30?), the fact that accurate predictions can be made for both Neptune and
Pluto is strong evidence that the latter is a true planet. The signature of a true planet
should be that it obeys the correctly understood form of the TitusBode Law, which
this article has shown Pluto to do with an acceptable degree of accuracy.
11. Planets beyond
Pluto?
It remains an open question as yet whether the Solar System has more than
nine true planets (as opposed to Kuiper Belt objects that may be mistaken for them). It is an
active research topic amongst astronomers. BBC News reported on 13 October, 1999 work by Dr
John Murray^{22} of the UK’s Open University that suggests a large planet orbits
the Sun a thousand times further away than Pluto, that is, thirty thousand times further
than Earth. Murray, who studied the motion of socalled ‘long period’ comets, analysed the
orbits of thirteen comets in the Oort Cloud, a region of the Solar System about 50,000 AU
from the Sun and about onethird the distance to the nearest star containing an estimated
100 billion comets that spend millions of years before being deflected by collisions into an
orbit bringing them into the inner Solar System. He detected signs of a single massive
object that was disturbing all of them. Although not yet observed, the planet is several
times bigger than Jupiter, the largest known planet in the Solar System. At three thousand
billion miles from the Sun, it would take almost six million years to orbit it. It would not
have been already found because it is too faint and moves too slowly. It has been suggested
that the object is a planet that has escaped from another star because it orbits the Sun in
the opposite direction to that of the other
38
planets. Professor John Matese of the University of Louisiana at Layfayette
came to the same conclusion in a similar study.
It will be difficult to establish that any planetoid or object
found* to be orbiting the Sun further from Pluto is a natural planet
(that is, one formed at the time of the known planets) and not some Kuiper Belt object that
escaped and was forced gradually into a heliocentric orbit. This article predicts that the
next planet after Pluto should have a mean distance from the Sun of 58 AU – exactly that
measured for the recently reported object called “Buffy,” although it remains to be
determined whether this is more than coincidence.
The Kuiper Belt is a diskshaped region between 30 and 100 AU from the Sun
that contains many icy bodies. The predicted average distance from the Sun of the fourth member
of the second octet brings its orbit well into the Kuiper Belt, as does Pluto’s orbit. It would
be outside the socalled “classical KBO” orbit near about 50 AU, making a small planet
difficult to detect among the thousands of objects already observed in the Kuiper Belt. But one
such object precisely satisfying the modified Titius Bode Law has now been detected, although
regarded at the moment by astronomers as a Kuiper belt object. Being so far away, it would
likely obey the TitiusBode Law more accurately than Pluto, whose small deviation from the
version formulated here is probably due in part to the gravitational pull of its large
neighbour Neptune. This, indeed, is the case with Buffy, whose current distance agrees
precisely with prediction, although it spends all its time between 52 and 62 AU from the
Sun.
12. Satellite
evidence
Using satellites of the planets within the Solar System to test the
TitiusBode Law is complicated by the fact that many small satellites are likely captured
bodies that did not form directly along with the present planet or may have had their orbits
drastically changed during the early evolution of the Solar System. The smaller the satellite
in the sample, the more uncertain is its eligibility. In order to test the idea proposed in
this article that planets beyond the eighth one belong to a rescaled octet, it is necessary to
have a test sample of at least nine satellites for a given planet. This is because the
changeover to the new octave described by its own version of the law — the form that needs
testing — occurs with the eighth planet and so the ninth one is the first proper instance of
the new version. However, it is these more distant satellites whose origin may be uncertain
enough to render meaningless any disagreement between prediction and observation. Testing the
law is therefore problematic for n = 8–15.
Jupiter is the most obvious planet to analyze because its four Galilean
satellites are thought to be natural, being much larger than its other, much lighter, orbiting
bodies. Uranus also has four or possibly five satellites that stand out from its other
smaller
____________________________
* A planetoid 8001100 miles in diameter and as far away as 84 billion
miles was announced in March 2004. Called "Sedna," it is the largest object seen since Pluto
was discovered in 1930.
39
satellites. Both Saturn and Neptune, however, have only one large satellite,
making it difficult to test the rule. The smaller planets have few or no satellites, making
testing impossible. Therefore, only the Jovian and Uranian systems provide enough information
to test distance rules.
Table 15
Satellite 
Distance
(1,000 km)

Distance
(relative)

d_{n}

n

Io 
422

1.00

1.0

0

Europa 
671

1.6

1.5

1

Ganymede 
1,070

2.5

2.5

2

Callisto 
1,885

4.5

4.5

3

Howard L. Cohen pointed out^{23} in 1996 that the four Galilean moons of Jupiter obey:
d_{n} = 0.5 + 2^{n1}

(n = 0, 1, 2, 3, …)

(43)

Table 15 compares the actual and predicted distances of these satellites.
The agreement is good. Almathea, the next Jovian satellite to be discovered after Galileo
found those listed in the table, orbits 180,000 Km from Jupiter, or 0.43 of Io’s distance.
Cohen pointed out that, if n = ∞ for this satellite, its predicted distance is 0.5, which
is another good agreement. However, as in the case of Mercury, this value of n is
counterintuitive and should actually refer not to a satellite but to the asymptotic centre
of the spiral that would have generated Jupiter and its natural moons. Moreover, Equation 43
is not a proper expression of the general TitiusBode Law as the values of n should start
with 1, not 0. Instead, therefore, let us write for these five satellites the form of the
rule:
d_{n} = a + b×2^{n1},


(44)

where n = 1 for Io, i.e., a is the distance of Almathea from Jupiter (for the moment, we use
the standard interpretation). Let us work with the actual distances of the five satellites
rather than make assumptions about which satellite should provide the unit of distance. Using
Almathea and Io to determine a and b, then a = 180 and b = 242. Table 16 compares the actual and predicted distances of the test
satellites:
Table 16
Satellite

Actual distance
(1,000 Km)

Predicted distance
(1,000 Km)

Europa 
671

664

Ganymede 
1,070

1,148

Callisto 
1,885

2,116

We find that, whilst the agreement is good for Europa, it is poor for
Ganymede and Callisto. A progression of powersof2 requiring values of n that start from the
sensible value n = 1 therefore shows good agreement only for one of the five satellites.
However, Table 15 shows that the spacings between Io, Europa, Ganymede and Callisto
do increase as powers of 2. The predicted spacings between AlmatheaIo, IoEuropa,
EuropaGanymede and GanymedeCallisto are, respectively, 242, 242, 484 & 968, comparing
poorly with the measured values of 242, 249, 399 & 815. This means
40
therefore that the satellite Almathea does not fit the historical form of
TitiusBode Law because its use for determining values of a and b leads to a false, poor fit
for other satellites.
Table 17
Satellite

Actual distance
(1,000 Km)

Predicted distance
(1,000 Km)

Ganymede 
1,070

920

Callisto 
1,885

1,418

Suppose, instead, that Io and Europa are used to fix a and b. Table 17 compares the actual and predicted distances for the remaining two
test satellites. The fit is even poorer. If they conform to a general TitiusBode Law, the
spacings between Io, Europa, Ganymede and Callisto should be in the proportions 1:2:4.
Instead, they are in the ratio 1:1.60:3.27, which is a poor match.
Let us check, however, whether the reason for the nonBode behaviour of the
five satellites is that it is wrong to assume that ‘a’ in Equation 41 is the distance of the
nearest natural satellite from the planet. The theory presented here indicates that this is the
case, for this distance should be 3a/4, not a. The predicted correct form of the law
is
d_{n} = 4a + 3a×2^{n1},

(n = 1, 2, 3, etc)

(45)

This means that 3a = 180 for Almathea and a = 60 for the new law, whereas 4a = 180 and a = 45
for the old law. Table 18 compares the actual and predicted distances for the four Galilean
satellites:
Table 18
Satellite

Actual distance
(1,000 Km)

Predicted distance (old)
(1,000 Km)

Predicted distance (new)
(1,000 Km)

Io 
422

7×45 = 315

7×60 = 420

Europa 
671

10×45 = 450

10×60 = 600

Ganymede 
1,070

16×45 = 720

16×60 = 960

Callisto 
1,885

7×60 = 420

28×60 = 1680

For all four satellites, the new form of the law gives a far superior
fit than the traditional version. In the case of Io, it is excellent. Astronomers have
falsely concluded that the satellites of Jupiter do not fit the TitiusBode Law, whereas the
true reason for this is that they have misunderstood it by believing that its first term
represented the distance of the first planet or satellite from, respectively, the Sun or
planet, whereas it really denoted their distance from the asymptotic centre of the logarithmic
spiral. The difference in the case of the planets amounts to 0.4 – 0.3 = 0.1 AU, which was
small enough compared with 0.4 AU not to distort significantly the agreement between the
planetary data and the wrong form of the law. However, a similar 25% error in the assumed value
of the nearest satellite’s distance underestimates the total distance by 25% because
the deduced value of a appears in both terms of Equation 45 and so the discrepancy
gets larger, the further the satellite is from the planet, as Table 18 indicates. That said, the measured spacings 249, 399 and 815
between Io, Europa, Ganymede and Callisto still show 1028% difference from the values 180,
360 and 720 predicted by the new form of the TitiusBode Law. This is not large enough to be
certain that they do not obey even the correctly interpreted law, but it is also not small
enough
41
to be sure that they do. The four major satellites are known to be locked
into orbital periods that are each twice that of the next inner satellite. It is believed
that the reason for this is that tidal drag is forcing them outwards to lock to the
period of the
Table 19
Satellite

Semimajor axis
(Uranian radii)

Difference

Miranda 
5.08




2.40

Ariel 
7.48




2.93

Umbriel 
10.41




6.64

Titania 
17.05




5.74

Oberon 
22.79


outermost, large satellite, Callisto. This factor distorts the picture and
makes a test of any theory of the TitiusBode Law based upon satellites
inconclusive.
The major satellites of Uranus also have nonBode spacings (Table 19). The spacing between Titania and Oberon is less than that between
Umbriel and Titania, despite
42
Oberon being further out from Uranus than Titania is. The lesser satellites
of Uranus have mean distances of the order of 23 Uranian radii or several hundred radii. Their
spacings decrease, then increase, showing no signs of obeying a TitiusBode rule, whether old
or new.
13. Conclusion
As a planetary nebula collapses into an accretion disk centred on a
protostar, it aggregates into vortical currents or streams of material flowing along a
logarithmic spiral. Logarithmic spirals are ubiquitous in nature as the geometrical shape of
biological systems. One such example is the golden spiral found in seashells such as the
Nautilus, successive radii of whose curves are in the proportion of the Golden Ratio. The
cochlea is another example (Fig. 21), although its spiralling is not confined to a plane. The asymptotic
centre of the logarithmic spiral does not coincide with the nucleus of the disk — the future
Sun — because the swirling is not a stable state like elliptical orbital motion produced by
gravity. Mutual attraction between bodies in the spirals causes the latter to break into
separate sections, the material in each spreading eventually into an elliptical ring of
matter bound to the protostar. The relative sizes of the rings are set by the geometry of
these logarithmic spiral sections, which double in radius from one to the next. Eventually,
each ring aggregates into planetesimals and then a planet whose orbital mean distance is the
arithmetic mean of the radii of the edges of the annulus. This explains why the average
distances of the planets from the Sun are perfect fourths of successive octaves of
undertones. A modified TitiusBode Law emerges from this scenario that fits the data better
than the historical version. The predicted distance of Pluto from Earth differs from the
known value by only 0.05%, which is very impressive. It explains why Mercury seems a
singular case of the geometric progression of distances by correcting what astronomers
mistook in the relationship as the mean distance of Mercury from the Sun, whereas this term
really denotes the distance of the asymptotic centre of the logarithmic spiral from the Sun.
The musical analogy suggests that planets distribute themselves in groups of eight like the
notes of the Pythagorean musical scale. This eightfold pattern had long been signalled by
the breakdown of the relationship for Neptune and Pluto, the ninth and tenth members of the
Solar System, but astronomers failed to realise its significance. Instead, they wrongly
concluded that the good fit to the TitiusBode Law was merely coincidental and that this
regularity was not a true law. On the contrary, both planets fit the modified law. Moreover,
the recent detection of a body about half the size of Pluto at precisely the distance
predicted by the modified law for the next planet beyond it is evidence that — as with
Uranus, this law once again has predicted the mean distance of a new planet from the
Sun.
The logarithmic spiral distribution of matter that generated the planets may
be likened to what mathematicians understand as a ‘fractal’ in the sense that it was
selfsimilar, although not continuously so at all scales of distance but only for the
discrete series of scales set by each octet. Planetary systems do not contain an arbitrary
number of planets but, instead, are made up of octets of planets in which — like the tonic and
octave of the Pythagorean musical scale — the most distant member of one octet (octave) is the
first member (tonic) of the next outer octet. The geometry of each octet is the same, but
rescaled and shifted so that its asymptotic centre coincides with the average distance of the
last member of the previous octet. As a result, Neptune and Pluto obey, mathematically
speaking, the same kind of TitiusBode Law as the other planets do, but its parameters are
rescaled for the octet to which they belong.
Evidence that this rescaling actually exists (that is, apart from the better
fit to the
43
relationship by all ten known members of the Solar System) appears
in the form of beautiful, mathematical properties displayed by their theoretical distances that
cannot, plausibly, be due to chance because their inherent Pythagorean character would not then
have manifested. This is supported by the system of Hebrew Godnames, which have been shown
elsewhere to prescribe the mathematics of superstrings and their spacetime structure. The way
in which the Divine Names determine the octet of planets is unmistakably clear. Amazingly, the
predicted distance (in tenths of an AU) between the crossing points of the outer spirals of
Earth and Uranus is 248 — precisely the number of transmitter particles predicted by the five
superstring theories to mediate their forces! That this, too, cannot be coincidence is
indicated by the presence also of the grouptheoretical numbers 112 and 128 associated
with the superstring gauge symmetry group E_{8}, which denote distances between pairs
of planets. Together with the unique Pythagorean character of the TitiusBode Law when
expressed in terms of the average EarthSun distance, this has an extraordinary implication for
the planet Earth. It is as if, when God the Architect decided to design the Solar System, He
had wished to leave the following clue for its future inhabitants about the nature of the
subatomic world: placing the end of a tape measure at the edge of the particular spiral arm of
the planetary nebula from which Earth was to form, God extended the tape until it reached a
spiral arm at the mark numbered 248, at which point He finished His task of sizing the nebula
for the first octet of planets and used spirals left over for all remaining planets. In this
sense, Earth occupies a unique position amongst the family of planets.
The beautiful, mathematical properties of the Solar System revealed through
this article’s presentation of the theory underlying the TitiusBode Law is evidence that the
Solar System is not some galactic backwater created by random chance. Instead, it is a cradle
for life — the Divine Life — that bears the signature of its Designer in the geometry of its
sacred Tree of Life blueprint, shaped according to the mathematical archetypes embodied in the
ten ancient Hebrew, Divine Names. The Pythagoreans taught that form is number. Their principle
of apeiron, or the Unlimited (Plato’s principle of the Indefinite Dyad) is at work
sizing the arms of the logarithmic spiral according to powers of 2, whilst their principle of
peras (Limit) shows itself in the TitiusBode Law as the number 3 and its
multiplication by powers of 2 to define the actual distances of planets relative to
the asymptotic centre of the logarithmic spiral. It is the perpetual interplay and balance of
these opposite but complementary principles that creates harmonia — the true
Pythagorean “music of the spheres” — in the form of the ten octaves of perfect fourths
of undertones whose wavelengths are the average distances of the planets from this centre.
“We shall not cease from exploration, and the end of all our exploring
will be to arrive where we started and know the place for the first time.”
T.S.
Eliot
References
^{1} Johann Daniel Titius, Betrachtung über die Natur, vom Herrn Karl
Bonnet (Leipzig, 1766), pp. 7–8; transl. by Stanley Jaki in “The early history of the
TitiusBode Law,'' American Journal of Physics, vol. 40 (1972), pp. 1014–23.
^{2} http://en.wikipedia.org/wiki/Ellipse.
^{3} http://home.att.net/~numericana/answer/geometry.htm#elliptic.
44
^{4} Ibid.
^{5} Phillips, Stephen M. Article 11: “Plato’s Lambda — Its Meaning,
Generalisation and Connection to the Tree of Life,” (WEB, PDF).
^{6} Ibid., pp. 7–9.
^{7} http://www.space.com/scienceastronomy/gemini_keck_020107.html.
^{8} http://www.space.com/scienceastronomy/astronomy/brown_dwarf_001122–1.html.
^{9} http://www.solarviews.com/eng/orionnebula1.htm.
^{10} http://www.seds.org/hst/OriProp4.html.
^{11} Phillips, Stephen M. Article 15:
“The Mathematical Connection Between Superstrings and Their Micropsi Description: A Pointer
Towards Mtheory,” (WEB, PDF).
^{12} Phillips, Stephen M. Article 16: “The Tone Intervals of the Seven
Octave Species and Their Correspondence with Octonion Algebra and Superstrings,” (WEB, PDF).
^{13} See announcement at http://www.cfeps.astrosci.ca/4b7/index.html.
^{14} See articles at http://www.smphillips.mysite.com.
^{15} “The Mathematical Connection between Religion and Science,” Stephen
M. Phillips (Antony Rowe Publishing, England, 2009).
^{16} Phillips, Stephen M. Article 1: “The Pythagorean Nature of
Superstring and Bosonic String Theories,” (WEB, PDF).
^{17} Refs. 13 & 14.
^{18} In a press release dated Feb. 3, 1999, the International Astronomical
Union stated, "No proposal to change the status of Pluto as the ninth planet in the solar
system has been made by any Division, Commission or Working Group of the IAU responsible for
solar system science. Lately, a substantial number of smaller objects have been discovered
in the outer solar system, beyond Neptune, with orbits and possibly other properties similar
to those of Pluto. It has been proposed to assign Pluto a number in a technical catalogue or
list of such TransNeptunian Objects (TNOs) so that observations and computations concerning
these objects can be conveniently collated. This process was explicitly designed to not
change Pluto's status as a planet." (see: http://www.iau.org/PlutoPR.html). Subsequently, a
decision was made, forced by a minority of astronomers, to relegate Pluto to the status of a
dwarf planet. According to the demonstration in this article that its mean distance fits
well the generalised TitiusBode law, this undemocratic and unscientific decision is
incorrect.
^{19} http://www.astro.rug.nl/~mwester/aos/aose.html.
^{20} http://www.ifa.hawaii.edu/faculty/jewitt/kb.html.
^{21} http://www.ifa.hawaii.edu/faculty/jewitt/papers/ESO/ESO.pdf.
^{22} http://www.ras.org.uk/html/press/pn9932.htm.
^{23} “The TitiusBode Relation Revisited,” Howard L. Cohen. http://www.fluridastars.org/960coke.html.
45
