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Why Twelve ?

By
Koenraad Elst

The present paper deals with a question of
symbolism: what is so special about the number 12? Historically, the
preference for the number 12 goes back to the Zodiac. Thus, the twelvestar
flag of the European Union was designed, in a public contest, by a devotee
of the Virgin Mary, who thought of the Apocalypse passage where a celestial
virgin appears in a circle of twelve stars; and these "twelve stars", in
Hebrew mazzalot (whence mazzel!, "good luck", originally "lucky star",
"beneficial stellar configuration"), were a standard expression referring to
the Zodiac, the division of the Ecliptic in twelve equal parts, each one of
them represented by a symbol: Aries, Taurus, Gemini, Cancer, Leo, Virgo,
Libra, Scorpio, Sagittarius, Capricorn, Aquarius, Pisces.

Though we acknowledge the intimate
connection between astrology and the symbolic structure of the Zodiac, it is
outside the scope of this paper to comment on the merits claimed for
astrology. Indeed, we assume that stellar lore including the Zodiac precedes
its use as a tool for divination, and that it is worth analyzing purely as a
symbolic construct, regardless of its use by diviners. Contrary to what some
astrologers claim, astronomy is very much older than astrology. But unlike
astrology, the natural tendency to read "faces in the clouds", or in this
case images in the stellar groupings, is probably as old as stargazing
itself.

**
Not-so-special properties of twelve**

The relationship of the number 12 with other
numbers is interesting, but not really unique. Thus, it is said that 12 = 3
x 4, with the added explanation that "3 represents time" while "4 represents
space". All very good, but then 10 = 2 x 5, which is not bad either and just
as pregnant with number symbolism. And note that in both cases, the factors
when added (rather than multiplied) yield 7, that mystical number. So, for a
unique property, we must look elsewhere.

In number theory, we do meet 12 in
intriguing places. It is the sum of the first three natural numbers
satisfying Pythagoras's (actually Baudhayana's) theorem, 3˛ + 4˛ = 5˛, and
also figures in the next Pythagorean threesome: 5˛ + 12˛ = 13˛. In
Fibonacci's series, the 12th number happens to be 12˛, or 144; it is the
only number to have this property except for 1 (for the first power, the
property is shared by the numbers 1 and 5, which stands at the 5th place;
for the third power, there is none). There are twelve multiplications of
natural numbers equalling 360 (1x360, 2x180, 3x120, 4x90, 5x72, 6x60, 8x45,
9x40, 10x36, 12x30, 15x24, 18x20). All very interesting, but less telling
and unique than the properties of 12 conceived as a geometrical entity, viz.
as the division of the circle into 12 equal parts.

**
How to divide the Ecliptic?**

The Ecliptic can be divided into any number
of zones. Wellknown is the division of 27 or 28 moonstations of about 13°
each, marking the angular distance covered daily by the moon. The division
in lunar mansions links an astronomical phenomenon, the moon's movement,
with a division of space. The same principle probably underlies the division
in twelve: it seems to be based on the approximately twelve lunation cycles
in the solar year (whose quarterperiods of roughly seven days may also be
related to the division in weeks).

But why should immutable space be subjected
to divisions suggested by the coincidental and highly impermanent data of
the moon's motion? There could well have been no moon at all (as is the case
for Venusians), or mankind could have come into existence and designed a
Zodiac millions of years ago, when the moon was closer to the earth and its
cycle as expressed in earthly days or fractions of earthly years much
shorter.

Freeing ourselves from the suggestions
emanating from accidental circumstances, we want to construct a division of
the circle based on nothing but the abstract circle itself, considered as a
geometrical figure, hence part of a continuum of geometrical constructions.
Which division of the circle is intrinsically most meaningful to the whole
project of symbolically representing the diverse aspects of the universe
with the sections of the ecliptical circle?

**
World models**

The Zodiac is devised and understood as a
world model, a simplification of the infinite complexity of the phenomenal
world to a scheme with a finite number of elements, which nonetheless
approximates the structure of the real world in that it embodies all worldly
oppositions. In a rational world model (e.g. the four/five elements), if we
have an element meaning "big" or "cold", than we must have one which means
"small" c.q. "hot", just as in reallife natural cycles, a sunrise is
counterbalanced with a sunset. The symmetry of a circle and of its rational
divisions is already a good metaphor for this general symmetry requirement
of credible world models.

A world model replaces the practically
infinite multiplicity of phenomena with a finite set, just as a regular
polygon inscribed in a circle replaces the infinite division of the circle
into infinitely small sections with a finite division into discreet and
finite sections. If we study the surface of these polygons, we find that
practically all of them, just like the circle itself (surface = pi, assuming
radius = 1), have a surface numerically represented by a number reaching
decimally into infinity (in practice represented by a finite sum involving
at least one root), although the surface values of the polygons, unlike
those of the circle, are not transcendent numbers (meaning numbers which can
only be analyzed into infinite sums, e.g. pi = 4/1 4/3 + 4/5 4/7... ad
infinitum).

For our project of replacing unmanageable
infinity with more manageable finiteness, we find polygons with surface
values consisting of a finite combination of roots and rational numbers a
great improvement visāvis transcendent numbers, but we would prefer polygons
with even more finite and manageable measurements, viz. those which have a
rational surface value. Best of all are those with a natural number as
magnitude of its surface. There are three of them: the bronze medal is for
the inscribed square with surface = 2, silver for the circumscribed square
with surface = 4, and gold for the inscribed dodecagon with surface = 3, the
natural surface value most closely approaching pi. This way, the division
into twelve is not just one in a series, it is quite special and corresponds
in a neat metaphorical way with the whole project of devising a world model.

**
Squaring the circle**

The second unique property of the division
into twelve is that it somehow "squares the circle". At least, it bridges
the gap between straight and circular, radius and circumference. As anyone
who studied trigonometry knows, the sinus of 30° is Ŋ. This means that,
alone among the angles into which a circle can be divided, the angle of 30°
combines a rational division of the circle (into 12, or of the quartercircle
into 3) with a rational division of the radius (into 2). This is a truly
unique intrinsic property of the division into 12.

**
Effortlessly dividing the circle**

A third special property of the division in
12 is that it is the most natural division of the circle, i.e. the one which
does not require any other data (c.q. geometrical instruments and
magnitudes) to get constructed except those already used in the construction
of the circle itself, viz. compas width equal to the radius. If one
constructs a samesize circle with any point of the first circle's perimeter
as the centre, one obtains a perimeter passing through the first circle's
centre and intersecting its perimeter twice at 60° of the new circle's
centre. Next, these two intersection points become the centres of new
samesize circles, and so on. The result is a set of six samesize circles
symmetrically distributed around the original circle, with 13 intersection
points: one in the original centre, six on the original perimeter at 60°
intervals, and six outside the circle. The straight lines connecting the
latter six with the original centre intersect the original perimeter exactly
halfway in the said 60° intervals. This way, the circle is neatly divided in
12 x 30°.

Moreover, this entirely natural construction
reveals a specific structure: the division in alternating "positive" and
"negative" signs, being the intersection points on c.q. outside the original
perimeter. The twelve intersection points can also be connected to form the
pattern known as Sri Cakra or Magen David, i.e. a straightstanding triangle
intertwined with an inverted triangle.