(Upon request, I republish an old article.)
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
twelve-star 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 in their turn, 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 Zodiacal constellations’ names 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 star-gazing 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
equally, 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, 3² + 4² = 5², and also figures in the next Pythagorean threesome: 5² +
12² = 13². In Fibonacci's series, the
12th number happens to be 12², 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 equaling 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. Well-known is the division of
27 or 28 moon-stations 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 quarter-periods
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 fundamental
worldly relations, starting with oppositions.
In a rational world model (e.g. the four/five elements), if we have an
element meaning "big" or "cold", then we must have one
which means "small" c.q. "hot", just as in real-life
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 of classes, 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 irrational 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 vis-à-vis 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 these: 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 1/2. 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 quarter-circle
into 3, the dynamic number befitting the orbit of motion) with a rational
division of the radius (into 2, the static number befitting the radius as the
“skeleton” of the circle). 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 same-size 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 same-size circles, and so
on. The result is a set of six same-size
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 in India since millennia as the Sri Cakra (circle of the
Goddess of Plenty) or in Jewish circles since a few centuries as the Magen
David (David’s shield), i.e. a straight-standing triangle intertwined with
an inverted triangle.
One of the side-effects, as it were, of
this natural construction of the division into 12 is that it somehow achieves
three trisections. Though some
hare-brained “trisectors” refuse to accept it, the ancient Greek dream of
dividing a given angle into three equal angles with ruler and compass has been
proven to be an impossibility. In the
present case, we have therefore not tried to divide the square angle (which
wasn’t given in the first place) into three equal angles of 30°, nor that of 180° into 3 x 60°, nor even that of 360° into 3 x 120°. Yet, it so happens that we have
ended up with the angles of 120°, 60° and 30° explicitated in our construction.
What was otherwise impossible, here it’s been done.
An open-ended appeal
We welcome feedback, as we may have
overlooked something. Possibly more
special geometrical facts can be mustered to show that the division into 12
(which a temporary coincidence may be credited with suggesting, viz. through
the moon's cycle as a fraction of the solar year) has a more profound, more
stable and more universal mathematical basis.
But the three properties outlined here are, in our opinion, decisive.
Copyright:
author, 1989.
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