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HORUS VOL I. Issue 1
The Pentagram of Venus
by William J. Douglas
Since Venus is in inner planet, it passes between the Sun and the Earth each time it
overtakes the latter. An inferior conjunction occurs when Venus is on a straight line
between the Sun and the Earth. And the time between two successive inferior conjunctions
is 584 days (to the nearest integer), which is called the synodic period. Using a 365-day
year (as did the Mayans and Egyptians), five circlings of Venus is very nearly equal to
eight Earth years, since 5 x 584 = 8 x 365 = 2920 days. Venus is not visible at the time
conjunctions occur, but her first appearance as a Morning Star (heliacal rising) occurs within a few
days after inferior conjunction, indicating completion of a synodic period.
The eight-year period defined by five synodic revolutions of Venus is the basis for
the eight-year (and later, four-year) festivals of Mexico, Greece, and Egypt. The eight-year
period is divided into two four-year periods by the superior conjunctions, that is, when
Venus passes behind the Sun.
Using the integer values given above, each inferior conjunction would occur 144 degrees
from the previous one, the fifth one coinciding with the starting point. Starting with the
first conjunction just within Sagittarius, the following ones occur in Leo, Pisces, Libra,
Taurus, and then back to the starting point in Sagittarius. These points, when joined by
straight lines in sequence, form a pentagram, as shown by the solid lines in Figure 1. When
today's actual sidereal values we used (583.92 and 365.256) successive conjunctions are
separated by 144.48 degrees, the fifth being 2.4 degrees from the starting point, as shown
by the dotted lines in Figure 1: each successive period of eight years forms another pentagram
rotated 2.4 degrees. The pentagram was the secret symbol of the Pythagoreans, and provides a basis for linking
Egyptian, Greek, and Babylonian mythology and astronomy.
Conjunctions of other planets also define geometric figures, such as the Trigon of Great
Conjunctions or Jupiter and Saturn (Figure 2).
[*!* Image: Figure 1 - Pentagramma Veneris]
[*!* Image: Figure 2 - The Trigon of Great Conjunctions]
In round numbers, the conjunctions occur every 20 years, a Trigon (triangle) is formed in
60 years, and it takes 2400 years for a complete rotation. The Trigon will be considered in a future issue.
Even long before the Greeks, observation of the planets was primarily concerned with the
stationary points where their retrograde loops reversed, and with their heliacal risings and
settings. According to an ancient observational rule, the latter stood in constant intervals
of time relative to the station points. Thus they considered the planets as moving from
one point to another, rather than as one continuous movement as we do today. For Venus,
the pentagram summarizes her first appearance as a Morning Star around the zodiac.
Five Belongs To Venus
For the Pythagoreans, numbers were points having position, and were represented with dots.
The numbers one, two, three, and four comprised the Tetraktys, which in turn relate to the
geometric concepts of a point, a straight line, a triangle, and a solid. The Tetraktys, upon
which the Pythagorean number theory was based, could be impressed in wet clay using the tips
of one, two, three or four fingers. And the imprint of the fingers and the thumb spread in a
circle, when joined in the same fashion as for the five heliacal risings of Venus, form an
asymmetrical pentagram, which resembles the secret recognition symbol used by the Pythagoreans.
[*!* Image: The Tetraktys]
The Pentagram was given the name "Health" by Pythagoras, each point being one of the five
Greek letters involved. Tracing the earliest records of the pentagram to ancient Mesopotamia,
H. E. Stapleton suggests that "Health" implied bodily existence in peaceful relationship with
the unseen controller of human fife, and that the pentagram may have been originally regarded
as the hand of some Deity. Looking for parallels in Mesoamerica we find that an open hand
(Mayan hieroglyph T713a) with prefix T24 and suffix T181 (Figure 3) occurs throughout the
pages of the Dresden Codex as a verb which relates the object god to Venus, who is the actor
or initiator of action. T713a itself depicts a hand (in human terms) indicating offering or
giving, in contrast with the hand of a deity a suggested by Stapleton.
[*!* Image: Figure 3 - The Venus Verb Glyph]
Five was also associated with Venus in the sphere of twelve pentagons" -the dodecahedron,
the fifth of the regular solids (Figure 4). Each face also represents a pentagram, which can
be inscribed by drawing fines between alternating vertices. But the dodecahedron is the symbol
for Aphrodite Urania (Ourania), the Starry Mantle, the Framework of the Whole, Parmenides'
Daemon Lady who steers the Heavens. And the Starry Mantle is Ishtar/Innin, who is Venus.*
[*!* Image: Figure 4 - The Dodecahedron, and a Pentagon with an inscribed Pentagram]
These heliacal risings of the Morning Star, this point of light, shifting position by itself,
causing the pentagram to flash forth in the order of time, was the physical image-one that had been known and
worshipped for ages.
The Eight Year Festivals
The Venus Pentagram was first discovered in modern times by M. Knapp, who supposed that
the Sothic period pertained to Venus, rather than to Sirius. Taking as his model the famous
Keplerian time diagram of the successive conjunctions of Jupiter and Saturn, he tried the
same with Venus-and got a pentagram, but his discovery was ignored by the historians of
Astronomy. It was rediscovered in 1963 by 0. Hinze and G. de Santillana when the latter was
investigating "the insistent recurrence of the eight-year cycles in so many inscriptions and
seal cylinders of civilizations East and West". This cycle was the eight solar years of the
five synodic revolutions of Venus, which is confirmed by the eight-pointed star on countless
Mesopotamian seal cylinders, borderstones, etc., which every time means Ishtar/Venus.
In his comparison of the Ptolemaic and Copernican systems, Wm. Stahlman mentions Ptolemy's
use of the fact "that Venus makes five complete cycles of anomaly in about eight Egyptian
years." He notes that although the eight year period for Venus is well attested from many
sources in antiquity, including usage in a completely different context in Babylonian astronomy,
little is known about its origin.
The eight-year period is also the basis of the oktaeteris, which was introduced into
Greece by Eudoxus. It was a lunar-solar calendar since there are 99 lunar synodic months
of 29.53 days in eight years (in round figures), the oktaeteris was used extensively until
it was superseded by the 19 year Metonic cycle.
Looking for parallels in Mesoamerica, we find that one pass through the Venus table of
the Mayan Dresden Codex covers a period of eight solar years or five synodic revolutions
of Venus. The table is repeated thirteen times to fit the 260-day augural calendar to the
synodic revolutions of Venus. We find in the Mayan codices Glyph T510b, which is known as
the Star Glyph, and is used in the Dresden Codex with the prefix "Great", designating Venus
as the "Great-Star". T510b itself is not infrequently taken to mean Venus wherever it appears (Figure 5).
[*!* Image: Figure 5 - The Great Star Glyph]
But the eight-pointed star was not used in Egypt for Venus. De Santillana informs us that
the Egyptians, laying the accent on the five, have been even more 'Venus minded'; the five
revolutions are hers, after all, she draws the pentagram". They used a five pointed star
'[*!* Image: 5-pointed star],' as a determinative in words for time measurements and the
names of constellations and stars. (A determinative is a sign used to differentiate between
words with the same spelling, which happens frequently when the only alphabetic signs are
consonants.) Especially interesting is the Egyptian [*!* Image: heiroglyph Spdt] their
rendition of Sothis. (Spdt is the transliteration of Egyptian consonants into the English alphabet.)
The Golden Section
Each leg of the pentagram is divided by the intersecting leg according to the Golden Section;
the ratio of the larger segment to the smaller is equal to the ratio of the whole to
the larger. This proportion, also known as 0 (the Greek letter phi), was considered by
Plato in his Timaeus to be the key to the physics of the cosmos. The value of 0
is an irrational number; it cannot be expressed as the ratio of two integers. But an
increasingly accurate approximation to it can be obtained by the additive series:
1, 2, 3, 5, 8, 13, 21, 34, ....
in which each term is the sum of the preceding two. Dividing any number in the series
by the preceding one gives an approximate value for 0. Accuracy improves as the series is
extended (Figure 6), converging onto the correct value of 1.6180. . . This series is known
as the Fibonacci series. Leonardo Fibonacci learned of it during his travels in Egypt, and
introduced it to Europe. It was known and used in Old Kingdom times, as indicated by the
frequent use of 0 in the proportions and measurements of Egyptian architecture, such as the
Temple of Luxor, and the Great Pyramid of Cheops.
The above series is of interest here, since during the eight years required for five
synodic revolutions, Venus makes thirteen revolutions around the Sun-and the numbers 5, 8,
and 13 are successive terms in the series. The Venus/Earth orbital ratio of 13/8 = 1.625,
which is fairly close to today's actual value of 1.6255. The Babylonian Ninsianna tablets
imply that Venus made 313 revolutions around the Sun for 193 by the Earth. The resulting
orbital ratio is 313/193 1.62176, which falls halfway between today's value and 0.
The full significance of the Pentagram and the additive series based on the Tetraktys
can best be demonstrated by the famous Pythagorean Theorem, which states that the diagonal
of a right triangle is equal to the square root of the sum of the squares of the two sides.
This was known to be true in antiquity for selected numbers, such as 3, 4, and 5, but a
general solution awaited the Pythagoreans, who showed that in many cages exact solutions
for the hypotenuse could be expressed as the ratio of two integers. But they proved that
there were exceptions, such as the diagonal of a square, which they called irrationals. (Note
that the term "irrational" originated in this context.) Having discovered this flaw in numbers
through a geometric problem, they developed geometric algebra, which eliminated the difficulty
with irrationals, thus laying the foundation for Greek mathematics.
Rooted in the Tetraktys, the series leads to the pentagram, a geometric construction of
the Golden Section, and a rational approximation to it. Could any symbol be more appropriate
than the pentagram for those who discovered the irrational and how to cope with it?
The series applies to Venus due to the phenomenon of orbital resonance. The asteroid
Toro is in 5:8 resonance with the Earth's orbit. One asteroid is in 3:1, and three are in
2:1 resonance with Jupiter. Saturn is in 2:5 near resonance with Jupiter, producing the
rotating triangle shown in Figure 2. Other resonances have probably existed in the past.
Resonances frequently involve the above series, since small integers are favored.
Ancient Astronomical Data
Figure 6 shows a comparison of the orbital ratio of Venus and Earth with successive
approximations to 0. The Fibonacci series is shown across the top. Horizontal dashed
lines between each pair of the series are placed vertically according to the decimal
value of their ratio. Note how successive approximations swing above and below 0,
but closer each time.
[*!* Image: Figure 6 - Decimal Values of Approximation to 0.
The dashed lines show the decimal values (vertical scale) derived from their number
pairs indicated by the ends (horizontal scale). The first entry, 8/5 = 1.6, gives 0 accurate
to one decimal place. Note that successive approximation swing above and below 0, converging
onto it. LABELS: FIBONACCI SERIES. Current (1.62552) Dresden Codex (1.6250).
Artificial Insertion (1.622222) Ninsianna Tablets (1.62176).]
Along the right side are listed the decimal values of the Venus/Earth orbital ratio
from various sources. The top pair of values corresponds to the fixed and rotating
pentagrams of Figure 1. The fixed pentagram relates to Mayan astronomical data, since
the Venus Tables in the Dresden Codex give 584 days for Venus' synodic period, and their
calendar system used a 365-day year.
Halfway between the top pair and the actual value of 0 appears the value for the
Ninsianna tablets mentioned above. With a minor adjustment, the 313/193 ratio simplifies
to 73/45, which are members of another additive series starting with I and 5:
1, 5, 6, 11, 17, 28, 45, 73. . . .
This case corresponds to 28 synods in 45 Earth years with 73 revolutions of Venus,
successive conjunctions being separated by 141.43 degrees. In this case the "pentagram"
rotates 12.86 degrees every fifth conjunction, returning to the starting point at the
28th synod, providing a fixed reference framework for the Ninsianna data like the Dresden
Codex pentagram does for modern data (Figure 7). The 28 points invite comparison with the
lunar zodiac and the lunar sidereal period of 27.32 days. Since the moon was observed
relative to the stellar background in those times, this suggests that they may have linked the
Venusian and Lunar calendars. And granting that the 56 (2 x 28) Aubrey holes at the Stonehenge
represented the zodiac, we can envision a stone every 22nd hole to mark the conjunctions of Venus.
[*!* Image: Figure 7 - The Artificial Insertion]
The Venus/Earth orbital ratio of 73/45 1.622222... Since 365/225 gives exactly the
same value, we use the synodic equation to compute the corresponding synodic period.
1/225 - 1/365 = 1/586.6
which is 586.6 days, whereas observational data from Section I of the Ninsianna Tablets
gives 586.64 days, and the Artificial insertion gives 587 days. So far, this is not remarkable,
since Rose and Vaughan [KRONOS V:4 1980] have shown that the Artificial Insertion could have
been derived from a version of Section 1; a close agreement is expected. What is peculiar,
though, is the coincidence that when the nearest integer values for the orbital periods of
Venus and Earth today are used in the synodic equation, we get the same value for the
synodic period as in the Ninsianna tablets.
However, it is tempting to speculate whether the Babylonians understood the synodic
equation, and were observing orbital periods, rather than the synodic period, which varies
considerably, even today. This possibility is suggested by later Babylonian mathematics,
which used tables of reciprocals, and was concerned with various types of means, including
the harmonic mean, which bears a striking resemblance to the synodic equation.
In considering the above, the reader should keep in mind that using the nearest integer
number of days does not necessarily indicate sloppiness, since observational data is based on a day count.
The sequence of 13, 21, 34 from the original series is of interest, even though there
is no supporting data for such a sequence. It would correspond to 13 synods in 21 Earth
years with 34 revolutions of Venus. The orbital ratio is 34/21 = 1.6190, which is very
close to 0. It forms a figure with 13 points (Figure 8) separated by 360/13 = 27.69 degrees.
The fact that 13 and 28 apply to Figures 7 and 8 with the role of separation and number of
points interchanged emphasizes the interdependence of cycles of 13 and 28 in calendar systems which include lunar cycles.
If we divided the circle into 364 days rather than 360 degrees, the interrelationship
between 13 and 28 would apply exactly. There is provision for a 364-day computing year in
the Mayan Dresden Codex, which is especially concerned with Venus and the Moon, and Alban
Wall has shown evidence that the Stonehenge was used as a Lunar/Solar calendar with a 364
day counter. Considering that the Chinese once divided the circle into 365 1/4 equal parts,
and that Vedic astronomy in India measured time rather than angles, the 364 part division
of the circle does not seem out of order.
Is there significance in aft this for our own times? On the one hand, the cross-cultural
quantitative evidence suggests great antiquity for knowledge of the value of 0, the Golden
Section, and the related Fibonacci series. Further, these seem to have been incorporated in
both astral-religious symbolism and the calendric systems associated with Venus observations.
These relationships form part of a comparative quantitative platform for astronomical research.
Their inclusion in prominent mythico-religious art aids objective study in this field as well.
Most of all, perhaps, it calls attention to the sophistication exercised by ancient
civilizations in their respective systems of astronomy and mensuration. These practices
were simultaneously motivated by the belief that the visible astronomical bodies, their
positional and temporal cycles, and the seeming verities in numbers and geometry were
expressions of the divine. We better appreciate, therefore, the care with which the
specialists of antiquity tracked the celestial paths of their gods, studied the relationships
between their measured cycles, and sought to integrate these with their achievement in quantification.
We are also the more nonplused by the strange discrepancies between ancient observations
and those expected from the present order of nature. Unless we negate the evidence for the
inferred precision of their methods and records, the case for a nonuniformitarian interpretation
of natural history takes on a renewed credibility.
[*!* Image: Figure 8 - The Golden Section]
*These names for the Queen of Heaven we set forth by Giorgio de Santillana ("Prologue to
Parmenides", Reflections on Men and Ideas, 1968, pp. 86, 108-116) where he shows that Five is Venus.