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Updated -- 9 November 2006

Of all the single digit numbers, nine (9) may be the most profound.  Composed of three trinities (3 times 3 equals 9), nine represents the principles of the sacred Triad taken to their utmost expression.  The Chaldeans believed 9 to be sacred, and kept it apart in their numerology from the other numbers. Nine has been and in some cases still is considered thrice sacred and represents perfection, balance, order -- in effect, the supreme superlative.

More information on this topic can be found at the Halexandria Forums.

(9/22/9) A late addition to the marvelous world of Nines is a three-dimensional magic... double tetrahedron (or star tetrahedron). See, for example, the forum thread at 09/09/09.

In Numerology, the positive characteristics of nine (9) are selflessness, fulfillment, completion, universality, universal understanding, interrelatedness, compassion, idealism tolerance, forgiveness, generosity, benevolence, humanitarianism, emotionalism, and justice.  Nine is also associated with accomplished artists and thinkers who are inspired by universal truths.  Simultaneously, 9 can represent negative characteristics, from selfishness to extravagance to vulgarity -- essentially the opposites of the positive characteristics.

In the base 10 system, where all numbers are represented by ten distinct forms (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), nine is the final number.  As such, it becomes a limit, a bound, or the ultimate attainment.  The Greeks called “nine” the horizon, where the Ennead, or the nothing/void lay beyond.  Expressions such as “a cat has nine lives”, a “cat-o’-nine-tails”, “the whole nine yards”, “cloud nine”, “dressed to the nines”,  “a stitch in time saves nine”, and “possession is nine point (or nine-tenths) of the law” are all variations of the concept of the ninth level being the nth degree, the highest level, or the maximum possibility.

Ancient and modern traditions are replete with ninefold symbolism.  The Norse God Odin, ruler of the 9 Norse worlds, hung 9 days on the world axis or Yggdrasil tree to win the secrets of wisdom for mankind.  In Scandinavia, 9 day fertility feasts were held every 9 years.  There were 9 Norse giantesses, who strode 9 paces at a time and lived at the edge of the sea and land.  The city of Troy in Homer’s Iliad and Oddessey was besieged for 9 years, while Odysseus wandered for 9 more years in trying to return home.  The Greek goddess, Demeter, was depicted with 9 ears of wheat and searched 9 days for her daughter Persephone.  The birth of Apollo and Artemis by Leto took 9 days and nights (Artemis becoming the midwife in the process and later choosing two 9-year old girls as her companions).  The Greeks also honored 9 muses, while the Egyptians honored a company of 9 “gods” or neteru.  Egyptian pharaohs, meanwhile, were often symbolized by 9 bows.  Celtic traditions talk of 9 Celtic maidens and 9 virgins attending Bridget, while the sacred Beltane fire rites were attended by a cycle of 9 groups of 9 men.  Aztec, Mayan, and Native American myths describe 9 cosmic levels (four above, earth, and four below). As the most auspicious number of celestial power in ancient Chinese, 9 became the rule in 9 great social laws, 9 classes of officials, 9 sacred rites, and 9-story pagodas.  The festival of the “double yang” was held on the 9th hour of the 9th day of the 9th month. In Christian symbolism, there are 9 orders of angelic choirs in 9 circles of heaven and 9 orders of devils within 9 rings of hell -- possibly accounting for the fact that it took 9 days for Lucifer and his angels to fall from heaven.  And speaking of fallen angels and/or hell, there are 9 justices of the United States Supreme Court!

There were 9 Gods of the Sabines (an ancient tribe of Italy): Aeneas, Bacchus, Esculapius, Fides, Fortuna, Hercules, Romulus, Santa, and Vesta.  Medieval theology listed 9 orders of Angels, i.e., Seraphim, Cherubim, Thrones, Dominions, Virtues, Powers, Principalities, Archangels, and Angels; 9 Stones: Sapphire, Emerald, Carbuncle, Beryl, Onyx, Chrysolite, Jasper, Topaz, and Sardis; and 9 Beatitudes.  In heraldry, there are 9 accepted places on the shield that signify the heralidic arms.  There are even 9 Worthies, famous individuals comparable to the Seven Wonders of the Ancient World:  Alexander (the Great), Hector, Julius Caesar, Joshua, David, Judas Maccabaeus, King Arthur, Charlemagne, and Godfrey of Bouillon.  [See Clovis I to Godfroi and Crusades and Secret Societies.]  There are even 9 magnitudes of the Richter Earthquake Scale (the latter theoretically possible but has never occured during human history -- it would amount to probably four times the intensity of the 1964 Alaskan earthquake).

According to one source, <http://www.esotericarchives.com/agrippa/agripp2b.htm> Heinrich Cornelius Agrippa (1486-1535) has described the Number 9 by noting that it is dedicated to the Muses.  Considering the nine movable spheres (the planets of antiquity), and the nine Muses -- Calliope, Urania, Polymnia, Terpsichore, Clio, Melpomene, Erato, Euterpe, Thalia -- Agrippa considered which nine Muses were appropriated to the nine Spheres.  He did this by noting that the first resembles the supreme Sphere (the Primum mobile), and descending in order to the Sphere of the Moon, he determined that Calliope is appropriated to the Primum mobile; Urania to the Starry Heaven, Polymnia to Saturn, Terpsichore, to Jupiter, Cleo to Mars, Melpomene to the Sun, Erato to Venus, Euterpe to Mercury, and Thalia to the Moon.  [Thalia is the Muse of Comedy, my personal favorite.]

Beethoven wrote 9 symphonies, after which he died.  To this day, a superstition among many musical composers forbids the numbering of a symphony past the number 9.  Mahler wrote more symphonies, but never named any one of them, number 9.  Equally superstitious, Baseball has 9 innings and 9 players (often playing at 9-figure salaries -- if we include the decimals), and figures the “bottom of the 9th” to be the last chance to win.

On a higher level, Abraham was 99 when the Lord spoke to him, Islam acknowledges the 99 Beautiful Names of God, and “amen” (from the Hebrew “so be it”) transforms in the Greek alphabet into the number 99.  And if you want to get thoroughly mathematical...

The 3 by 3 Magic Square is revered in the cultures of Islam, Jains              4    9    2

of India, Tibetan Buddhism, Celts, African, Shamanic, and Jewish               3    5    7

mysticism.  In this arrangement, the three columns, three rows and               8    1    6

two diagonals always add up to fifteen.  In Feng Shui the numbers

within each cell of the magic square have specific significance for working with the earth’s subtle creative energies for the good of society and the environment.           

[NOTE:  In the below mathematics, there are numerous, relevant conclusions, which do not require a facility in mathematics to understand.  These important inferences are marked with a v to allow the non-mathematician to quickly isolate them.]  

Degrees in a Circle  

There are 360 “degrees” in a circle.  Which is a puzzlement.  Like why 360?  Keep in mind that the “degree” is nothing more than a measurement of arc which is determined by the first statement.  More precisely, a degree can be thought of as being defined to be an arc whose length is 1/360th of a circle.  Clearly, the circle could just have easily been divided into 100 units or 1000 units.  So why, you might ask, are there 360 degrees?

Obviously 360 degrees approximates the number of days in an Earth year, 365.24.  There is, of course, the minor inconvenience of there being 5.24 extra days for which there are no corresponding degrees.  This accounts for the need of leap years every four years -- except for century marks which are not divisible by 400 (i.e. 2000 was a leap year, 1900 was not). There is also evidence to support the contention that prior to 1500 B.C.E., the number of days in the year was very nearly 360.  (During the Ages in Chaos, it changed!)

Brian Stokes of New Zealand has noted that "the Babylonian counting system was based on the number 60, which was almost certainly chosen because it was the lowest number to have 2, 3, 4, 5 and 6 as factors (you need to get to 420 to get 7, and 60 also has as factors 10, 12, 15, 20 and 30)." He has also noted that "the relationship with days in the year was much more clearly stated by the Maya who had a five-day festival to make up the days to 365. In fact their whole counting system was based on 20, which should have gone from 20 to 20x20, 20x20x20, etc. but actually went from 20, to 20x18 (= 360) and then powers of 20 again – 360x20, 360x20x20, and so on.

Be this as it may, a more interesting possibility is that 360 can be equally divided (with the quotient being a whole integer) by 10 of the first 12 numbers (whereas 100, for example, can only be equally divided by 5 of the first 12 numbers).  To illustrate this, consider the following table:

Number 100 / Number 360 / Number
1 100 360
2 50 180
3 33.33333... 120
4 25 90
5 20 72
6 16.66666... 60
7 14.28571... 51.42857 `42857...
8 12.5 45
9 11.11111... 40
10 10 36
11 9.09090... 32.72727272727...
12 8.33333... 30

This may not appear to be of overriding importance, but a society -- ancient and otherwise -- which does not have ready access to hand-held computers might have found this to be of singular importance and ease.  There is also a curious connection to the number 9.

There is a proliferation of 9’s which keep cropping up in the 360 degree circle -- as well as 3’s and 6’s, the natural components of 9.  This is particularly true when we apply the tool of reduction from Numerology.  For example, 666 can be reduced to 6+6+6=18, and 1+8=9.  Any number can be similarly reduced (although, repeated numbers such as 22 or 55 are often referred to as “Master Numbers” and are not always automatically reduced by numerologists).  The numbers in the table above, for example, reduce in the manner of: 360=9; 180=9; 120=3; 90=9; 72=9; 60=6; 45=9; 40=4; 36=9, and 30=3.  Notice that the only number not 3, 6, or 9 is 4 (but was derived from dividing 360 by 90)!

But notice something even more curious.  Both quotients of 7 and 11 yield a series of repeating numbers which numerologically reduce to 9, i.e. 1+4+2+8+5+7=27; and 2+7=9.  At the same time, the missing numbers in the 7-sequence are 3, 6, and 9!

The singular importance of the divisors 7 and 11 has been pointed out by Michael Glickman [1]. In addition, Michael Schneider [2] has pointed out that any digit divided by 9 yields a repeating sequence of only one number:  For example:

Number Divided by 7 Divided by 9
1 0.142857 142857... 0.11111111111...
2 0.2857 142857 14... 0.22222222222...
3 0.42857 142857 1... 0.33333333333...
4 0.57 142857 1428... 0.44444444444...
5 0.7 142857 14285... 0.55555555555...
6 0.857 142857 142... 0.66666666666...
7 1.0000000000000... 0.77777777777...
8 1.142857 142857... 0.88888888888...
9 1.2857 142857 14... 0.99999999999... *

[* One reader, who shall remain nameless, has pointed out that 9 divided by 9 equals 1 -- and not 0.999999999...! However the Wonders of Math might suggest otherwise.]

An astounding result is that any number other than a multiple of 7 will result in the same 142857 sequence as a quotient when being divided by 7!  This explains why 7 was considered sacred by the ancients -- along with the number 12; 7 being the terrestrial holy number and 12 the heavenly holy number.  It also explains the proliferation of 7 and 12 in our calendars, ancient traditions and everyday usage -- from 7 Days of the Week and 12 months of the year, to 7 or 12 chakras, 7 colors of the rainbow and 12 astrological signs, to 7 dwarfs and 12 disciples.  The list goes on endlessly, including aspects of the Kali Yuga, and connections to Tarot (3 and/or 7), Numerology (9), and Astrology (12).           

v On the other hand, 9 seems to be particularly associated with the process of human birth.  In addition to the 9 months of gestation, and 9 openings of the body (all of which operate in some fashion in connection with conception and gestation), there is also the fact that the tail of the sperm is made of 9 twisted threads, which after uniting with the egg, forms a centriole, which is a circle of 9 parallel tubes.           

The 9 connection becomes even stranger when we extend the first table of numbers in order to consider the following quotients:  

                        360 divided by:

                                    13        =          27.692307692307692307692307692307...

                                    14        =          25.714285714285714285714285714285...

                                    15        =          24

                                    16        =          22.5

                                    17        =          21.1764705882352941176470588235294...

                                    18                 20

                                    19        =          18.94736842105263157894736842105263157...

                                    20        =          18

                                    21        =          17.142857142857142857142857142857...

                                    22        =          16.3636363636363636363636363636...

                                    23        =          15.65217391304347826086956521739130434...

                                    24        =          15

                                    25        =          14.4

                                    26        =          13.84615384615384615384615384615...

                                    27        =          13.33333333333333333333333333333...

                                    28        =          12.85714285714285714285714285714...

                                    29        =          12.41379310344827586206896551724137931...

                                    30        =          12

                                    31        =          11.61290322580645161290322580645161290...

                                    32        =          11.25

                                    33        =          10.90909090909090909090909...

                                    34        =          10.58823529411764705882352941176470...

                                    35        =          10.285714285714285714285714285714...

                                    36        =          10  

From this table and some mental gymnastics, we can derive some very interesting rules.  Clearly, any whole number divisible by two will always yield a quotient which is either an integer or a half-integer.  The same can be said of four and eight (which are, of course, multiples of 2), except that the half-integer may instead be a quarter or eighth of an integer.  Numbers divisible by five have the same characteristic as those divided by two -- due to the fact that the base 10 system is 5 times 2.  Numbers divisible by three or six will always yield an integer or two forms of an infinite sequence: .333333...or .666666...

Finally, 9 will yield an integer, OR one of the infinite sequences of a single repeating digit.  It is only numbers divisible by seven or higher prime numbers which turn out to be really interesting. [A prime number is any number which is not equally divisible by any number other than itself -- or to be a bit more mathematically precise, a prime number is "a number with exactly two factors" (namely 1 and the number itself). Thus the first prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37...; i.e. 1 is not a prime. There is, incidentally, only one possible set of primes multiplying to any number (for example, 2x2x3x5 = 60. This would not be true if 1 was a prime, in that multiplying by 1 leaves the number unchanged.

[There is also the decidedly non-mathematical suggestion that we are all ONE, or as the Zen Master requested of the hamburger vendor, "Make me one with everything." But that's another and far more elaborate story. See, for example, Lothar Schafer's essay, "Quantum Reality and the Consciousness of the Universe" [Zygon, Journal of Religion and Science, Volume 41, Number 3, September 2006], where he (and others) argue that the nonlocality of nature implies a universe of "undivided wholeness."]

In reviewing the above tables, we note an interesting phenomena:  The quotient is either a whole number, a finite fraction, or a number which generates a repeating Infinite Series.  The thing is that for all quotients which are a finite fraction (for example, 360 divided by 16, 25, or 32), the fraction reduces to 9!  I.e. 360/32 = 11.25; where 1+1+2+5=9.  Meanwhile, all of the infinite series are a repeating sequence of numbers (underlined in the above table), where the repeating sequences reduce to 9!  The first obvious example is 360 (or any other number -- but not a multiple of 7), when divided by 7, 14, 21, 28, 35 will yield the same 7-sequence of numbers, 142857, which is reduced by adding 1+4+2+8+5+7 to obtain 27 (and 2+7= 9). [Note that there are no 3s, 6s, or 9s in the sequence.]

360 divided by 49 is something altogether different.  This might not be too surprising, inasmuch as 49 is 7x7 -- the second 7 adding an additional complexity.  In any case, 360/49 equals 7.3469387755102040816326530612244897959183673469...  Now... guess what the total of the 41 digits reduces to.  Sorry.  The total is 187, which reduces to 9+7 or 7.  But the 9 does make an appearance.    We obtain even wilder numbers if we divide 360 by 343 (7x7x7) or higher powers of 7.  The number seven adds a complexity which is essentially multiplied to the nth (9th) degree!

Meanwhile the quotients of 13, 26, 39... always yield one of two 13-sequences, 769230 (which reduces to 7+6+9+2+3+0=27/9) and 153846 (which also reduces to 9)!  Note also the first half of both the 7- and 13-sequences, when added to the corresponding number of the second half, equals 9.  (In other words, 769230 becomes 7+2, 6+3, 9+0.)

The quotients of 11, 22, 33, 44... always yield a two-digit number (e.g. 27, 63, 90, 81...) which all reduce to 9 [which, incidentally also works whatever the number being divided].  The remaining quotients have longer repeating sequences, but incredibly the sequences still reduce to 9!  For example, the sums of the underlying sequences for 17 and 34 yield 72/9; for 19 and 38, 81/9; 23 and 46 yield 99/9; 29 and 58 yield 126/9, and 31 and 62 give 54/9.  Simultaneously, we note that 17 has a (17-1) 16 digit sequence, 19 an (19-1) 18 digit sequence, 23 a (23-1) 22 digit sequence, and 29 a (29-1) 28 digit sequence.  Plus which, in all of the sequences listed above -- with the exception of 31 -- the first half of the sequences, when added to the last half of the sequences, yield 9999999...  It is probably relevant that 30 yields a 15 digit sequence, where two of the sequences raise the sequence total to 30 -- and continue the previous sequential sizes in some not totally clear fashion.

There is another curious factor involving the infinite sequences.  This is the fact that if you subtract the decimal equivalent of the sequence from 1.00, you obtain the same sequence again, but in a different order.  Again however, this does not work with 31. Brian Stokes has noted that 31 is the smallest prime number with an odd number of digits in the sequence -- which may account for... something.

One other possibility is to divide 360 by 77!  Recall that 7 and 11 were our first clues to the inherent intrigue of integer indexes -- so why not check it out!?  The answer turns out to be a short sequence of: 4.675324675324675324... which has the curious quality that the six repeating digits include 2,3,4,5,6, and 7; and the whole thing reduces to 27/9.

But consider this 7-11 sequence for a moment and multiply all of the numbers together.  (You can also include 1 as a multiplier and arrive at the same result, but you now have the nicety of having the first 7 numbers multiplied together in the mathematical fashion known as Seven Factorial and written as 7!.)  The answer is 1x2x3x4x5x6x7 = 5040.  At the same time, 7x8x9x10 = 5040.

Besides reducing to 9, 5040 is a very interesting number.  So interesting in fact that Plato used the number as the chief symbol of an ideal city, modeled on the “patterns in the heavens”, which he describes in the Laws. But before we delve into strange world of 5040, we might take the next logical step and multiply 8x9x10x11.  The answer is 7920.  7920 is another intriguing number and, as it turns out, represents the diameter of the Earth in miles!  (The Earth is actually slightly out of round and has a diameter at the equator of 7927 miles and at the poles of 7900 miles.)

It has been pointed out in Harmony of the Spheres that the mile, like the degree, is an arbitrary unit of measurement.  You might think that there are 5,280 feet in a mile (or 8 furlongs), but then this just defines the mile on the basis of another arbitrary unit of measurement.  Ultimately, we may find it far more enlightening to turn the point around and state that the mile is defined on the basis of the Earth’s diameter, and thus the diameter of the Earth is 7,920 of these “mile” units of length measurement.  With the mile thus defined, we can note the curious fact of the Earth’s diameter being equal to 8x9x10x11 miles as fairly ho-hum.

Again, Brian Stokes -- obviously a trouble maker -- has suggested that the mile is not totally arbitrary, noting that "the foot is exactly 5/7 of the Biblical cubit and there are ancient measurements in Jerusalem which equate precisely to the mile." However, the converse argument is that the cubit may have been derived from the mile, the latter which was derived by their being 8x9x10x11 miles in the Earth's mean diameter.

Somewhat more dramatically, Brian has noted that, "The more recent metric system is actually based on the polar circumference (40 million meters, actually 40,008,600 -- or an error of one in 5000). If we take a circumference of one rod, pole or perch -- 5.5 yards -- the diameter is 63 inches. Multiply this by 25 million and you get 24,857.95 miles, against the actual polar circumference of 24,860.2, an error of one in 10,000 -- better than metric! Was this known or is it co-incidence? (There is a fascinating section on the English mile in "The Templars' Secret Island" by Erling Haagensen and Henry Lincoln [the latter of Holy Blood, Holy Grail fame] -- if you are interested in Sacred Geometry, this is the book for you." [And having a copy myself, I must agree with Brian. Note in particular Chapter Six!]

We might also note that 7920 is just 7200 plus 720.  Alternatively, 720 times 11 is 7200 + 720 = 7920.  720 is also 6! (i.e. 6x5x4x3x2x1).  Meanwhile 720 is exactly twice 360!  (And everything in sight reduces to 9!  Naturally.)  This leads us to consider the radius of the Earth, which is half the diameter of the Earth and equals 3960 miles.  Obviously, we can note that 3960 = 3600 + 360, or 360 times 11 (or 6!-2!).  But let’s return to our number of 5040, derived from either 7x8x9x10 or 1x2x3x4x5x6x7, or even 10! - 6!.  5,040 miles less the Earth’s radius (3,960 miles) equals 1,080 miles.  Guess what the radius of the Moon is!  If you guessed 1,080 miles, you get an “A+” for the day.

Some of these meandering thoughts are included in the below table, i.e.:  

1x2x3 6  
1x2x3x4 24 Hours in an Earth Day
1x2x3x4x5 120 (see Genesis 6:3)
3x120 or 720/2 360 Degrees in a Circle
1x2x3x4x5x6 720 Degrees in electron orbit+
360+720 (or 3x360) 1080 Radius of Moon (miles)
3x720 2160 Diameter of Moon (miles) *25,920 years/12
200x2160 432,000 Length of the Kali Yuga)
11x360 720+1080+2160 3960 Radius of Earth (miles)
v                     (radius/diameters of Earth and Moon have an exact 11/3 ratio)

v   (radius/diameters of Earth and Mercury as well as their orbits, have a 2.618/1 ratio, i.e. Earth’s radius = (1 + f) x Mercury’s radius)

v         1x2x3x4x5x6x7    5040 3,960 + 1080
v         7x8x9x10 5040 Earth radius plus Moon radius (miles)
8x9x10x11 7920 Diameter of Earth (miles)

 (11! / 7! = 7920, while 10! / 6! = 5040)

2160 + 7920 10,080 Earth plus Moon diameters (miles)
9x10x11x12 11,880 Earth radius + diameter (miles)

also (= 10,080 + 1,800 = 12! / 8!)

10x11x12x13 17,160  
17,160 - 11,880 5,280 Feet in a mile
11x12x13x14 24,024  
10x11x12x13x14 240,240 Approximate Earth-Moon distance
(which actually varies from 221,460 to 252,700 miles)
v         12!  479,001,600 Approximate Sun-Jupiter distance (459,800,000 to 506,800,000)

*The period of time (years) for a complete revolution of the precession of the Earth’s axis. + 720 is the number of degrees in an electron's "full circuit" i.e. for an electron, 360 is not a full circle, 720 is. Ref. http://www.metaparticles.com/page15a.htm.

Any product or three or more whole numbers in sequence will reduce to a 3, 6, or 9.  If one of the numbers in the sequence reduces to 9, the sequence will always reduce to 9.  Any product of six or more whole numbers in sequence will reduce to a 9 (inasmuch as in any sequence of six, there is always two numbers which reduce to 3, 6, or 9).

Note that the mile is already defined by the Earth’s dimension, and the fact that the Moon connects into the Earth in such a way as to fulfill the fondest desires of the numbers is just slightly beyond belief.  But then it becomes even more intriguing!

According to John Martineau [3] (using “John Michell’s famous construction”), we note that the relationship between the Earth and her Moon can be derived by forming a 3-4-5 right triangle (the shape of a right triangle where all three sides are the three lowest whole integers -- i.e. according to the Pythagorean Theorem, 32 + 42 = 52).  If we then place the Moon within a square whose side is three, and add two 3-4-5 right triangles on either side, we obtain a side of 11 (3+4+4). The Earth fits into a square with a side of 11 with over a 99.9% accuracy!  Another way of saying this is that the Moon’s radius is 360 times 3 and the Earth’s radius is 360 times 11.  Thus the ratio of their radii equals 11/3 exactly! In this way the Earth squares the circle of the Moon.

vvv The profound implication of all of this is that the Moon is mathematically sized to the Earth, the choice of 360 degrees in a circle is not arbitrary, and if we multiply the radius of the Earth (3,960 miles) by 60 (the result of multiplying the sides of the 3-4-5 triangle, and a nice complement to 360 degrees) we obtain 237,600 miles.  Guess how far the Moon is from the Earth!

Well, not exactly.  The lunar orbit is not perfectly circular such that the Moon varies in its distance from the Earth, ranging from 221,460 miles (perigee -- closest point of approach) to 252,700 miles (apogee).  The average is 237,080 miles.  This represents an error of approximately 0.218855%.  Inasmuch as the Moon is very, very slowly moving away from the Earth, it’s only a matter of time before its average distance is exactly 60 times the radius of the Earth (or 30 times the diameter).

v Finally, we might also add that the angular diameter of the Moon (which varies from 29’ 22” to 33’ 31”) is effectively the same as the Sun -- a fact which accounts for solar eclipses as viewed by Earth being so limited in their duration and location on the surface of the globe.

Furthermore.  If we draw a circle representing the Moon and place it tangent to a circle drawn to scale of the Earth, we find that a circle whose center is the center of the Earth’s circle and whose circumference passes through the center of the Moon’s circle will have a circumference approximately equal to four times the diameter of the Earth.  This requires that p be set equal to 22/7 (=3.1428571428... vice 3.14159265358979323846...).

[It might be noted that p is an irrational, Transcendental Number, defined as an infinite series, where one never reaches a repeating sequence such as encountered in dividing a number by 7.  Two other such irrational, transcendental numbers are f (also known as the Golden Mean and equal to 1.61803398875...) and e (the base of the natural logarithm and equal to 2.71828...; = 2+1/2!+1/3!+1/4!+...+1/n!).  All three of these numbers turn out to be phenomenally important in Sacred Geometry and should therefore be instantly memorized -- although perhaps not to more than 20 decimal places as in the case of p.]

v Oh by the way, the eight 135 degree corner angles of an octagon add to... 1,080, the radius of the Moon in miles!

v John Martineau [3], in his book entitled A Book of Coincidence (where the word “coincidence” means “co-in-siding” -- congruent, synchronistic, or simultaneous), shows that the bodies of the solar system and their orbits are related to each other more or less precisely by a series of basic geometrical figures.  For example, if we take an equilateral five-pointed star, sized such that the Earth’s orbit is drawn touching the points, then a circle drawn touching the intersections of the lines connecting the points of the star will, to scale, approximate Mercury’s orbit (to within 99% accuracy).  The same effect results when an equilateral 30-pointed star is used to relate Saturn’s orbit to that of Earth’s.

vv Martineau [3] goes on to show that all of the planets of our Solar System are related to all of the other planets by simple geometrical relationships.  What is absolutely astounding, however, is the fact that the 5- and 30- pointed stars used to connect Earth’s orbit with that of Mercury’s and Saturn’s, respectively, can also connect, in the same way, the relative physical sizes of the three planets! 

This incredible “coincidence” only occurs when Earth is compared to either Mercury or Saturn (although, obviously, a five pointed star inside the 30-pointed star would connect Mercury and Saturn in this dual manner).  If we note that Mercury and Saturn are the innermost and outermost of the 9 medieval planets (the Sun and Moon were included as planets in medieval times), one might begin to suspect that within this Solar System of 9 planets, Earth is special or unique!  This flies in the face of modern science and astronomy, but then again, most everything worth talking about flies in the face of modern science and astronomy!

Let us not fail to mention that the five-pointed star and its component parts are intimately related to f, the Golden Mean briefly mentioned above. Meanwhile, the 30-pointed star connecting Earth and Saturn is obviously related to the 360 degree circle, the 60 multiple of Earth radius and Moon distance, and is very nearly the orbital period of Saturn in Earth years (actually 29.457 (9!) Earth years).

v There are also thirty precise divisions in Stonehenge’s outer trilithon circle, and the physical sizes of the Moon, Earth, Mercury, Venus, Mars, Saturn and Jupiter can be determined by relative sizing of the Earth to one or two of Stonehenge’s circles (see A Book of Coincidence).  The fact the ancient builders (circa 2000 B.C.E.) apparently knew the physical sizes of the visible planets might be something to ponder!  (Another fly in the face of modern science, perhaps?)

With respect to the five-pointed star and internal regular pentagon, one might note that the internal angles of the pentagon is 108 degrees (3x36), and that 1,080 (3x360) has 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 50, 72, and many other numbers as factors (i.e. their quotients are integers).  And obviously 1080, 3960, 5040, 7920... all reduce to 9.  Slowly we are beginning to see why geometry means “earth measure”.

While we’re on the subject -- and since all subjects are related and interconnected, we can safely say that we are always on the subject -- we might mention the planets Venus and Mars have an interesting relationship, one that is related to the Earth-Moon connection.  The ratio of the perigee of Venus-Mars (when both planets are nearest each other -- on the same side of the Sun and with Mars at its closest point to the Sun) to the apogee of Venus-Mars (when both planets are furthest from each other -- on opposite sides of the Sun and Mars at its furthest point from the Sun) is 3:11 -- the same as the ratio of radii of the Moon and Earth!  Clearly Earth is sandwiched in some esoteric manner between Venus and Mars, i.e. between Love and War.  And maybe Beauty and Sex.  Or both.

It should be mentioned that Venus is an exceptional planet, both in its relationship to Earth and in its own right.  The geometries between Earth and Venus, which John Martineau [3] has so brilliantly highlighted, include a circle inscribed in a square inscribed in a circle (within 99.9% accuracy), a pentagon or five-pointed star connection, a double five-pointed star, eight circles (or “halos”), and has a mass of 0.81 that of Earth (9!).

[Incidentally, the mass of the Earth is approximately 81 times that of the Moon.]

The uniqueness of Venus, on the other hand, includes the fact that it is the only planet (or god/goddess) whose “birth” is described in the ancient texts, has the most nearly circular orbit of all the planets (suggesting it’s the newest of the brood?), the only planet which has a retrograde motion (turns on its axis in an opposite direction to every other planet and the vast majority of moons in the Solar System), its axis is very nearly perpendicular to the plane of its orbit (i.e., there will be no marked “season” on Venus), and has an unusually small orbital inclination of only 3 degrees and 24 minutes (9!).

Alex B. Geddes has noted the circular radii of the planets have an intriguing symmetry, including Venus/Neptune = 1.204 Mercury/Uranus; Mercury/Saturn = 1.208 Earth/ Neptune; and Earth/Jupiter = 1.206 Mars/Saturn.  Also, Earth/Venus = 2.872 Mars/ Mercury; and Jupiter/Saturn = 2.876 Neptune/Uranus (which approximates e = 2.718...).

v Finally, and most wonderfully, Mercury times Earth divided by (Venus times Mars) = 1.001 Jupiter times Uranus divided by (Saturn times Neptune).  Notice how the 1st and 3rd planet from the sun, divided by the 2nd and 4th, equals approximately the 5th and 7th divided by the 6th and 8th -- odds over evens.  It’s all clearly magical!

But lest we go too far afield, let us return to Earth and note the fact that it is tilted!  The angle of tilt (which is primarily responsible for the seasons) equals 23 degrees 27 minutes and 8.26 seconds.  Approximately.  The decimal equivalent (to the last significant decimal place consistent with the accuracy of + or - 0.01 seconds or arc) is: 23.45229... degrees.  Obviously, this angle reduces to...

v But there’s more.  Naturally.  The axis of the Earth points to Polaris, the North Star.  But this is not consistent.  There is something called the Precession of the Axes, where because of the Earth’s tilt (23.45229... degrees), the Earth’s axis very slowly rotates and carves a circle in the Northern skies.  On this circle lie Polaris, Alpha Draconis and Vega -- the three Pole Stars which dominate different eras.  The period of time for the Precession of the Axes is 25,920 years.  This is a long time, and represents a number which reduces to 9.  But you knew that, right?  And you also probably guessed that each astrological age (of which there are twelve -- with us progressing from Pisces to Aquarius) lasts for 2,160 (9!) years.  If we divide 2,160 by 3 (number of pole stars), we get 720/9.

There is no end to this magic!  If we look further into long periods of time, we can recall the Yugas from the Hindu Tradition, where the “Four-Fold Golden Age Yuga” lasts some 1,728,000 years, the “Three-Fold Age of Knowledge Yuga” lasts 1,296,000 years, the “Two-Fold Age of Sacrifice Yuga” lasts 864,000 years; and the “Age of Discord Yuga” (the “Kali Yuga”) lasts 432,000 years.  The sum total is 4,320,000 years.  And all reduce to 9!  Furthermore, the geologic record (the Annals of Earth) has shown a mass extinction of species, which occurs roughly every 25.92 (9!) million years.

More recently, in 1996 for example, there were dozens of crop circle formations appearing in Southern England (basically southwest of London).  Their locations were typical of previous years and included Stonehenge, Windmill Hill (near Avebury), Oliver’s Castle, Roundway and Roundway Ridge, Etchilhampton, Liddington Castle, Alton Barnes, and Ashbury.  The curious thing is that when these particular 9 crop circle locations were connected on a map, all of the angles and distances (within the accuracy of the map) reduced to 9!  This amounts to at least 36 measures of angles and distances between the formations!  Furthermore, two of the really unusual formations included glyphs -- shapes which were carved into equilateral triangles measuring 56.7 (9) feet on a side.  Associated with these glyphs were teardrop shaped formations which were each 27 feet in length, subtending an angle of 36 degrees, and offset from the centerline of the equilateral triangles by 13.5 degrees (all 9’s).

And for those enamored with Richard Hoagland’s observation that numerous very important events have been occurring on July 20th on various years -- including the first lunar landing, the Priore de Sion (1000 year old secret society) coming out of the closet, the Clementine Mission diversion to Mars, etceteras, etceteras.  We might point out that July 20th is often written 7-20... which -- you’ll never believe this -- reduces to 9.

And so, as we complete (what was orginally) the 9th page of this tome at the 9th hour on the 9th day of March 2001, we bid you all a...     Well...    You know...    Completion.

Alternatively, try checking out the 432 number, forging ahead to Fibonacci Numbers and Sacred Geometry, or back to Sacred Mathematics.  OR exciting cyberspace, shutting down the computer, and taking a walk in the sun.  We’ll still be here when you get back.   

More information on this topic can be found at the Halexandria Forums.


Sacred Mathematics         Infinite Series         Magic Squares

Forward to:

Wonders of Math

432         666

Fibonacci Numbers         Sacred Geometry



[1] Glickman, Michael, Presentation at Crop Circle Symposium, Denver, 2/22/97.

[2] Schneider, Michael S., A Beginner’s Guide to Constructing the Universe, Harper Perrenial, New York, 1995.

[3]  Martineau, John, A Book of Coincidence, Wooden Books, Powys, Wales, 1995.



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