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A

MAZE

IN

ZAZAZA ENTERS AZAZAZ

AZAZAZAZAZAZAZZAZAZAZAZAZAZA

ZAZAZAZAZAZAZAZAZAAZAZAZAZAZAZAZAZAZ

THE

MAGICALALPHABET

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26
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14
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+
=
115
1+1+5
=
7
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2
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2
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351
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=
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=
126
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=
9
-
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26
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occurs
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12
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21
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occurs
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=
18
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9
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45
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126
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4+5
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BEYOND THE VEIL ANOTHER VEIL ANOTHER VEIL BEYOND

 

 

A

HISTORY OF GOD

Karen Armstrong 1993

The God of the Mystics

Page 250

"Perhaps the most famous of the early Jewish mystical texts is the fifth century Sefer Yezirah (The Book of Creation). There is no attempt to describe the creative process realistically; the account is unashamedly symbolic and shows God creating the world by means of language as though he were writing a book. But language has been entirely transformed and the message of creation is no longer clear. Each letter of the Hebrew alphabet is given a numerical value; by combining the letters with the sacred numbers, rearranging them in endless configurations, the mystic weaned his mind away from the normal connotations of words."

 

Page 250

THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY THE ACCOUNT

IS UNASHAMEDLY SYMBOLIC AND SHOWS GOD CREATING THE WORLD BY MEANS OF LANGUAGE AS

THOUGH HE WERE WRITING A BOOK. BUT LANGUAGE HAS BEEN ENTIRELY TRANSFORMED AND THE

MESSAGE OF CREATION IS NO LONGER CLEAR EACH LETTER OF THE HEBREW ALPHABET IS GIVEN

A NUMERICAL VALUE BY COMBINING THE LETTERS WITH THE SACRED NUMBERS REARRANGING

THEM IN ENDLESS CONFIGURATIONS THE MYSTIC WEANED THE MIND AWAY FROM THE NORMAL

CONNOTATIONS OF WORDS

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

....

 

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ME
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ME
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9
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I
ME
I
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I
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I
ME
1

 

 

 

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LIGHT AND LIFE

Lars Olof Bjorn 1976

Page 197

"By writing the 26 letters of the alphabet in a certain order one may put down almost any message (this book 'is written with the same letters' as the Encyclopaedia Britannica and Winnie the Pooh, only the order of the letters differs). In the same way Nature is able to convey with her language how a cell and a whole organism is to be constructed and how it is to function. Nature has succeeded better than we humans; for the genetic code there is only one universal language which is the same in a man, a bean plant and a bacterium."

"FOR THE GENETIC CODE THERE IS ONLY ONE UNIVERSAL LANGUAGE"

 

DNA AND DNA DNA AND DNA DNA AND DNA

DNA AND DNA DNA AND DNA DNA AND DNA

 

"BY WRITING THE 26 LETTERS OF THE ALPHABET IN A CERTAIN ORDER

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

FINGERPRINTS OF THE GODS

A QUEST FOR THE BEGINNING AND THE END

Graham Hancock 1995

Chapter 32

Speaking to the Unborn

Page 285

"It is understandable that a huge range of myths from all over the ancient world should describe geological catastrophes in graphic detail. Mankind survived the horror of the last Ice Age, and the most plausible source for our enduring traditions of flooding and freezing, massive volcanism and devastating earthquakes is in the tumultuous upheavals unleashed during the great meltdown of 15,000 to 8000 BC. The final retreat of the ice sheets, and the consequent 300-400 foot rise in global sea levels, took place only a few thousand years before the beginning of the historical period. It is therefore not surprising that all our early civilizations should have retained vivid memories of the vast cataclysms that had terrified their forefathers.
Much harder to explain is the peculiar but distinctive way the myths of cataclysm seem to bear the intelligent imprint of a guiding hand.l Indeed the degree of convergence between such ancient stories is frequently remarkable enough to raise the suspicion that they must all have been 'written' by the same 'author'.
Could that author have had anything to do with the wondrous deity, or superhuman, spoken of in so many of the myths we have reviewed, who appears immediately after the world has been shattered by a horrifying geological catastrophe and brings comfort and the gifts of civilization to the shocked and demoralized survivors?
White and bearded, Osiris is the Egyptian manifestation of this / Page 286 / universal figure, and it may not be an accident that one of the first acts he is remembered for in myth is the abolition of cannibalism among the primitive inhabitants of the Nile Valley.2 Viracocha, in South America, was said to have begun his civilizing mission immediately after a great flood; Quetzalcoatl, the discoverer of maize, brought the benefits of crops, mathematics, astronomy and a refined culture to Mexico after the Fourth Sun had been overwhelmed by a destroying deluge.
Could these strange myths contain a record of encounters between scattered palaeolithic tribes which survived the last Ice Age and an as yet unidentified high civilization which passed through the same epoch?
And could the myths be attempts to communicate?

A message in the bottle of time

'Of all the other stupendous inventions,' Galileo once remarked,

what sublimity of mind must have been his who conceived how to communicate his most secret thoughts to any other person, though very distant either in time or place, speaking with those who are in the Indies, speaking to those who are not yet born, nor shall be this thousand or ten thousand years? And with no greater difficulty than the various arrangements of two dozen little signs on paper? Let this be the seal of all the admirable inventions of men.3

If the 'precessional message' identified by scholars like Santillana, von Dechend and Jane Sellers is indeed a deliberate attempt at communication by some lost civilization of antiquity, how come it wasn't just written down and left for us to find? Wouldn't that have been easier than encoding it in myths? Perhaps.
Nevertheless, suppose that whatever the message was written on got destroyed or worn away after many thousands of years? Or suppose that the language in which it was inscribed was later forgotten utterly (like the enigmatic Indus Valley script, which has been studied closely for more than half a century but has so far resisted all attempts at decoding)? It must be obvious that in such circumstances a written / Page 287 / legacy to the future would be of no value at all, because nobody would be able to make sense of it.
What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them - and the city of Teotihuacan may be the calling-card of a lost civilization written in the eternal language of mathematics.
Geodetic data, related to the exact positioning of fixed geographical points and to the shape and size of the earth, would also remain valid and recognizable for tens of thousands of years, and might be most conveniently expressed by means of cartography (or in the construction of giant geodetic monuments like the Great Pyramid of Egypt, as we shall see).
Another 'constant' in our solar system is the language of time: the great but regular intervals of time calibrated by the inch-worm creep of precessional motion. Now, or ten thousand years in the future, a message that prints out numbers like 72 or 2160 or 4320or 25,920 should be instantly intelligible to any civilization that has evolved a modest talent for mathematics and the ability to detect and measure the almost imperceptible reverse wobble that the sun appears to make along the ecliptic against the background of the fixed stars..."

"What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them"

"WRITTEN IN THE ETERNAL LANGUAGE OF MATHEMATICS"

 

 

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THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY

THE ACCOUNT IS SYMBOLIC AND SHOWS GOD CREATING THE WORLD BY MEANS OF LANGUAGE

AS THOUGH WRITING A BOOK BUT LANGUAGE ENTIRELY TRANSFORMED

THE MESSAGE OF CREATION IS CLEAR EACH LETTER OF

THE

ALPHABET

IS

GIVEN

A

NUMERICAL

VALUE BY COMBINING THE LETTERS WITH THE SACRED NUMBERS

REARRANGING THEM IN ENDLESS CONFIGURATIONS

THE MYSTIC WEANED THE MIND AWAY FROM THE NORMAL CONNOTATIONS OF WORDS

 

....

 

 

 

 

 

THE DEATH OF GODS IN ANCIENT EGYPT

Jane B. Sellars 1992

Page 204

"The overwhelming awe that accompanies the realization, of the measurable orderliness of the universe strikes modern man as well. Admiral Weiland E. Byrd, alone In the Antarctic for five months of polar darkness, wrote these phrases of intense feeling:

Here were the imponderable processes and forces of the cosmos, harmonious and soundless. Harmony, that was it! I could feel no doubt of oneness with the universe. The conviction came that the rhythm was too orderly. too harmonious, too perfect to be a product of blind chance - that, therefore there must be purpose in the whole and that man was part of that whole and not an accidental offshoot. It was a feeling that transcended reason; that went to the heart of man's despair and found it groundless. The universe was a cosmos, not a chaos; man was as rightfully a part of that cosmos as were the day and night.10

Returning to the account of the story of Osiris, son of Cronos god of' Measurable Time, Plutarch takes, pains to remind the reader of the original Egyptian year consisting of 360 days.

Phrases are used that prompt simple mental. calculations and an attention to numbers, for example, the 360-day year is described as being '12 months of 30 days each'. Then we are told that, Osiris leaves on a long journey, during which Seth, his evil brother, plots with 72 companions to slay Osiris: He also secretly obtained the measure of Osiris and made ready a chest in which to entrap him.

The, interesting thing about this part of the-account is that nowhere in the original texts of the Egyptians are we told that Seth, has 72 companions. We have already been encouraged to equate Osiris with the concept of measured time; his father being Cronos. It is also an observable fact that Cronos-Saturn has the longest sidereal period of the known planets at that time, an orbit. of 30 years. Saturn is absent from a specific constellation for that length of time.

A simple mathematical fact has been revealed to any that are even remotely sensitive to numbers: if you multiply 72 by 30, the years of Saturn's absence (and the mention of Osiris's absence prompts one to recall this other), the resulting product is 2,160: the number of years required, for one 30° shift, or a shift: through one complete sign of the zodiac. This number multplied by the / Page205 / 12 signs also gives 25,920. (And Plutarch has reminded us of 12)

If you multiply the unusual number 72 by 360, a number that Plutarch mentions several times, the product will be 25,920, again the number of years symbolizing the ultimate rebirth.

This 'Eternal Return' is the return of, say, Taurus to the position of marking the vernal equinox by 'riding in the solar bark with. Re' after having relinquished this honoured position to Aries, and subsequently to the to other zodiacal constellations.

Such a return after 25,920 years is indeed a revisit to a Golden Age, golden not only because of a remarkable symmetry In the heavens, but golden because it existed before the Egyptians experienced heaven's changeability.

But now to inform the reader of a fact he or she may already know. Hipparaus did: not really have the exact figures: he was a trifle off in his observations and calculations. In his published work, On the Displacement of the Solstitial and Equinoctial Signs, he gave figures of 45" to 46" a year, while the truer precessional lag along the ecliptic is about 50 seconds. The exact measurement for the lag, based on the correct annual lag of 50'274" is 1° in 71.6 years, or 360° in 25,776 years, only 144 years less than the figure of 25,920.

With Hipparchus's incorrect figures a 'Great Year' takes from 28,173.9 to 28,800 years, incorrect by a difference of from 2,397.9 years to 3,024.

Since Nicholas Copernicus (AD 1473-1543) has always been credited with giving the correct numbers (although Arabic astronomer Nasir al-Din Tusi,11 born AD 1201, is known to have fixed the Precession at 50°), we may correctly ask, and with justifiable astonishment 'Just whose information was Plutarch transmitting'

AN IMPORTANT POSTSCRIPT

Of course, using our own notational system, all the important numbers have digits that reduce to that amazing number 9 a number that has always delighted budding mathematician.

Page 206

Somewhere along the way, according to Robert Graves, 9 became the number of lunar wisdom.12

This number is found often in the mythologies of the world. the Viking god Odin hung for nine days and nights on the World Tree in order to acquire the secret of the runes, those magic symbols out of which writing and numbers grew. Only a terrible sacrifice would give away this secret, which conveyed upon its owner power and dominion over all, so Odin hung from his neck those long 9 days and nights over the 'bottomless abyss'. In the tree were 9 worlds, and another god was said to have been born of 9 mothers.

Robert Graves, in his White Goddess, Is intrigued by the seemingly recurring quality of the number 72 in early myth and ritual. Graves tells his reader that 72 is always connected with the number 5, which reflects, among other things, the five Celtic dialects that he was investigating. Of course, 5 x 72= 360, 360 x 72= 25,920. Five is also the number of the planets known to the ancient world, that is, Saturn, Jupiter, Mars, Venus Mercury.

Graves suggests a religious mystery bound up with two ancient Celtic 'Tree Alphabets' or cipher alphabets, which as genuine articles of Druidism were orally preserved and transmitted for centuries. He argues convincingly that the ancient poetry of Europe was ultimately based on what its composers believed to be magical principles, the rudiments of which formed a close religious secret for centuries. In time these were-garbled, discredited and forgotten.

Among the many signs of the transmission of special numbers he points out that the aggregate number of letter strokes for the complete 22-letter Ogham alphabet that he is studying is 72 and that this number is the multiple of 9, 'the number of lunar wisdom'. . . . he then mentions something about 'the seventy day season during which Venus moves successively from. maximum eastern elongation 'to inferior conjunction and maximum western elongation'.13

Page 207

"...Feniusa Farsa, Graves equates this hero with Dionysus. Farsa has 72 assistants who helped him master the 72 languages created at the confusion of Babel, the tower of which is said to be built of 9 different materials

We are also reminded of the miraculous translation into Greek of the Five Books of Moses that was done by 72 scholars working for 72 days, Although the symbol for the Septuagint is LXX, legend, according to the fictional letter of Aristeas, records 72. The translation was done for Ptolemy Philadelphus (c.250 BC), by Hellenistic Jews, possibly from Alexandra.14

Graves did not know why this number was necessary, but he points out that he understands Frazer's Golden Bough to be a book hinting that 'the secret involves the truth that the Christian dogma, and rituals, are the refinement of a great body of primitive beliefs, and that the only original element in Christianity- is the personality of Christ.15

Frances A. Yates, historian of Renaissance hermetisma tells, us the cabala had 72 angels through which the sephiroth (the powers of God) are believed to be approached, and further, she supplies the information that although the Cabala supplied a set of 48 conclusions purporting to confirm the Christian religion from the foundation of ancient wisdom, Pico Della Mirandola, a Renaissance magus, introduced instead 72, which were his 'own opinion' of the correct number. Yates writes, 'It is no accident there are seventy-two of Pico's Cabalist conclusions, for the conclusion shows that he knew something of the mystery of the Name of God with seventy-two letters.'16

In Hamlet's Mill de Santillana adds the facts that 432,000 is the number of syllables in the Rig-Veda, which when multiplied by the soss (60) gives 25,920" (The reader is forgiven for a bit of laughter at this point)

The Bible has not escaped his pursuit. A prominent Assyriologist of the last century insisted that the total of the years recounted mounted in Genesis for the lifetimes of patriarchs from the Flood also contained the needed secret numbers. (He showed that in the 1,656 years recounted in the Bible there are 86,400 7 day weeks, and dividing this number yields / Page 208 / 43,200.) In Indian yogic schools it is held that all living beings exhale and inhale 21,600 times a day, multiply this by 2 and again we have the necessary 432 digits.

Joseph Campbell discerns the secret in the date set for the coming of Patrick to Ireland. Myth-gives this date-as-the interesting number of AD.432.18

Whatever one may think-of some of these number coincidences, it becomes difficult to escape the suspicion that many signs (number and otherwise) - indicate that early man observed the results of the movement of Precession and that the - transmission of this information was considered of prime importance.

With the awareness of the phenomenon, observers would certainly have tried for its measure, and such an endeavour would have constituted the construction-of a 'Unified Field Theory' for nothing less than Creation itself. Once determined, it would have been information worthy of secrecy and worthy of the passing on to future adepts.

But one last word about mankind's romance with number coincidences.The antagonist in John Updike's novel, Roger's Version, is a computer hacker, who, convinced, that scientific evidence of God's existence is accumulating, endeavours to prove it by feeding -all the available scientific information. into a comuter. In his search for God 'breaking, through', he has become fascinated by certain numbers that have continually been cropping up. He explains them excitedly as 'the terms of Creation':

"...after a while I noticed that all over the sheet there seemed to hit these twenty-fours Jumping out at me. Two four; two, four. Planck time, for instance, divided by the radiation constant yields a figure near eight times ten again to the negative twenty-fourth, and the permittivity of free space, or electric constant, into the Bohr radius ekla almost exactly six times ten to the negative twenty-fourth. On positive side, the electromagnetic line-structure constant times Hubble radius - that is, the size of the universe as we now perceive it gives us something quite close to ten to the twenty-fourth, and the strong-force constant times the charge on the proton produces two point four times ten to the negative eighteenth, for another I began to circle twenty-four wherever it appeared on the Printout here' - he held it up his piece of stripped and striped wallpaper, decorated / Page 209 / with a number of scarlet circles - 'you can see it's more than random.'19
This inhabitant of the twentieth century is convinced that the striking occurrences of 2 and 4 reveal the sacred numbers by which God is speaking to us.

So much for any scorn directed to ancient man's fascination with number coincidences. That fascination is alive and well, Just a bit more incomprehensible"

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

Our minds seem to be organised to search for relationships and sequences. We look for hidden orders.

These intuitions seem to be more important than the facts themselves, for there is always the thrill at finding something, a pattern, it is a discovery - what was unknown is now revealed. Imagine looking up at the stars and finding the zodiac!

Searching out patterns is a pure delight.

Suddenly the counters fall into place and a connection is found, not necessarily a geometric one, but a relationship between numbers, pictures of the mind, that were not obvious before. There is that excitement of finding order in something that was otherwise hidden.

And there is the knowledge that a huge unseen world lurks behind the facades we see of the numbers themselves."

 

....

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

 

THE MAGIC MOUNTAIN

Thomas Mann 1875-1955

Page 466

"Had not the normal, since time was, lived on the achievements of the abnormal? Men consciously and voluntarily descended into disease and madness, in search of knowledge which, acquired by fanaticism, would lead back to health; after the possession and use of it had ceased to be conditioned by that heroic and abnormal act of sacrifice. That was the true death on the cross, the true Atonement."

 

THE TRUE DEATH ON THE CROSS THE TRUE AT ONE MENT

 

 

ATONEMENT

 

-
-
-
-
A
-
-
-
-
-
-
-
-
T
-
-
-
-
-
-
-
-
O
-
-
-
-
-
-
-
-
N
-
-
-
-
A
T
O
N
E
M
E
N
T
-
-
-
-
M
-
-
-
-
-
-
-
-
E
-
-
-
-
-
-
-
-
N
-
-
-
-
-
-
-
-
T
-
-
-
-

 

 

ADVENT 486 ADVENT

 

THE BUDDHA
BUDDHA
BUDDHA DHARMA SANGHA
BUDDHA DHARMA SANGHA
BODHISATTVA, B, ODH, I, SA, TTVA
BODHISATTVA, BUDDHA
BUDDHA, DHARMA, SANGHA
EARTH AIR FIRE WATER
EARTH AIR FIRE WATER
EARTH AIR FIRE WATER HUMAN SPIRIT
FIRE
DODECAHEDRON
THE TENTH AVATAR
THE TENTH AVATAR
DREAMER, READ ME
HEROIC, HERO, I, C
HERO, I, SEE, HERO I SEE, HEROIC
LUCIFER, THE LIGHT, BEARER
LIGHT, ISIS
LUCIFER, LUCIFRE, LUC, I, FRE
LUCIFER, LU, CIFRE
IMHOTEP, I, MHO, TEP
PHI
PHI, P, H, I
SAMSARA
CHESS

 

 

-
9
T
H
E
-
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
-
+
=
16
1+6
=
7
=
7
=
7
-
9
T
H
E
-
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
2
3
4
4
-
1
+
=
21
2+1
=
3
=
3
=
3
-
-
20
-
5
-
2
21
4
4
-
1
+
=
57
5+7
=
12
1+2
3
=
3
-
9
T
H
E
-
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
2
21
4
4
8
1
+
=
73
7+3
=
10
1+0
1
=
1
-
-
2
8
5
-
2
3
4
4
8
1
+
=
37
3+7
=
10
1+0
1
=
1
-
9
T
H
E
-
B
U
D
D
H
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
4
4
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
5
-
--
-
-
-
-
-
-
-
5
occurs
x
1
=
4
=
5
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
22
9
T
H
E
-
B
U
D
D
H
A
-
-
23
-
-
9
-
37
-
28
2+2
-
-
-
-
-
--
-
-
-
-
-
-
-
2+3
-
-
-
-
3+7
-
2+8
4
9
T
H
E
-
B
U
D
D
H
A
-
-
5
-
-
9
-
10
-
10
-
-
2
8
5
-
2
3
4
4
8
1
-
-
-
-
-
-
-
1+0
-
1+0
4
9
T
H
E
-
B
U
D
D
H
A
-
-
5
-
-
9
-
1
-
1

 

 

9
T
H
E
-
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
-
-
-
8
-
+
=
16
1+6
=
7
=
7
=
7
9
T
H
E
-
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
-
2
-
5
-
2
3
4
4
-
1
+
=
21
2+1
=
3
=
3
=
3
-
20
-
5
-
2
21
4
4
-
1
+
=
57
5+7
=
12
1+2
3
=
3
9
T
H
E
-
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
2
21
4
4
8
1
+
=
73
7+3
=
10
1+0
1
=
1
-
2
8
5
-
2
3
4
4
8
1
+
=
37
3+7
=
10
1+0
1
=
1
9
T
H
E
-
B
U
D
D
H
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
-
2
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
4
4
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
5
-
--
-
-
-
-
-
-
-
5
occurs
x
1
=
4
=
5
-
-
8
-
-
-
-
-
-
8
-
-
-
8
occurs
x
2
=
16
1+6
7
9
T
H
E
-
B
U
D
D
H
A
-
-
23
-
-
9
-
37
-
28
-
-
-
-
-
--
-
-
-
-
-
-
-
2+3
-
-
-
-
3+7
-
2+8
9
T
H
E
-
B
U
D
D
H
A
-
-
5
-
-
9
-
10
-
10
-
2
8
5
-
2
3
4
4
8
1
-
-
-
-
-
-
-
1+0
-
1+0
9
T
H
E
-
B
U
D
D
H
A
-
-
5
-
-
9
-
1
-
1

 

 

-
6
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
-
-
-
-
8
-
+
=
8
-
=
8
=
8
-
6
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
2
3
4
4
-
1
+
=
14
1+4
=
5
=
5
-
-
2
21
4
4
-
1
+
=
32
3+2
=
5
=
5
-
6
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
2
21
4
4
8
1
+
=
40
4+0
=
4
=
4
-
-
2
3
4
4
8
1
+
=
22
2+2
=
4
=
4
-
9
B
U
D
D
H
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
4
-
-
-
-
4
occurs
x
2
=
8
5
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
27
6
B
U
D
D
H
A
-
-
18
-
-
6
-
22
2+7
-
--
-
-
-
-
-
-
-
1+8
-
-
-
-
2+2
9
6
B
U
D
D
H
A
-
-
9
-
-
6
-
4
-
-
2
3
4
4
8
1
-
-
-
-
-
-
-
-
9
6
B
U
D
D
H
A
-
-
9
-
-
6
-
4

 

 

-
6
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
-
-
-
-
8
-
+
=
8
-
=
8
=
8
-
6
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
2
3
4
4
-
1
+
=
14
1+4
=
5
=
5
-
-
2
21
4
4
-
1
+
=
32
3+2
=
5
=
5
-
6
B
U
D
D
H
A
-
-
-
-
-
--
-
-
-
-
2
21
4
4
8
1
+
=
40
4+0
=
4
=
4
-
-
2
3
4
4
8
1
+
=
22
2+2
=
4
=
4
-
9
B
U
D
D
H
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
4
-
-
-
-
4
occurs
x
2
=
8
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
27
6
B
U
D
D
H
A
-
-
18
-
-
6
-
22
2+7
-
--
-
-
-
-
-
-
-
1+8
-
-
-
-
2+2
9
6
B
U
D
D
H
A
-
-
9
-
-
6
-
4
-
-
2
3
4
4
8
1
-
-
-
-
-
-
-
-
9
6
B
U
D
D
H
A
-
-
9
-
-
6
-
4

 

 

B
U
D
D
H
A
-
D
H
A
R
M
A
-
S
A
N
G
H
A
2
3
4
4
8
1
-
4
8
1
9
4
1
-
1
1
5
7
8
1
-
-
-
-
-
1
-
-
-
1
-
-
1
-
1
1
-
-
-
1
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
4
-
-
-
4
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
8
-
-
-
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
B
U
D
D
H
A
-
D
H
A
R
M
A
-
S
A
N
G
H
A

 

 

1
occurs
x
6
=
6
-
-
6
2
occurs
x
1
=
2
-
-
2
3
occurs
x
1
=
3
-
-
3
4
occurs
x
4
=
16
1+6
=
7
5
occurs
x
1
=
5
-
-
5
7
occurs
x
1
=
7
-
-
7
8
occurs
x
3
=
24
2+4
=
6
9
occurs
x
1
=
9
-
-
9

 

 

-
BUDDHA DHARMA SANGHA
-
-
-
-
B+U+D
27
9
9
-
D
4
4
4
-
H+A
9
9
9
-
D
4
4
4
-
H+A
9
9
9
-
R
18
9
9
-
M+A+S+A+N+G
55
19
1
-
H+A
9
9
9
18
BUDDHA DHARMA SANGHA
135
72
54
1+8
-
1+3+5
7+2
5+4
9
BUDDHA DHARMA SANGHA
9
9
9

 

 

-
18
B
U
D
D
H
A
-
D
H
A
R
M
A
-
S
A
N
G
H
A
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
-
-
-
-
-
1
-
5
-
8
-
+
=
30
3+0
=
3
=
3
=
3
-
-
-
-
-
-
8
-
-
-
8
-
-
-
-
-
19
-
14
-
8
-
+
=
48
4+8
=
12
1+2
3
=
3
-
18
B
U
D
D
H
A
-
D
H
A
R
M
A
-
S
A
N
G
H
A
-
-
-
-
-
--
-
-
-
-
-
-
2
3
4
4
-
1
-
4
-
1
9
4
1
-
-
1
-
7
-
1
+
=
42
4+2
=
6
=
6
=
6
-
-
2
21
4
4
-
1
-
4
-
1
18
13
1
-
-
1
-
7
-
1
+
=
60
6+0
=
6
=
6
=
6
-
18
B
U
D
D
H
A
-
D
H
A
R
M
A
-
S
A
N
G
H
A
-
-
-
-
-
--
-
-
-
-
-
-
2
21
4
4
8
1
-
4
8
1
18
13
1
-
19
1
14
7
8
1
+
=
135
1+3+5
=
9
=
9
=
9
-
-
2
3
4
4
8
1
-
4
8
1
9
4
1
-
1
1
5
7
8
1
+
=
72
7+2
=
9
=
9
=
9
-
18
B
U
D
D
H
H
A
D
H
A
R
M
A
-
S
A
N
G
H
A
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
-
1
1
-
-
-
1
-
-
1
occurs
x
6
=
6
=
6
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
4
4
-
-
-
4
-
-
-
4
-
-
-
-
-
-
-
-
-
-
4
occurs
x
4
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
=
5
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
-
8
-
-
-
8
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
6
18
B
U
D
D
H
H
A
D
H
A
R
M
A
-
S
A
N
G
H
A
-
-
39
-
-
18
-
72
-
45
-
1+8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
12
-
-
1+8
-
7+2
-
4+5
6
9
B
U
D
D
H
A
-
D
H
A
R
M
A
-
S
A
N
G
H
A
-
-
3
-
-
9
-
9
-
9
-
-
2
3
4
4
8
1
-
4
8
1
9
4
1
-
1
1
5
7
8
1
+
=
72
7+2
=
9
=
9
=
9
6
9
B
U
D
D
H
A
-
D
H
A
R
M
A
-
S
A
N
G
H
A
-
-
3
-
-
9
-
9
-
9

 

 

11
BODHISATTVA
-
-
-
-
B
2
2
2
-
O+D+H
27
18
9
-
I
9
9
9
-
S+A
20
11
2
-
T+T+V+A
63
9
9
11
BODHISATTVA
-
-
-

 

 

11
BODHISATTVA
121
49
4
6
BUDDHA
40
22
4

 

 

6
BUDDHA
40
22
4
6
DHARMA
45
27
9
6
SANGHA
50
23
5
18
Add to Reduce
135
72
18
1+8
Reduce to Deduce
1+3+5
7+2
1+8
9
 Essence of Number
9
9
9

 

 

A BUDDHIST BIBLE

SELECTIONS FROM SANSKRIT SOURCES

THE LANKAVATARA SCRIPTURE

Compiled by Dwight Goddard 1938

Page 317

"Then said Mahamati to the Blessed One: Pray tell us, Blessed One, concerning the characteristics of deep attachments to existence and as to how we may become detached from existence?
The Blessed One replied: When one tries to understand the significance of things by means of words and discriminations, there follow immeasurably deep-seated attachments to existence. For instance: there are the deep-seated attachments to signs of individuality, to causation, to the notion of being and non­being, to the discrimination of birth and death, of doing and not-doing, to the habit of discrimination itself upon which the philosophers are so dependent.
There are three attachments that are especially deep-seated in the minds of all: greed, anger and infatuation, which are based on lust, fear and pride. Back of these lies discrimination and desire which is procreative and is accompanied with excitement and avariciousness and love of comfort and desire for eternal life; and, following, is a succession of rebirths on the five paths of existence and a continuation of attachments. But if these attachments are broken off, no signs of attachment nor of detachment will remain because they are based on things that are non-existent; when this truth is clearly understood the net of attachment is cleared away.
But depending upon and attaching itself to the triple combination which works in unison there is the rising and the con­tinuation of the mind-system incessantly functioning, and be / Page 318 / cause of it there is the deeply-felt and continuous assertion of the will-to-live. When the triple combination that causes the functioning of the mind-system ceases to exist, there is the triple emancipation and there is no further rising of any combination. When the existence and the non-existence of the ex­ternal world are recognised as rising from the mind itself, then the Bodhisattva is prepared to enter into the state of imagelessness and therein to see into the emptiness which characterises all discrimination and all the deep-seated attachments resulting therefrom. Therein he will see no signs of deep-rooted attachment nor detachment; therein he will see no one in bondage and no one in emancipation, except those who themselves cherish bondage and emancipation, because in all things there is no "substance" to be taken hold of.
But so long- as these discriminations are cherished by the ignorant and simple-minded they go on attaching themselves to them and, like the silkworm, go on spinning their thread of discrimination and enwrapping themselves and others, and are charmed with their prison. But to the wise there are no signs of attachment nor of detachment; all things are seen as abiding in solitude where there is no evolving of discrimination. Mahamati, you and all the Bodhisattvas should have your abode where you can see all things from the view-point of solitude.
Mahamati, when you and other Bodhisattvas understand well the distinction between attachment and detachment, you will be in possession of skillful means for avoiding becoming at­tached to words according to which one proceeds to grasp meanings. Free from the domination of words you will be able to establish yourselves where there will be a "turning- about" in the deepest seat of consciousness by means of which you will attain self-realisation of Noble Wisdom and be able to enter into all the Buddha-lands and assemblies. There you will be stamped with the stamp of the powers, self-command, the psychic faculties, and will be endowed with the wisdom and the power of the ten inexhaustible vows, and will become ra­diant with the variegated rays of the Transformation Bodies. Therewith you will shine without effort like the moon, the sun, the magic wishing-jewel, and at every stage will view things as being- of perfect oneness with yourself, uncontaminated by any self-consciousness. Seeing that all things are like a dream, you will be able to enter into the stage of the Tathagatas and be / Page 319 / able to deliver discourses on the Dharma to the world of beings in accordance with their needs and be able to free them from all dualistic notions and false discriminations.
Mahamati, there are two ways of considering self-realisation: namely, the teachings about it, and the realisation itself. The teachings as variously given in the nine divisions of the doc­trinal works, for the instructions of those who are inclined to­ward it, by making use of skillful means and expedients, are intended to awaken in all beings a true perception of the Dharma. The teachings are designed to keep one away from all the dualistic notions of being and non-being and oneness and otherness.
Realisation itself is within the inner consciousness. It is an inner experience that has no connection with the lower mind­system and its discriminations of words, ideas and philosophical speculations. It shines out with its own clear light to reveal the error and foolishness of mind-constructed teachings, to render impotent evil influences from without, and to guide one unerringly to the realm of the good non-outflowings. Mahamati, when the earnest disciple and Bodhisattva is provided with these requirements, the way is open to his perfect attain­ment of self-realisation of Noble Wisdom, and to the full enjoyment of the fruits that arise therefrom."

 

 

5
EARTH
52
25
7
3
AIR
28
19
1
4
FIRE
38
29
2
5
WATER
67
22
4
17
First Total
185
95
14
1+7
Add to Reduce
1+8+5
9+5
1+4
8
Second Total
14
14
5
-
Reduce to Deduce
1+4
1+4
-
8
Essence of Number
5
5
5

 

 

A
=
1
3

AIR
28
19
1
F
=
6
4

FIRE
38
29
2
E
=
5
5

EARTH
52
25
7
W
=
5
5

WATER
67
22
4
H
=
8
5

HUMAN
57
21
3
S
=
1
6

SPIRIT
91
37
1
-
-
26
28

-
333
153
18
-
-
2+6
10

-
3+3+3
1+5+3
1+8
-
-
8
1

-
9
9
9

 

 

E
=
5
-
5
EARTH
52
25
7
F
=
6
-
4
FIRE
38
29
2
A
=
1
-
3
AIR
28
19
1
W
=
5
-
5
WATER
67
22
4
-
-
17
-
17
Add
185
95
14
-
-
1+7
-
1+7
Reduce
1+8+5
9+5
1+4
-
-
8
-
8
 Total
14
14
5
-
-
-
-
-
Deduce
1+4
1+4
-
-
-
8
-
8
Essence
5
5
5

 

 

--
17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
8
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
+
=
26
2+6
=
8
=
8
=
8
-
-
-
-
-
--
8
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
+
=
26
2+6
=
8
=
8
=
8
--
17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
5
1
9
2
-
-
6
-
9
5
-
1
-
9
-
5
1
2
5
9
+
=
69
6+9
=
15
1+5
6
=
6
-
-
5
1
18
20
-
-
6
-
18
5
-
1
-
18
-
23
1
20
5
18
+
=
159
1+5+9
=
15
1+5
6
=
6
--
17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
5
1
18
20
8
-
6
9
18
5
-
1
9
18
-
23
1
20
5
18
+
=
185
1+8+5
=
14
1+4
5
=
5
-
-
5
1
9
2
8
-
6
9
9
5
-
1
9
9
-
5
1
2
5
9
+
=
95
9+5
=
14
1+4
5
=
5
-
17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
--
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
=
4
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
9
-
-
-
-
9
9
-
-
-
9
9
-
-
-
-
-
9
-
-
9
occurs
x
6
=
36
3+6
9
14
17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
31
-
-
17
-
95
-
32
1+4
1+7
-
-
9
-
-
-
-
9
9
-
-
-
9
9
-
-
-
-
-
9
-
-
3+1
-
-
1+7
-
9+5
-
3+2
5
8
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
4
-
-
8
-
14
-
5
-
-
5
1
9
2
8
-
6
9
9
5
-
1
9
9
-
5
1
2
5
9
-
-
-
-
-
-
-
1+4
-
3+2
5
8
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
4
-
-
8
-
5
-
5

 

 

17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
--
8
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
+
=
26
2+6
=
8
=
8
=
8
-
-
-
-
--
8
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
+
=
26
2+6
=
8
=
8
=
8
17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
-
-
-
--
-
-
-
-
-
5
1
9
2
-
-
6
-
9
5
-
1
-
9
-
5
1
2
5
9
+
=
69
6+9
=
15
1+5
6
=
6
-
5
1
18
20
-
-
6
-
18
5
-
1
-
18
-
23
1
20
5
18
+
=
159
1+5+9
=
15
1+5
6
=
6
17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
-
-
-
--
-
-
-
-
-
5
1
18
20
8
-
6
9
18
5
-
1
9
18
-
23
1
20
5
18
+
=
185
1+8+5
=
14
1+4
5
=
5
-
5
1
9
2
8
-
6
9
9
5
-
1
9
9
-
5
1
2
5
9
+
=
95
9+5
=
14
1+4
5
=
5
17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
--
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
=
4
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
-
-
-
9
9
-
-
-
9
9
-
-
-
-
-
9
-
-
9
occurs
x
6
=
36
3+6
9
17
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
31
-
-
17
-
95
-
32
1+7
-
-
9
-
-
-
-
9
9
-
-
-
9
9
-
-
-
-
-
9
-
-
3+1
-
-
1+7
-
9+5
-
3+2
8
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
4
-
-
8
-
14
-
5
-
5
1
9
2
8
-
6
9
9
5
-
1
9
9
-
5
1
2
5
9
-
-
-
-
-
-
-
1+4
-
3+2
8
E
A
R
T
H
-
F
I
R
E
-
A
I
R
-
W
A
T
E
R
-
-
4
-
-
8
-
5
-
5

 

 

17
E
A
R
T
H
F
I
R
E
A
I
R
W
A
T
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
--
8
-
9
-
-
-
9
-
-
-
-
-
-
+
=
26
2+6
=
8
=
8
=
8
-
-
-
-
--
8
-
9
-
-
-
9
-
-
-
-
-
-
+
=
26
2+6
=
8
=
8
=
8
17
E
A
R
T
H
F
I
R
E
A
I
R
W
A
T
E
R
-
-
-
-
-
--
-
-
-
-
-
5
1
9
2
-
6
-
9
5
1
-
9
5
1
2
5
9
+
=
69
6+9
=
15
1+5
6
=
6
-
5
1
18
20
-
6
-
18
5
1
-
18
23
1
20
5
18
+
=
159
1+5+9
=
15
1+5
6
=
6
17
E
A
R
T
H
F
I
R
E
A
I
R
W
A
T
E
R
-
-
-
-
-
--
-
-
-
-
-
5
1
18
20
8
6
9
18
5
1
9
18
23
1
20
5
18
+
=
185
1+8+5
=
14
1+4
5
=
5
-
5
1
9
2
8
6
9
9
5
1
9
9
5
1
2
5
9
+
=
95
9+5
=
14
1+4
5
=
5
17
E
A
R
T
H
F
I
R
E
A
I
R
W
A
T
E
R
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
--
-
-
-
1
-
-
-
1
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
=
4
-
5
-
-
-
-
-
-
-
5
-
-
-
5
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
-
-
9
9
-
-
9
9
-
-
-
-
9
-
-
9
occurs
x
6
=
36
3+6
9
17
E
A
R
T
H
F
I
R
E
A
I
R
W
A
T
E
R
-
-
31
-
-
17
-
95
-
32
1+7
-
-
9
-
-
-
9
9
-
-
9
9
-
-
-
-
9
-
-
3+1
-
-
1+7
-
9+5
-
3+2
8
E
A
R
T
H
F
I
R
E
A
I
R
W
A
T
E
R
-
-
4
-
-
8
-
14
-
5
-
5
1
9
2
8
6
9
9
5
1
9
9
5
1
2
5
9
-
-
-
-
-
-
-
1+4
-
3+2
8
E
A
R
T
H
F
I
R
E
A
I
R
W
A
T
E
R
-
-
4
-
-
8
-
5
-
5

 

 

-
4
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
-
-
9
-
-
+
=
9
-
=
9
=
9
=
9
-
4
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
-
6
-
9
5
+
=
20
2+0
=
2
=
2
=
2
-
-
6
-
18
5
+
=
29
2+9
=
11
1+1
2
=
2
-
4
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
-
6
9
18
5
+
=
38
3+8
=
11
1+1
2
=
2
-
-
6
9
9
5
+
=
29
2+9
=
11
1+1
2
=
2
-
4
F
I
R
E
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
4
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
2
=
18
1+8
9
25
4
F
I
R
E
-
-
20
-
-
4
-
29
-
20
2+5
-
-
9
9
-
-
-
2+0
-
-
-
-
2+9
-
2+0
7
4
F
I
R
E
-
-
2
-
-
4
-
11
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
1+1
-
-
7
4
F
I
R
E
-
-
2
-
-
4
-
2
-
2

 

 

-
4
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
-
-
9
-
-
+
=
9
-
=
9
=
9
=
9
-
4
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
-
6
-
9
5
+
=
20
2+0
=
2
=
2
=
2
-
-
6
-
18
5
+
=
29
2+9
=
11
1+1
2
=
2
-
4
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
-
6
9
18
5
+
=
38
3+8
=
11
1+1
2
=
2
-
-
6
9
9
5
+
=
29
2+9
=
11
1+1
2
=
2
-
4
F
I
R
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
9
9
-
-
-
9
occurs
x
2
=
18
1+8
9
25
4
F
I
R
E
-
-
20
-
-
4
-
29
-
20
2+5
-
-
9
9
-
-
-
2+0
-
-
-
-
2+9
-
2+0
7
4
F
I
R
E
-
-
2
-
-
4
-
11
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
1+1
-
-
7
4
F
I
R
E
-
-
2
-
-
4
-
2
-
2

 

 

N
=
5
-
5
NORTH
75
30
3
S
=
1
-
5
SOUTH
83
20
2
E
=
5
-
4
EAST
45
9
9
W
=
5
-
4
WEST
67
13
4
-
-
17
-
18
Add
270
72
18
-
-
1+7
-
1+8
Reduce
2+7+0
7+2
1+8
-
-
8
-
9
Deduce
9
9
9

 

 

-
18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
-
5
6
-
-
8
-
1
6
-
-
8
-
-
-
1
-
-
-
-
1
-
+
=
36
3+6
=
9
=
9
=
9
-
-
14
15
-
-
8
-
19
15
-
-
8
-
-
-
19
-
-
-
-
19
-
+
=
117
1+1+7
=
4
=
9
=
9
-
18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
9
2
-
-
-
-
3
2
-
-
5
1
-
2
-
5
5
-
2
+
=
36
3+6
=
9
=
9
=
9
-
-
-
-
18
20
-
-
-
-
21
20
-
-
5
1
-
20
-
23
5
-
20
+
=
153
1+5+3
=
9
=
9
=
9
-
18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
-
14
15
18
20
8
-
19
15
21
20
8
-
5
1
19
20
-
23
5
19
20
+
=
270
2+7+0
=
9
=
9
=
9
-
-
5
6
9
2
8
-
1
6
3
2
8
-
5
1
1
2
-
5
5
1
2
+
=
72
7+2
=
9
=
9
=
9
-
18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
1
-
-
-
-
1
-
-
-
1
occurs
x
4
=
4
=
4
-
-
-
-
-
2
-
-
-
-
-
2
--
-
-
-
-
2
-
-
-
-
2
-
-
2
occurs
x
4
=
8
=
8
-
-
-
-
-
-
-
-
--
-
3
--
-
-
-
--
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
6
-
-
-
-
-
6
--
-
--
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
11
18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
34
-
-
18
-
72
-
36
1+1
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+4
-
-
1+8
-
7+2
-
3+6
2
9
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
7
-
-
9
-
9
-
9
-
-
5
6
9
2
8
-
1
6
3
2
8
-
5
1
1
2
-
5
5
1
2
-
-
-
-
-
-
-
-
-
-
2
9
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
7
-
-
9
-
9
-
9

 

 

18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
5
6
-
-
8
-
1
6
-
-
8
-
-
-
1
-
-
-
-
1
-
+
=
36
3+6
=
9
=
9
=
9
-
14
15
-
-
8
-
19
15
-
-
8
-
-
-
19
-
-
-
-
19
-
+
=
117
1+1+7
=
4
=
9
=
9
18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
-
-
9
2
-
-
-
-
3
2
-
-
5
1
-
2
-
5
5
-
2
+
=
36
3+6
=
9
=
9
=
9
-
-
-
18
20
-
-
-
-
21
20
-
-
5
1
-
20
-
23
5
-
20
+
=
153
1+5+3
=
9
=
9
=
9
18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
14
15
18
20
8
-
19
15
21
20
8
-
5
1
19
20
-
23
5
19
20
+
=
270
2+7+0
=
9
=
9
=
9
-
5
6
9
2
8
-
1
6
3
2
8
-
5
1
1
2
-
5
5
1
2
+
=
72
7+2
=
9
=
9
=
9
18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
1
-
-
-
-
1
-
-
-
1
occurs
x
4
=
4
=
4
-
-
-
-
2
-
-
-
-
-
2
--
-
-
-
-
2
-
-
-
-
2
-
-
2
occurs
x
4
=
8
=
8
-
-
-
-
-
-
-
--
-
3
--
-
-
-
--
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
5
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
6
-
-
-
-
-
6
--
-
--
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
8
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
18
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
34
-
-
18
-
72
-
36
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+4
-
-
1+8
-
7+2
-
3+6
9
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
7
-
-
9
-
9
-
9
-
5
6
9
2
8
-
1
6
3
2
8
-
5
1
1
2
-
5
5
1
2
-
-
-
-
-
-
-
-
-
-
9
N
O
R
T
H
-
S
O
U
T
H
-
E
A
S
T
-
W
E
S
T
-
-
7
-
-
9
-
9
-
9

 

 

18
N
O
R
T
H
S
O
U
T
H
E
A
S
T
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
5
6
-
-
8
1
6
-
-
8
-
-
1
-
-
-
1
-
+
=
36
3+6
=
9
=
9
=
9
-
14
15
-
-
8
19
15
-
-
8
-
-
19
-
-
-
19
-
+
=
117
1+1+7
=
4
=
9
=
9
18
N
O
R
T
H
S
O
U
T
H
E
A
S
T
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
-
-
9
2
-
-
-
3
2
-
5
1
-
2
5
5
-
2
+
=
36
3+6
=
9
=
9
=
9
-
-
-
18
20
-
-
-
21
20
-
5
1
-
20
23
5
-
20
+
=
153
1+5+3
=
9
=
9
=
9
18
N
O
R
T
H
S
O
U
T
H
E
A
S
T
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
14
15
18
20
8
19
15
21
20
8
5
1
19
20
23
5
19
20
+
=
270
2+7+0
=
9
=
9
=
9
-
5
6
9
2
8
1
6
3
2
8
5
1
1
2
5
5
1
2
+
=
72
7+2
=
9
=
9
=
9
18
N
O
R
T
H
S
O
U
T
H
E
A
S
T
W
E
S
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
1
-
-
-
1
-
-
-
1
occurs
x
4
=
4
=
4
-
-
-
-
2
-
-
-
-
2
--
-
-
-
2
-
-
-
2
-
-
2
occurs
x
4
=
8
=
8
-
-
-
-
-
-
--
-
3
--
-
-
--
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
5
-
-
-
-
-
-
-
-
-
5
-
-
-
5
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
6
-
-
-
-
6
--
-
--
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
8
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
18
N
O
R
T
H
S
O
U
T
H
E
A
S
T
W
E
S
T
-
-
34
-
-
18
-
72
-
36
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+4
-
-
1+8
-
7+2
-
3+6
9
N
O
R
T
H
S
O
U
T
H
E
A
S
T
W
E
S
T
-
-
7
-
-
9
-
9
-
9
-
5
6
9
2
8
1
6
3
2
8
5
1
1
2
5
5
1
2
-
-
-
-
-
-
-
-
-
-
9
N
O
R
T
H
S
O
U
T
H
E
A
S
T
W
E
S
T
-
-
7
-
-
9
-
9
-
9

 

 

-
12
D
O
D
E
C
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
-
-
8
-
-
-
6
5
+
=
25
2+5
=
7
=
7
=
7
-
`-
-
15
-
-
-
-
8
-
-
-
15
14
+
=
52
5+2
=
7
=
7
=
7
-
12
D
O
D
E
C
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
-
4
-
4
5
3
1
-
5
4
9
-
-
+
=
35
3+5
=
8
=
8
=
8
-``
`-
4
-
4
5
3
3
-
5
4
18
-
-
+
=
44
4+4
=
8
=
8
=
8
-
12
D
O
D
E
C
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
`-
4
15
4
5
3
3
8
5
4
18
15
14
+
=
96
9+6
=
15
1+5
6
=
6
-
-
4
6
4
5
3
1
8
5
4
9
6
5
+
=
60
6+0
=
6
=
6
=
6
-
12
D
O
D
E
C
A
H
E
D
R
O
N
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
--
4
-
4
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
-
5
-
-
-
5
--
-
5
occurs
x
3
=
15
1+5
6
-
--
-
6
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
9
12
D
O
D
E
C
A
H
E
D
R
O
N
-
-
36
-
-
12
-
60
-
33
-
1+1
-
-
-
-
-
-
-
-
-
9
-
-
-
-
3+6
-
-
1+2
-
6+0
-
3+3
9
3
D
O
D
E
C
A
H
E
D
R
O
N
-
-
9
-
-
3
-
6
-
6
-
-
4
6
4
5
3
1
8
5
4
9
6
5
-
-
3+6
-
-
1+2
-
6+0
-
-
9
3
D
O
D
E
C
A
H
E
D
R
O
N
-
-
9
-
-
3
-
6
-
6

 

 

12
D
O
D
E
C
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
-
6
-
-
-
-
8
-
-
-
6
5
+
=
25
2+5
=
7
=
7
=
7
`-
-
15
-
-
-
-
8
-
-
-
15
14
+
=
52
5+2
=
7
=
7
=
7
12
D
O
D
E
C
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
4
-
4
5
3
1
-
5
4
9
-
-
+
=
35
3+5
=
8
=
8
=
8
`-
4
-
4
5
3
3
-
5
4
18
-
-
+
=
44
4+4
=
8
=
8
=
8
12
D
O
D
E
C
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
`-
4
15
4
5
3
3
8
5
4
18
15
14
+
=
96
9+6
=
15
1+5
6
=
6
-
4
6
4
5
3
1
8
5
4
9
6
5
+
=
60
6+0
=
6
=
6
=
6
12
D
O
D
E
C
A
H
E
D
R
O
N
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
--
4
-
4
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
-
5
-
-
-
5
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
--
-
6
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
--
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
12
D
O
D
E
C
A
H
E
D
R
O
N
-
-
36
-
-
12
-
60
-
33
1+1
-
-
-
-
-
-
-
-
-
9
-
-
-
-
3+6
-
-
1+2
-
6+0
-
3+3
3
D
O
D
E
C
A
H
E
D
R
O
N
-
-
9
-
-
3
-
6
-
6
-
4
6
4
5
3
1
8
5
4
9
6
5
-
-
3+6
-
-
1+2
-
6+0
-
-
3
D
O
D
E
C
A
H
E
D
R
O
N
-
-
9
-
-
3
-
6
-
6

 

 

-
14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
5
-
8
-
-
-
-
-
-
-
+
=
21
2+1
=
3
=
3
=
3
-
`-
-
8
-
-
-
-
14
-
8
-
-
-
-
-
-
-
+
=
30
3+0
=
3
=
3
=
3
-
14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
2
5
-
2
-
-
1
4
1
2
1
9
+
=
34
3+4
=
7
=
7
=
7
-
`-
20
-
5
-
20
5
-
20
-
-
1
22
1
20
1
18
+
=
133
1+3+3
=
7
=
7
=
7
-
14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
-
-
-
--
-
-
-
-
-
`-
20
8
5
-
20
5
14
20
8
-
1
22
1
20
1
18
+
=
163
1+6+3
=
10
1+0
1
=
1
-
-
2
8
5
-
2
5
5
2
8
-
1
4
1
2
1
9
+
=
55
5+5
=
10
1+0
1
=
1
-
14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-T
-
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
1
-
1
-
1
-
-
-
1
occurs
x
3
=
3
=
3
-
-
2
-
-
-
2
-
-
2
-
-
-
-
-
2
-
-
-
-
2
occurs
x
4
=
8
=
8
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
5
-
-
5
5
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
--
-
8
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
=
9
16
14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
29
-
-
14
-
55
-
37
1+6
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
2+9
-
-
1+4
-
5+5
-
3+7
7
5
T
H
E
-
T
E
N
T
H
-
A
V
A
T
O
R
-
-
11
-
-
5
-
10
-
10
-
-
2
8
5
-
2
5
5
2
8
-
1
4
1
2
1
9
-
-
1+1
-
-
-
-
1+0
-
1+0
7
5
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
2
-
-
5
-
1
-
1

 

 

14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
-
5
-
8
-
-
-
-
-
-
-
+
=
21
2+1
=
3
=
3
=
3
`-
-
8
-
-
-
-
14
-
8
-
-
-
-
-
-
-
+
=
30
3+0
=
3
=
3
=
3
14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
-
-
-
--
-
-
-
-
-
2
-
5
-
2
5
-
2
-
-
1
4
1
2
1
9
+
=
34
3+4
=
7
=
7
=
7
`-
20
-
5
-
20
5
-
20
-
-
1
22
1
20
1
18
+
=
133
1+3+3
=
7
=
7
=
7
14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
-
-
-
--
-
-
-
-
`-
20
8
5
-
20
5
14
20
8
-
1
22
1
20
1
18
+
=
163
1+6+3
=
10
1+0
1
=
1
-
2
8
5
-
2
5
5
2
8
-
1
4
1
2
1
9
+
=
55
5+5
=
10
1+0
1
=
1
14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-T
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
1
-
1
-
1
-
-
-
1
occurs
x
3
=
3
=
3
-
2
-
-
-
2
-
-
2
-
-
-
-
-
2
-
-
-
-
2
occurs
x
4
=
8
=
8
--
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
5
5
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
--
-
8
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
=
9
14
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
29
-
-
14
-
55
-
37
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
2+9
-
-
1+4
-
5+5
-
3+7
5
T
H
E
-
T
E
N
T
H
-
A
V
A
T
O
R
-
-
11
-
-
5
-
10
-
10
-
2
8
5
-
2
5
5
2
8
-
1
4
1
2
1
9
-
-
1+1
-
-
-
-
1+0
-
1+0
5
T
H
E
-
T
E
N
T
H
-
A
V
A
T
A
R
-
-
2
-
-
5
-
1
-
1

 

 

16
THE TENTH AVATAR
-
-
-
-
THE
33
15
6
-
TENTH
67
22
4
-
AVATAR
63
18
9
16
THE TENTH AVATAR
163
55
19
1+6
-
1+6+3
5+5
1+9
7
THE TENTH AVATAR
10
10
10
-
-
1+0
1+0
1+0
7
THE TENTH AVATAR
1
1
1

 

 

7
DREAMER
64
46
1
6
READ ME
46
28
1

 

 

6
HEROIC
-
-
-
-
HERO
46
28
1
-
I
9
9
9
-
C
3
3
3
6
HERO
58
40
13
-
-
5+8
4+0
1+3
6
HERO
13
4
4
-
-
1+3
-
-
6
HERO
4
4
4

 

 

4
HERO
46
28
1
1
I
9
9
9
3
SEE
29
11
2
-
-
-
-
-
-
-
-
-
-
8
HERO I SEE
84
48
3
6
HEROIC
58
40
4

 

 

7
LUCIFER
65
38
2
3
THE
33
15
6
5
LIGHT
56
29
2
6
BEARER
49
31
4

 

 

5
LIGHT
56
29
2
4
ISIS
56
20
2

 

 

7
LUCIFER
65
38
2
7
LUCIFRE
-
-
-
-
L+U+C
36
9
9
-
I
9
9
9
-
F+R+E
29
20
2
7
LU-CIFRE
-
-
-

 

 

7
LUCIFER
-
-
-
-
L+U
33
6
6
-
C+I+F+R+E
41
32
5
7
LUCIFER
65
38
11
-
-
6+5
3+8
1+1
7
LUCIFER
11
11
2
-
-
1+1
1+1
-
7
LUCIFER
2
2
2

 

 

7
IMHOTEP
-
-
-
-
I
9
9
9
-
M+H+O
36
18
`9
-
T+E+P
41
14
5
7
IMHOTEP
86
41
23
-
-
8+6
4+1
2+3
7
IMHOTEP
14
5
5
-
-
1+4
-
-
7
IMHOTEP
5
5
5

 

 

-
PHI
-
-
-
-
P
16
7
7
-
H
8
8
8
-
I
9
9
9
3
PHI
33
24
24
-
-
3+3
2+4
2+4
3
PHI
6
6
6

 

►Golden ratio - Wikipedia, the free encyclopedia Mark Barr (20th century) suggests the Greek letter phi (φ), the initial ...
en.wikipedia.org/wiki/Golden_ratio -

Show more results from wikipedia.orgPhi: 1.618. The best source to the golden section, golden mean ... The best information on the Golden Section, Divine Proportion, Fibonacci series and phi. Explore its application to art, design, life, beauty, mathematics, ...
www.goldennumber.net/ -

the golden ratio = 1.61803399
More about calculator.

Search ResultsPhi - Wikipedia, the free encyclopediaPhi (uppercase Φ, lowercase φ or math symbol ϕ), pronounced /ˈfaɪ/ or sometimes /ˈfiː/ in English, and [ˈfi] in modern Greek, is the 21st letter of the ...
en.wikipedia.org/wiki/Phi -

PhiFrom Wikipedia, the free encyclopediaJump to: navigation, search
For other uses, see Phi (disambiguation).

Greek alphabet
Αα Alpha Νν Nu
Ββ Beta Ξξ Xi
Γγ Gamma Οο Omicron
Δδ Delta Ππ Pi
Εε Epsilon Ρρ Rho
Ζζ Zeta Σσς Sigma
Ηη Eta Ττ Tau
Θθ Theta Υυ Upsilon
Ιι Iota Φφ Phi
Κκ Kappa Χχ Chi
Λλ Lambda Ψψ Psi
Μμ Mu Ωω Omega
Other characters
Digamma Stigma
Heta San
Qoppa Sampi
Greek diacritics

Phi (uppercase Φ, lowercase φ or math symbol ϕ), pronounced /ˈfaɪ/ or sometimes /ˈfiː/ in English,[1] and [ˈfi] in modern Greek, is the 21st letter of the Greek alphabet. In modern Greek, it represents [f], a voiceless labiodental fricative. In Ancient Greek it represented [pʰ], an aspirated voiceless bilabial plosive (from which English ultimately inherits the spelling "ph" in words derived from Greek). In the system of Greek numerals it has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) arose from Φ.

Contents [hide]
1 Use as a symbol
2 Computing
3 See also
4 References

Use as a symbolThe lower-case letter (or often its variant, ) is often used to represent the following:

The golden ratio in mathematics, art, and architecture.
Euler's totient function φ(n) in number theory; also called Euler's phi function.
In algebra, group or ring homomorphisms
In probability theory, ϕ(x) = (2π)−½e−x2/2 is the probability density function of the normal distribution.
In probability theory, φX(t) = E[ eitX ] is the characteristic function of a random variable X.
An angle, typically the second angle mentioned, after θ (theta). Especially:
The argument of a complex number.
The phase of a wave in signal processing.
In spherical coordinates, mathematicians usually refer to phi as the polar angle (from the z-axis). The convention in physics is to use phi as the azimuthal angle (from the x-axis).
One of the dihedral angles in the backbones of proteins.
Internal or effective angle of friction.
Electric potential in physics.
The work function in electronics.
A shorthand representation for an aromatic functional group in organic chemistry.
The fugacity coefficient in thermodynamics.
The ratio of free energy destabilizations of protein mutants in phi value analysis.
In cartography and navigation, latitude.
In combustion engineering, equivalence ratio. The ratio between the actual fuel air ratio to the stoichiometric fuel air ratio.
A sentence in first-order logic.
Porosity in geology and hydrology.
Strength (or resistance) reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods.
The symbol for a voiceless bilabial fricative in the International Phonetic Alphabet (using the straight line variant character)
In economics, this is usually an additive term.[clarification needed]
In flight dynamics, the roll angle.
The upper-case letter Φ is used as a symbol for:

The golden ratio conjugate −0.618... in mathematics.
The magnetic flux and electric flux in physics, with subscripts distinguishing the two.
The cumulative distribution function of the normal distribution in mathematics and statistics.
Philosophy.[why?]
The number of phases in a power system in electrical engineering, for example 1Φ for single phase, 3Φ for three phase.
A common symbol for the parametrization of a surface in vector calculus.
In Lacanian algebra, Φ stands for the imaginary phallus and also represents phallic signification. -Φ stands in for castration.[2]
The diameter symbol in engineering, ⌀, is often incorrectly referred to as "phi". This symbol is used to indicate the diameter of a circular section, for example "⌀14", means the diameter of the circle is 14 units.

 

 

-
PHI
-
-
-
-
P
16
7
7
-
H
8
8
8
-
I
9
9
9
3
PHI
33
24
24
-
-
3+3
2+4
2+4
3
PHI
6
6
6

 

 

-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
7
-
-
+
=
7
-
=
7
=
7
-
`-
16
-
-
+
=
16
1+6
=
7
=
7
-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
16
8
9
+
=
33
3+3
=
6
=
6
-
-
7
8
9
+
=
24
2+4
=
6
=
6
-
3
P
H
I
-T
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
ONE
1
-
-
-
2
-
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
-
-
-
-
3
THREE
3
-
-
-
4
-
-
-
-
-
-
4
FOUR
4
-
-
-
5
-
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
-
6
SIX
6
-
-
-
-
--
7
-
-
-
-
7
occurs
x
1
=
7
-
-
-
8
-
-
-
8
occurs
x
1
=
8
-
-
-
-
9
-
-
9
occurs
x
1
=
9
21
3
P
H
I
-
-
24
-
-
3
-
24
2+1
-
-
-
9
-
-
2+4
-
-
-
-
2+4
3
3
P
H
I
-
-
6
-
-
3
-
6
-
-
7
8
9
-T
-
-
-
-
-
-
-
3
3
P
H
I
-
-
6
-
-
3
-
6

 

 

-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
7
-
-
+
=
7
-
=
7
=
7
-
`-
16
-
-
+
=
16
1+6
=
7
=
7
-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
16
8
9
+
=
33
3+3
=
6
=
6
-
-
7
8
9
+
=
24
2+4
=
6
=
6
-
3
P
H
I
-T
-
-
-
-
-
-
-
-
--
7
-
-
-
-
7
occurs
x
1
=
7
-
-
-
8
-
-
-
8
occurs
x
1
=
8
-
-
-
-
9
-
-
9
occurs
x
1
=
9
21
3
P
H
I
-
-
24
-
-
3
-
24
2+1
-
-
-
9
-
-
2+4
-
-
-
-
2+4
3
3
P
H
I
-
-
6
-
-
3
-
6
-
-
7
8
9
-T
-
-
-
-
-
-
-
3
3
P
H
I
-
-
6
-
-
3
-
6

 

 

-
PHI
-
-
-
-
P
16
7
7
-
H
8
8
8
-
I
9
9
9
3
PHI
33
24
24
-
-
3+3
2+4
2+4
3
PHI
6
6
6

 

 

Pi - Wikipedia, the free encyclopediaπ (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in the Euclidean plane; ...

Fundamentals - History - Open questions - Use in mathematics and science
en.wikipedia.org/wiki/Pi -

 

PiFrom Wikipedia, the free encyclopediaJump to: navigation, search
This article is about the number. For the Greek letter, see Pi (letter). For other uses, see Pi (disambiguation).

Mosaic depicting π at the entrance to the math building at Technische Universität BerlinPart of a series of articles on
the mathematical constant π

Uses
Area of disk · Circumference
Use in other formulae
Properties
Irrationality · Transcendence
Less than 22/7
Value
Approximations · Memorization
People
Archimedes · Zu Chongzhi
Madhava of Sangamagrama
William Jones · John Machin
John Wrench
History
Chronology · Book
In culture
Legislation · Holiday
Related topics
Squaring the circle · Basel problem
Other topics related to π
v · d · e

π (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in the Euclidean plane; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159265 in the usual decimal notation. Many formulae from mathematics, science, and engineering involve π, which makes it one of the most important mathematical constants.[1]

π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics. Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture.

Probably because of the simplicity of its definition, the concept of π has become entrenched in popular culture to a degree far greater than almost any other mathematical construct.[2] It is, perhaps, the most common ground between mathematicians and non-mathematicians.[3] Reports on the latest, most-precise calculation of π (and related stunts) are common news items.[4][5][6] The current record for the decimal expansion of π, if verified, stands at 5 trillion digits.[7]

The Greek letter π, often spelled out pi in text, was first adopted for the number as an abbreviation of the Greek word for perimeter "περίμετρος" (or as an abbreviation for "perimeter/diameter") by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse who provided an approximation of the number, although this name for the constant is uncommon in modern English-speaking contexts.

Contents [hide]
1 Fundamentals
1.1 The letter π
1.2 Geometric definition
1.3 Irrationality and transcendence
1.4 Decimal representation
1.5 Estimating π
2 History
2.1 Antiquity
2.2 Second millennium AD
2.3 Computation in the computer age
2.4 Pi and continued fraction
2.5 Memorizing digits
3 Open questions
4 Use in mathematics and science
4.1 Geometry and trigonometry
4.2 Complex numbers and calculus
4.3 Physics
4.4 Probability and statistics
4.5 Geomorphology and chaos theory
5 In popular culture
6 See also
7 References
8 External links

FundamentalsThe letter πMain article: pi (letter)

Lower-case π is used to symbolize the constantThe name of the Greek letter π is pi.[8] The name pi is commonly used as an alternative to using the Greek letter. As a mathematical symbol, the Greek letter is not capitalized (Π) even at the beginning of a sentence, and instead the lower case (π) is used at the beginning of a sentence. When referring to this constant, the symbol π is always pronounced "pie" in English, which is the conventional English pronunciation of the Greek letter. The constant is named "π" because "π" is the first letter of the Greek word περίμετρος (perimeter), probably referring to its use in the formula perimeter/diameter which is constant for all circles, the word "perimeter" being synonymous here with "circumference."[9] William Jones was the first to use the Greek letter in this way, in 1706,[10] and it was later popularized by Leonhard Euler in 1737.[11][12] William Jones wrote:

There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to ... 3.14159, &c. = π[13]

The capital letter pi (Π) has a completely different mathematical meaning; it is used for expressing products (notice that the word "product" begins with the letter "p" just like "perimeter/diameter" does). It can also refer to the osmotic pressure of a solution.

Geometric definitionIn Euclidean plane geometry, π is defined as the ratio of a circle's circumference C to its diameter d:[9]

The ratio C/d is constant, regardless of a circle's size. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d.

Alternatively π can be defined as the ratio of a circle's area A to the area of a square whose side is equal to the radius r of the circle:[9][14]

These definitions depend on results of Euclidean geometry, such as the fact that all circles are similar, and the fact that the right-hand-sides of these two equations are equal to each other (i.e. the area of a disk is Cr/2). These two geometric definitions can be considered a problem when π occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define π without reference to geometry, instead selecting one of its analytic properties as a definition. A common choice is to define π as twice the smallest positive x for which the trigonometric function cos(x) equals zero.[15

(Diagrams omitted)

Irrationality and transcendenceMain article: Proof that π is irrational
π is an irrational number, meaning that it cannot be written as the ratio of two integers. π is also a transcendental number, meaning that there is no polynomial with rational coefficients for which π is a root.[16] An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[17] This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity; many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.[18][19]

Decimal representationSee also: Numerical approximations of π
The decimal representation of π truncated to 50 decimal places is:[20]

π = 3.14159265358979323846264338327950288419716939937510...
Various online web sites provide π to many more digits.[21] While the decimal representation of π has been computed to more than a trillion (1012) digits,[22] elementary applications, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error of less than one millimetre, and the decimal representation of π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.[23][24]

Because π is an irrational number, its decimal representation does not repeat, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating π's properties.[25] Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of the decimal representation of π, no simple base-10 pattern in the digits has ever been found.[26] Digits of the decimal representation of π are available on many web pages, and there is software for calculating the decimal representation of π to billions of digits on any personal computer.

Estimating πMain article: Numerical approximations of π

Number system Approximation of π
Binary 11.00100100001111110110...[27]
Octal 3.11037552421026430215...
Decimal 3.14159265358979323846264338327950288...
Hexadecimal 3.243F6A8885A308D31319...[28]
Rational approximations 3, 22⁄7, 333⁄106, 355⁄113, 103993/33102, ...[29]
(listed in order of increasing accuracy)

Continued fraction [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1...][30]
(This fraction is not periodic. Shown in linear notation)

3.1415926535 8979323846 2643383279 5028841971 6939937510
5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959

An estimate of π accurate to 1120 decimal digits was obtained using a gear-driven calculator in 1948, by John Wrench and Levi Smith. This was the most accurate estimate of π before electronic computers came into use.[31]

The earliest numerical approximation of π is almost certainly the value 3.[32] In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle.

π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, attributed to Archimedes,[33] is to calculate the perimeter, Pn , of a regular polygon with n sides circumscribed around a circle with diameter d. Then compute the limit of a sequence as n increases to infinity:

This limit converges because the more sides the polygon has, the closer the ratio approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range:[34]

π can also be calculated using purely mathematical methods. Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions.[16] Formulas for calculating π using elementary arithmetic typically include series or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to π.[35] The more terms included in a calculation, the closer to π the result will get.

Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series:[36]

 

 

-
PI
-
-
-
-
P
16
7
7
-
I
9
9
9
2
PI
25
16
16
-
-
2+5
1+6
1+6
2
PI
7
7
7

 

 

-
2
P
1
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+7
=
8
=
8
-
-
-
9
+
=
9
1+7
=
8
=
8
-
2
P
1
-
-
-
-
-
-
-
-
-
-
7
-
+
=
7
-
=
7
=
7
-
-
16
-
+
=
16
1+6
=
7
=
7
-
2
P
1
-
-
-
-
-
-
-
-
-
-
16
9
+
=
25
2+5
=
7
=
7
-
-
7
9
+
=
16
1+6
=
7
=
7
-
2
P
I
-T
-
-
-
-
-
-
-
1
-
-
-
-
-
1
ONE
1
-
-
-
2
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
-
-
-
3
THREE
3
-
-
-
4
-
-
-
-
-
4
FOUR
4
-
-
-
5
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
29
2
P
I
-
-
16
-
-
2
-
16
2+9
-
-
9
-
-
2+4
-
-
-
-
1+6
11
2
P
I
-
-
6
-
-
2
-
7
1+1
-
7
9
-T
-
-
-
-
-
-
-
2
2
P
I
-
-
6
-
-
2
-
7

 

 

-
2
P
1
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+7
=
8
=
8
-
-
-
9
+
=
9
1+7
=
8
=
8
-
2
P
1
-
-
-
-
-
-
-
-
-
-
7
-
+
=
7
-
=
7
=
7
-
-
16
-
+
=
16
1+6
=
7
=
7
-
2
P
1
-
-
-
-
-
-
-
-
-
-
16
9
+
=
25
2+5
=
7
=
7
-
-
7
9
+
=
16
1+6
=
7
=
7
-
2
P
I
-T
-
-
-
-
-
-
-
1
-
-
-
-
-
1
ONE
1
-
-
-
2
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
-
-
-
3
THREE
3
-
-
-
4
-
-
-
-
-
4
FOUR
4
-
-
-
5
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
29
2
P
I
-
-
16
-
-
2
-
16
2+9
-
-
9
-
-
2+4
-
-
-
-
1+6
11
2
P
I
-
-
6
-
-
2
-
7
1+1
-
7
9
-T
-
-
-
-
-
-
-
2
2
P
I
-
-
6
-
-
2
-
7

 

 

-
2
P
1
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+7
=
8
=
8
-
-
-
9
+
=
9
1+7
=
8
=
8
-
2
P
1
-
-
-
-
-
-
-
-
-
-
7
-
+
=
7
-
=
7
=
7
-
-
16
-
+
=
16
1+6
=
7
=
7
-
2
P
1
-
-
-
-
-
-
-
-
-
-
16
9
+
=
25
2+5
=
7
=
7
-
-
7
9
+
=
16
1+6
=
7
=
7
-
2
P
I
-T
-
-
-
-
-
-
-
-
-
7
-
-
-
7
occurs
x
1
=
7
-
-
-
9
-
-
9
occurs
x
1
=
9
29
2
P
I
-
-
16
-
-
2
-
16
2+9
-
-
9
-
-
2+4
-
-
-
-
1+6
11
2
P
I
-
-
6
-
-
2
-
7
1+1
-
7
9
-T
-
-
-
-
-
-
-
2
2
P
I
-
-
6
-
-
2
-
7

 

 

-
7
S
A
M
S
A
R
A
-
-
-
-
-
--
-
-
-
-
1
-
-
1
-
-
-
+
=
2
-
=
2
=
2
-
-
19
-
-
19
-
-
-
+
=
38
3+8
=
11
1+1
2
-
7
S
A
M
S
A
R
A
-
-
-
-
-
--
-
-
-
-
-
1
4
-
1
9
1
+
=
16
1+6
=
7
=
7
-
-
-
1
13
-
1
18
1
+
=
34
3+4
=
7
=
7
-
7
S
A
M
S
A
R
A
-
-
-
-
-
--
-
-
-
-
19
1
13
19
1
18
1
+
=
72
7+2
=
9
=
9
-
-
1
1
4
1
1
9
1
+
=
18
1+8
=
9
=
9
-
7
S
A
M
S
A
R
A
-
-
-
-
-
--
-
-
-
-
1
1
-
1
1
-
1
-
-
1
occurs
x
5
=
5
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
31
7
S
A
M
S
A
R
A
-
-
14
-
-
7
-
18
3+1
-
-
-
-
-
-
9
-
-
-
1+4
-
-
-
-
1+8
4
7
S
A
M
S
A
R
A
-
-
5
-
-
7
-
9
-
-
1
1
4
1
1
9
1
+
=
18
1+8
=
9
=
9
4
7
S
A
M
S
A
R
A
-
-
5
-
-
7
-
9

 

 

7
S
A
M
S
A
R
A
-
-
-
-
-
--
-
-
-
1
-
-
1
-
-
-
+
=
2
-
=
2
=
2
-
19
-
-
19
-
-
-
+
=
38
3+8
=
11
1+1
2
7
S
A
M
S
A
R
A
-
-
-
-
-
--
-
-
-
-
1
4
-
1
9
1
+
=
16
1+6
=
7
=
7
-
-
1
13
-
1
18
1
+
=
34
3+4
=
7
=
7
7
S
A
M
S
A
R
A
-
-
-
-
-
--
-
-
-
19
1
13
19
1
18
1
+
=
72
7+2
=
9
=
9
-
1
1
4
1
1
9
1
+
=
18
1+8
=
9
=
9
7
S
A
M
S
A
R
A
-
-
-
-
-
--
-
-
-
1
1
-
1
1
-
1
-
-
1
occurs
x
5
=
5
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
7
S
A
M
S
A
R
A
-
-
14
-
-
7
-
18
-
-
-
-
-
-
9
-
-
-
1+4
-
-
-
-
1+8
7
S
A
M
S
A
R
A
-
-
5
-
-
7
-
9
-
1
1
4
1
1
9
1
+
=
18
1+8
=
9
=
9
7
S
A
M
S
A
R
A
-
-
5
-
-
7
-
9

 

 

-
5
C
H
E
S
S
-
-
-
-
-
-
-
-
-
-
-
8
-
1
1
+
=
10
1+0
=
1
=
1
-
-
-
8
-
19
19
+
=
46
4+6
=
10
1+0
1
-
5
C
H
E
S
S
-
-
-
-
-
-
-
-
-
-
3
-
5
-
-
+
=
8
-
=
8
=
8
-
-
3
-
5
-
-
+
=
8
-
=
8
=
8
-
5
C
H
E
S
S
-
-
-
-
-
-
-
-
-
-
3
8
5
19
19
+
=
54
5+4
=
9
=
9
-
-
3
8
5
1
1
+
=
18
1+8
=
9
=
9
-
5
C
H
E
S
S
-
-
-
-
-
-
-
-
-
-
-
1
1
1
1
-
-
1
occurs
x
2
=
1
2
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
4
-
-
-
-
-
3
occurs
x
1
=
3
4
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
9
9
-
-
-
-
-
9
occurs
x
1
=
9
27
5
C
H
E
S
S
-
-
18
-
-
5
-
18
2+7
-
9
9
-
-
-
-
-
1+8
-
-
-
-
1+8
9
5
C
H
E
S
S
-
-
9
-
-
5
-
9
-
-
3
8
5
1
1
-
-
-
-
-
-
-
-
9
5
C
H
E
S
S
-
-
9
-
-
5
-
9

 

PLATO 64 PLATO

 

5
C
H
E
S
S
-
-
-
-
-
-
-
-
-
-
8
-
1
1
+
=
10
1+0
=
1
=
1
-
-
8
-
19
19
+
=
46
4+6
=
10
1+0
1
5
C
H
E
S
S
-
-
-
-
-
-
-
-
-
3
-
5
-
-
+
=
8
-
=
8
=
8
-
3
-
5
-
-
+
=
8
-
=
8
=
8
5
C
H
E
S
S
-
-
-
-
-
-
-
-
-
3
8
5
19
19
+
=
54
5+4
=
9
=
9
-
3
8
5
1
1
+
=
18
1+8
=
9
=
9
5
C
H
E
S
S
-
-
-
-
-
-
-
-
-
-
1
1
1
1
-
-
1
occurs
x
2
=
1
-
3
4
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
-
9
9
-
-
-
-
-
9
occurs
x
1
=
9
5
C
H
E
S
S
-
-
18
-
-
5
-
18
-
9
9
-
-
-
-
-
1+8
-
-
-
-
1+8
5
C
H
E
S
S
-
-
9
-
-
5
-
9
-
3
8
5
1
1
-
-
-
-
-
-
-
-
5
C
H
E
S
S
-
-
9
-
-
5
-
9

 

 
 
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