Previous Page
   Next Page
 
Evokation
  
 
Index
  

 

A

MAZE

IN

ZAZAZA ENTER ZAZAZA

ZAZAZAZAZAZAZAAZAZAZAZAZAZAZ

ZAZAZAZAZAZAZAZAZAAZAZAZAZAZAZAZAZAZ

THE

MAGICALALPHABET

ABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBA

 12345678910111213141516171819202122232425262625242322212019181716151413121110987654321

 

 

F
=
6
9
FRATERNAL
95
41
5
G
=
7
9
GREETINGS
104
50
5
C
=
3
8
CHILDREN
73
46
1
O
=
6
2
OF
21
12
3
T
=
2
3
THE
33
15
6
R
=
9
7
RAINBOW
82
37
1
L
=
3
5
LIGHT
56
29
2
-
-
36
43
First Total
464
230
23
-
-
3+6
4+3
Add to Reduce
4+6+4
2+3+0
2+3
Q
-
9
7
Second Total
14
5
5
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
9
7
Essence of Number
5
5
5

 

 

 

WORK DAYS OF GOD

Herbert W Morris D.D.circa 1883

Page 22

"As all the words in the English language are composed out of the twenty-six letters of the alphabet,.."

 

 

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."

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

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

A
B
C
D
E
F
G
H
I
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
J
K
L
M
N
O
P
Q
R
10
11
12
13
14
15
16
17
18
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
S
T
U
V
W
X
Y
Z
I
19
20
21
22
23
24
25
26
9
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
ME
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
I
ME
I
ME
I
ME
I
ME
I
9
18
9
18
9
18
9
18
9
=
1+8
=
1+8
=
1+8
=
1+8
=
=
9
=
9
=
9
=
9
=
I
ME
I
ME
I
ME
I
ME
1
9
9
9
9
9
9
9
9
9
I
ME
I
ME
I
ME
I
ME
1

 

 

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

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
-
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

 

 

A

HISTORY OF GOD

Karen Armstrong

The God of the Mystics

Page 250

"(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."  

 

 

A
B
C
D
E
F
G
H
I
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
J
K
L
M
N
O
P
Q
R
10
11
12
13
14
15
16
17
18
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
S
T
U
V
W
X
Y
Z
I
19
20
21
22
23
24
25
26
9
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
ME
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
I
ME
I
ME
I
ME
I
ME
I
9
18
9
18
9
18
9
18
9
=
1+8
=
1+8
=
1+8
=
1+8
=
=
9
=
9
=
9
=
9
=
I
ME
I
ME
I
ME
I
ME
1
9
9
9
9
9
9
9
9
9
I
ME
I
ME
I
ME
I
ME
1

 

 

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

 

 

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
-
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

 

 

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 4320 or 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"

 

 

 

 

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
-
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

 

 

 

 

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
-
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN FINE AMAZE

THE

ZED ALIZ ZED

IN SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

AT THE THROW OF THE NINTH RAM WHEN IN CONJUNCTION SET

THE

FAR YONDER SCRIBE

MADE RECORD OF THEIR FALL

 

 

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 RAINBOW LIGHT
-
-
-
3
THE
33
15
6
7
RAINBOW
82
37
1
5
LIGHT
56
29
2
15
THE RAINBOW LIGHT
171
81
9
1+5
-
1+7+1
8+1
-
6
THE RAINBOW LIGHT
9
9
9

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
-
-
-
-
5
6
-
-
-
1
-
-
-
-
6
-
8
+
=
43
4+3
=
7
-
7
-
7
-
-
-
-
-
-
-
-
8
9
-
-
-
-
14
15
-
-
-
19
-
-
-
-
24
-
26
+
=
115
1+1+5
=
7
-
7
-
7
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
1
2
3
4
-
-
7
8
9
-
2
3
4
5
-
7
-
+
=
83
8+3
=
11
1+1
2
-
2
-
1
2
3
4
5
6
7
-
-
10
11
12
13
-
-
16
17
18
-
20
21
22
23
-
25
-
+
=
236
2+3+6
=
11
1+1
2
-
2
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
-
9
-
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
3
=
3
-
3
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
3
=
6
-
6
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
3
=
9
-
9
-
-
-
-
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
-
-
+
=
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
+
=
9
occurs
x
2
=
18
1+8
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
45
-
-
26
-
126
-
54
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
4+5
-
-
2+6
-
1+2+6
-
5+4
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
-
-
-
-
-
-
-
-
-
-
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9

 

 

3
3
-
A+B+C
6
6
6
2
-
2
D+E
9
9
9
3
3
-
F+G+H
21
21
3
1
-
1
I
9
9
9
3
3
-
J+K+L
33
6
6
2
-
2
M+N
27
9
9
2
2
-
O+P
31
13
4
3
-
3
QRS
54
18
9
3
-
3
TUV
63
9
9
3
-
3
WXY
72
18
9
1
1
-
Z
26
8
8
26
12
14
First Total
351
126
81
2+6
1+2
1+4
Add to Reduce
3+5+1
1+2+6
8+1
8
3
5
Reduce to Deduce
9
9
9

 

 

3
A+B+C
6
6
6
-
D+E
-
-
-
3
F+G+H
21
21
3
-
I
-
-
-
3
J+K+L
33
6
6
-
M+N
-
-
-
2
O+P
31
13
4
-
QRS
-
-
-
-
TUV
-
-
-
-
WXY
-
-
-
1
Z
26
8
8
12
First Total
117
54
27
2+6
Add to Reduce
1+1+7
5+4
2+7
8
Reduce to Deduce
9
9
9

 

 

-
A+B+C
-
-
-
2
D+E
9
9
9
-
F+G+H
-
-
-
1
I
9
9
9
-
J+K+L
-
-
-
2
M+N
27
9
9
-
O+P
-
-
-
3
QRS
54
18
9
3
TUV
63
9
9
3
WXY
72
18
9
-
Z
-
-
-
14
First Total
234
72
54
1+4
Add to Reduce
2+3+4
7+2
5+4
5
Reduce to Deduce
9
9
9

 

 

-

4

ZERO

64

28

1
1
3

ONE
34
16
7
2
3

TWO
58
13
4
3
5

THREE
56
29
2
4
4

FOUR
60
24
6
5
4

FIVE
42
24
6
6
3

SIX
52
16
7
7
5

SEVEN
65
20
2
8
5

EIGHT
49
31
4
9
4

NINE
42
24
6
45
40
-
522
225
45
4+5
4+0
-
5+2+2
2+2+5
4+5
9
4
-
9
9
9

 

 

ADVENT 663 ADVENT

 

 

THE

ZED ALIZ ZED

AGAIN IMAGES JUST THE RIGHT AMOUNT OF MAGI MAGIC

 

 

HAMLET'S MILL

AN ESSAY INVESTIGATING THE ORIGINS OF HUMAN KNOWLEDGE

AND ITS TRANSMISSION THROUGH MYTH

Giorgio De Santillana and Hertha Von Dechend 1969

Intoduction

Page 1 (number omitted)

"The unbreakable fetters which bound down the Great Wolf Fenrir had been cunningly forged by Loki from these: the footfall of a cat, the roots of a rock, the beard of a woman, the breath of a fish, the spittle of a bird. The Edda

Toute vue des choses qui n'est pas estrange est fausse. VALERY


THIS IS meant to be only an essay. It is a first reconnaissance of a realm well-nigh unexplored and uncharted. From whichever way onc enters it, one is caught in the same bewildering circular complexity, as in a labyrinth, for it has no deductive order in the abstract sense, but instead resembles an organism tightly closed in itself, or even better, a monumental "Art of the Fugue."

The figure of Hamlet as a favorable starting point came by chance. Many other avenues offered themselves, rich in strange symbols and beckoning with great images, but the choice went to Hamlet because he led the mind on a truly inductive quest through a familiar landscape-and one which has the merit of its literary setting. Here is a character deeply present to our awareness, in whom ambiguities and uncertainties, tormented self-questioning and dispassionate insight give a presentiment of the modern mind. His personal drama was that he had to be a hero, but still try to avoid the role Destiny assigned him. His lucid intellect remained above the conflict of motives-in other words, his was and is a truly con­/ Page 2 / temporary consciousness. And yet this character whom the poet made one of us, the first unhappy intellectual, concealed a past as a legendary being, his features predetermined, preshaped by long­standing myth. There was a numinous aura around him, and many clues led up to him. But it was a surprise to find behind the mask an ancient and all-embracing cosmic power-the original master of the dreamed-of first age of the world.
Yet in all his guises he remained strangely himself. The original Amlodhi, * as his name was in Icelandic legend, shows the same characteristics of melancholy and high intellect. He, too, is a son dedicated to avenge his father, a speaker of cryptic but inescapable truths, an elusive carrier of Fate who must yield once his mission is accomplished and sink once more into concealment in the depths of time to which he belongs: Lord of the Golden Age, the Once and Future King.

This essay will follow the figure farther and farther afield, from the Northland to Rome, from there to Finland, Iran, and India; he will appear again unmistakably in Polynesian legend. Many other Dominations and Powers will materialize to frame him within the proper order.

Amlodhi was identified, in the crude and vivid imagery of the Norse, by the ownership of a fabled mill which, in his own time, ground out peace and plenty. Later, in decaying times, it ground out salt; and now finally, having landed at the bottom of the sea, it is grinding rock and sand, creating a vast whirlpool, the Maelstrom (i.e., the grinding stream, from the verb mala, "to grind"), which is supposed to be a way to the land of the dead. This imagery stands, as the evidence develops, for an astronomical process, the secular shifting of the sun through the signs of the zodiac which determines world-ages, each numbering thousands of years. Each age brings a World Era, a Twilight of the Gods. Great structures collapse; pillars topple which supported the great fabric; floods and cataclysms herald the shaping of a new world.
The image of the mill and its owner yielded elsewhere to more / Page 3 / sophisticated ones, more adherent to celestial events. In Plato's powerful mind, the figure stood out as the Craftsman God, the Demiurge, who shaped the heavens; but even Plato did not escape the idea he had inherited, of catastrophes and the periodic rebuilding of the world.

Tradition will show that the measures of a new world had to be procured from the depths of the celestial ocean and tuned with the measures from above, dictated by the "Seven Sages," as they are often cryptically mentioned in India and elsewhere. They turn out to be the Seven Stars of Ursa, which are normative in all cosmological alignments on the starry sphere. These dominant stars of the Far North are peculiarly but systematically linked with those which are considered the operative powers of the cosmos, that is, the planets as they move in different placements and configurations along the zodiac. The ancient Pythagoreans, in their conventional language, called the two Bears the Hands of Rhea (the Lady of Turning Heaven), and called the planets the Hounds of Persephone, Queen of the Underworld. Far away to the south, the mysterious ship Argo with its Pilot star held the depths of the past; and the Galaxy was the Bridge out of Time. These notions appear to have been common doctrine in the age before history-all over the belt of high civilizations around our globe. They also seem to have been born of the great intellectual and technological revolution of the late Neolithic period.

The intensity and richness, the coincidence of details, in this cumulative thought have led to the conclusion that it all had its origin in the Near East. It is evident that this indicates a diffusion of ideas to an extent hardly countenanced by current anthropology. But this science, although it has dug up a marvelous wealth of details, has been led by its modern evolutionary and psychological bent to forget about the main source of myth, which was astronomy -the Royal Science. This obliviousness is itself a recent turn of events-barely a century old. Today expert philologists tell us that Saturn and Jupiter are names of vague deities, subterranean or atmospheric, superimposed on the planets at a "late" period; they neatly sort out folk origins and "late" derivations, all unaware that planetary periods, sidereal and synodic, were known and rehearsed / Page 4 / in numerous ways by celebrations already traditional in archaic times. If a scholar has never known those periods even from elementary science, he is not in the best position to recognize them when they come up in his material.

Ancient historians would have been aghast had they been told that obvious things were to become unnoticeable. Aristotle was proud to state it as known that the gods were originally stars, even if popular fantasy had later obscured this truth. Little as he believed in progress, he felt this much had been secured for the future. He could not guess that W. D. Ross, his modern editor, would condescendingly annotate: "This is historically untrue." Yet we know that Saturday and Sabbath had to do with Saturn, just as Wednesday and Mercredi had to do with Mercury. Such names are as old as time; as old, certainly, as the planetary heptagram of the Har­ranians. They go back far before Professor Ross' Greek philology. The inquiries of great and meticulous scholars such as Ideler, Lep­sius, Chwolson, BoIl and, to go farther back, of Athanasius Kircher and Petavius, had they only been read carefully, and noted, would have taught several relevant lessons to the historians of culture, but interest shifted to other goals, as can be seen from current anthro­pology, which has built up its own idea of the "primitive" and what came after.

One still reads in that most unscientific of records, the Bible, that God disposed all things by number, weight and measure; ancient Chinese texts say that "the calendar and the pitch pipes have such a close fit, that you could not slip a hair between them." People read it, and think nothing of it. Yet such hints might reveal a world of vast and firmly established complexity, infinitely different from ours. But the experts now are benighted by the current folk fantasy, which is the belief that they are beyond all this-critics without nonsense and extremely wise.

In 1959 I wrote:
The dust of centuries had settled upon the remains of this great world-wide archaic construction when the Greeks came upon the scene. Yet something of it survived in traditional rites, in myths and fairy tales no longer understood. Taken verbally, it matured the / Page 5 / bloody cults intended to procure fertility, based on the belief in a dark universal force of an ambivalent nature, which seems now to monopolize our interest. Yet its original themes could flash out again, preserved almost intact, in the later thought of the Pythagoreans and of Plato.

But they are tantalizing fragments of a lost whole. They make one think of those "mist landscapes" of which Chinese painters are masters, which show here a rock, here a gable, there the tip of a tree, and leave the rest to imagination. Even when the code shall have yielded, when the techniques shall be known, we cannot expect to gauge the thought of those remote ancestors of ours, wrapped as it is in its symbols.

Their words are no more heard again Through lapse of many ages. . .

We think we have now broken part of that code. The thought behind these constructions of the high and far-off times is also lofty, even if its forms are strange. The theory about "how the world began" seems to involve the breaking asunder of a harmony, a kind of cosmogonic "original sin" whereby the circle of the ecliptic (with the zodiac) was tilted up at an angle with respect to the equator, and the cycles of change came into being.

This is not to suggest that this archaic cosmology will show any great physical discoveries, although it required prodigious feats of concentration and computing. What it did was to mark out the unity of the universe, and of man's mind, reaching out to its farthest limits. Truly, man is doing the same today.

Einstein said: "What is inconceivable about the universe, is that it should be at all conceivable." Man is not giving up. When he discovers remote galaxies by the million, and then those quasi-stellar radio sources billions of light-years away which confound his speculation, he is happy that he can reach out to those depths. But he pays a terrible price for his achievement. The science of astro­physics reaches out on a grander and grander scale without losing its footing. Man as man cannot do this. In the depths of space he loses himself and all notion of his significance. He is unable to fit himself into the concepts of today's astrophysics short of schizophrenia. Modern man is facing the nonconceivable. Archaic man, however, kept a firm grip on the conceivable by framing within his cosmos / Page 6 / an order of time and an eschatology that made sense to him and reserved a fate for his soul. Yet it was a prodigiously vast theory, with no concessions to merely human sentiments. It, too, dilated the mind beyond the bearable, although without destroying man's role in the cosmos. It was a ruthless metaphysics.

Not a forgiving universe, not a world of mercy. That surely not. Inexorable as the stars in their courses, miserationis parcissimae, the Romans used to say. Yet it was a world somehow not unmindful of man, one in which there was an accepted place for everything, rightfully and not only statistically, where no sparrow could fall unnoted, and where even what was rejected through its own error would not go down to eternal perdition; for the order of Number and Time was a total order preserving all, of which all were members, gods and men and animals, trees and crystals and even absurd errant stars, all subject to law and measure.

This is what Plato knew, who could still speak the language of archaic myth. He made myth consonant with his thought, as he built the first modern philosophy. We have trusted his clues as landmarks even on occasions when he professes to speak "not quite seriously." He gave us a first rule of thumb; he knew what he was talking about.

Behind Plato there stands the imposing body of doctrine attributed to Pythagoras, some of its formulation uncouth, but rich with the prodigious content of early mathematics, pregnant with a sci­ence and a metaphysics that were to flower in Plato's time. From it come such words as "theorem," "theory," and "philosophy." This in its turn rests on what might be called a proto-Pythagorean phase, spread all over the East but with a focus in Susa. And then there was something else again, the stark numerical computing of Baby­Ion. From it all came that strange principle: "Things are numbers."

Once having grasped a thread going back in time, then the test of later doctrines with their own historical developments lies in their congruence with tradition preserved intact even if half under­stood. For there are seeds which propagate themselves along the jetstream of time.

Page 7

And universality is in itself a test when coupled with a firm design. When something found, say, in China turns up also in Babyionian astrological texts, then it must be assumed to be relevant, for it reveals a complex of uncommon images which nobody could claim had risen independently by spontaneous generation.

Take the origin of music. Orpheus and his harrowing death may be a poetic creation born in more than one instance in diverse places. But when characters who do not play the lyre but blow pipes get themselves flayed alive for various absurd reasons, and their identical end is rehearsed on several continents, then we feel we have got hold of something, for such stories cannot be linked by internal sequence. And when the Pied Piper turns up both in the medieval German myth of Hamelin and in Mexico long before Columbus, and is linked in both places with certain attributes like the color red, it can hardly be a coincidence. Generally, there is little that finds its way into music by chance.

Again, when one finds numbers like 108, or 9 x 13, reappearing under several multiples in the Vedas, in the temples of Angkor, in Babylon, in Heraclitus' dark utterances, and also in the Norse Valhalla, it is not accident.

There is one way of checking signals thus scattered in early data, in lore, fables and sacred texts. What we have used for sources may seem strange and disparate, but the sifting was considered, and it had its reasons. Those reasons will be given later in the chapter on method. I might call it comparative morphology. The reservoir of myth and fable is great, but there are morphological "markers" for what is not mere storytelling of the kind that comes naturally. There is also wonderfully preserved archaic material in "secondary" primitives, like American Indians and West Africans. Then there are courtly stories and annals of dynasties which look like novels: the Feng Shen Yen I, the Japanese Nihongi, the Hawaiian Kumulipo. These are not merely fantasy-ridden fables.

In hard and perilous ages, what information should a well-born man entrust to his eldest son? Lines of descent surely, but what else? The memory of an ancient nobility is the means of preserving the / Page 8 / arcana imperii, the arcana legis and the arcana mundi, just as it was in ancient Rome. This is the wisdom of a ruling class. The Polynesian chants taught in the severely restricted Whare-wananga were mostly astronomy. That is what a liberal education meant then.

Sacred texts are another great source. In our age of print one is tempted to dismiss these as religious excursions into homiletics, but originally they represented a great concentration of attention on material which had been distilled for relevancy through a long period of time and which was considered worthy of being committed to memory generation after generation. The tradition of Celtic Druidism was delivered not only in songs, but also in tree-lore which was much like a code. And in the East, out of complicated games based on astronomy, there developed a kind of shorthand which became the alphabet.

As we follow the clues-stars, numbers, colors, plants, forms, verse, music, structres-a huge framework of connections is re­vealed at many levels. One is inside an echoing manifold where everything responds and everything has a place and a time assigned to it. This is a true edifice, something like a mathematical matrix, a World-Image that fits the many levels, and all of it kept in order by strict measure. It is measure that provides the countercheck, for there is much that can be identified and redisposed from rules like the old Chinese saying about the pitch pipes and the calendar. When we speak of measures, it is always some form of Time that provides them, starting from two basic ones, the solar year and the octave, and going down from there in many periods and intervals, to actual weights and sizes. What modern man attempted in the merely conventional metric system has archaic precedents of great complexity. Down the centuries there comes an echo of Al-Biruni's wondering a thousand years ago, when that prince of scientists discovered that the Indians, by then miserable astronomers, calculated aspects and events by means of stars-and were not able to show him anyone star that he asked for. Stars had become items for them, as they were to become again for Leverrier and Adams, who never troubled to look at Neptune in their life although they had computed and discovered it in 1847. The Mayas and the Aztecs in their / Page 9 / unending calculations seem to have had similar attitudes. The connections were what counted. Ultimately so it was in the archaic universe, where all things were signs and signatures of each other, inscribed in the hologram, to be divined subtly. And Number dominated them all (appendix # I ).

This ancient world moves a little closer if one recalls two great transitional figures who were simultaneously archaic and modern in their habits of thought. The first is Johannes Kepler, who was of the old order in his unremitting calculations and his passionate devotion to the dream of rediscovering the "Harmony of the Spheres." But he was a man of his own time, and also of ours, when this dream began to prefigure the polyphony that led up to Bach. In somewhat the same way, our strictly scientific world view has its counterpart in what John Hollander, the historian of music, has described as "The Untuning of the Sky." The second transitional figure is no less a man than Sir Isaac Newton, the very inceptor of the rigorously scientific view. There is no real paradox in mentioning Newton in this connection. John Maynard Keynes, who knew Newton as well as many of our time, said of him:

Newton was not the first of the Age of Reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual world rather less than 10,000 years ago. . . Why do I call him a magician? Because he looked on the whole universe and all that is in it as a riddle, as a secret which could be read by applying pure thought to certain evidence, certain mystic clues which God had laid about the world to allow a sort of philosopher's treasure hunt to the esoteric brother­hood. He believed that these clues were to be found partly in the evidence of the heavens and in the constitution of elements (and that is what gives the false suggestion of his being an experimental natural philosopher), but also partly in certain papers and traditions handed down by the brethren in an unbroken chain back to the original cryptic revelation in Babylonia. He regarded the universe as a cryptogram set by the Almighty-just as he himself wrapt the discovery of the calculus in a cryptogram when he communicated with Leibniz. By pure thought, by concentration of mind, the riddle, he believed, would be revealed to the initiate.1

Page 10

Lord Keynes' appraisal, written ca. 1942, remains both unconventional and profound. He knew, we all know, that Newton failed. Newton was led astray by his dour sectarian preconceptions. But his undertaking was truly in the archaic spirit, as it begins to appear now after two centuries of scholarly search into many cul­tures of which he could have had no idea. To the few clues he found with rigorous method, a vast number have been added. Still, the wonder remains, the same that was expressed by his great predecessor Galileo:

But of all other stupendous inventions, what sublimity of mind must have been his who conccived how to communicate his most secret thoughts to any othcr person, though very far distant either in time or place, speaking with those who are in the In dies, 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 arrangement of two dozcn little signs upon paper? Let this be the seal of all the admirable inventions of man.

'Way back in the 6th century A.D., Gregoire de Tours was writing: "The mind has lost its cutting edge, we hardly understand the Ancicnts." So much more today, despite our wallowing in mathematics for the million and in sophisticated technology.
It is undeniable that, notwithstanding our Classics Departments' labors, the wilting away of classical studies, the abandonment of any living familiarity with Greek and Latin has cut the ompha­loessa, the umbilical cord which connected our cultUre-at least at its top level-with Greece, in the same manner in which men of the Pythagorean and Orphic tradition were tied up through Plato and a few others with the most ancient Near East. It is beginning to appear that this destruction is leading into a very up-to-date Middle Ages, much worse than the first. People will sneer: "Stop the World, I want to get off." It cannot be changed, however; this is the way it goes when someone or other tampers with the reserved knowledge that science is, and was meant to represent.
But, as Goethe said at the very onset of the Progressive Age, "Noch ist es Tag, da ruhre sich der Mann! Die Nacht tritt ein, wo niemand wirken kann." ("It is still day, let men get up and / Page 11 / going-the night creeps in, when there is nothing doing.") There might come once more some kind of "Renaissance" out of the hopelessly condemned and trampled past, when certain ideas come to life again, and we should not deprive our grandchildren of a last chance at the heritage of the highest and farthest-off times. And if, as looks infinitely probable, even that last chance is passed up in the turmoil of progress, why then one can still think with Poliziano, who was himself a master humanist, that there will be men whose minds find a refuge in poetry and art and the holy tradition "which alone make men free from death and turn them to eternity, so long as the stars will go on, still shining over a world made for­ever silent." Right now, there is still left some daylight in which to undertake this first quick reconnaissance. It will necessarily leave out great and significant areas of material, but even so, it will in­vestigate many unexpected byways and crannies of the past."

Page 2 Note *. The indulgence of specialists is asked for the form of certain transliterations throughout the text; for example, Amlodhi instead of Amlodi, Grotte instead of Grotti, etc. (Ed.)

Page 9 Note 1 1 "Newton the Man," in The Royal Society. Newton Tercentenary Celebrations (1947), p. 29.

 

 

-
CODE
-
-
-
2
CO
18
9
9
2
DE
9
9
9
4
CODE
27
18
18
-
-
2+7
1+8
1+8
4
CODE
9
9
9

 

 

HAMLET'S MILL

AN ESSAY INVESTIGATING THE ORIGINS OF HUMAN KNOWLEDGE

AND ITS TRANSMISSION THROUGH MYTH

Giorgio De Santillana and Hertha Von Dechend 1969

Page 162

"Finally, there is one remarkable and disturbing coincidence from the same direction. It is known that in the final battle of the gods, the massed legions on the side of "order" are the dead warriors, the "Einherier" who once fell in combat on earth and who have been transferred by the Valkyries to reside with Odin in Valhalla-a theme much rehearsed in heroic poetry. On the last day, they issue forth to battle in martial array. Says the Grimnismal (23): "Five hundred gates and forty more-are in the mighty building of Wal­halla-eight hundred 'Einherier' come out of each one gate-on the time they go out on defence against the Wolf."
That makes 432,000 in all, a number of significance from of old.
This number must have had a very ancient meaning, for it is also the number of syllables in the Rigveda. But it goes back to the basic figure 10,800, the number of stanzas in the Rigveda (40 syllables to a stanza) which, together with 108, occurs insistently in Indian tradition. 10,800 is also the number which has been given by Heraclitus for the duration of the Aion, according to Censorinus (De die natali 18), whereas Berossos made the Babylonian Great Year to last 432,000 years. Again, 10,800 is the number of bricks of the Indian fire-altar (Agnicayana). 32
"To quibble away such a coincidence," remarks Schroder, "or to ascribe it to chance, is in my opinion to drive skepticism beyond its limits."33 Shall one add Angkor to the list? It has five gates, and to each of them leads a road, bridging over that water ditch which surrounds the whole place. Each of these roads is bordered by a row of huge stone figures, 108 per avenue, 54 on each side, altogether 540 statues of Deva and Asura, and each row carries a huge Naga / Page 163 / serpent with nine heads. Only, they do not "carry" that serpent, they are shown to "pull" it, which indicates that these 540 statues are churning the Milky Ocean, represented (poorly, indeed) by the water ditch,34 using Mount Mandara as a churning staff, and Vasuki, the prince of the Nagas, as their drilling rope. (Just to prevent misunderstanding: Vasuki had been asked before, and had agreeably consented, and so had Vishnu's tortoise avatar, who was going to serve as the fixed base for that "incomparably mighty churn," and even the Milky Ocean itself had made it clear that it was willing to be churned.) The whole of Angkor thus turns out to be a colossal model set up for "alternative motion" with true Hindu fantasy and incongruousness to counter the idea of a continuous one-way Precession from west to east."

 

 

CITY OF REVELATION

John Michell 1972

Page 77

CHAPTER SEVEN

3168, The Perimeter of the Temple

"If the numbers of the sacred principles, mentioned by St John in connection with the New Jerusalem, are obtained from the Greek text by the cabalistic method of gematria, it is found that they correspond to the dimensions of the city, set out in Fig 16. (Figure omitted) For example, the perimeter of a hexagon contained within the circle representing the earth, 7920 feet in diameter, measures 2376 feet, and 2376 is the number of (Greek text omitted), the twelve apostles of the Lamb (Revelation 21.14). 2376 x 2 feet is equal to 1746 MY, and 1745 = (Greek text omitted), the twelve apostles. The names of the apostles are said to be in the twelve foundations of the wall of the city. The wall is the circle of diameter 7920 feet and 14,400 cubits in circumference, and the foundations are the twelve corners of the double hexagon inscribed within it, fonowing the customary pattern of an astrological chart. The position of the twelve apostles in the scheme is thus clearly defined.
Of all the canonical numbers the most notable is 3168. The New Jerusalem measures 48,000 furlongs or 31,680,000 feet round the perimeter of its four sides; the mean perimeter of the Stonehenge sarsen circle is 316.8 feet; the perimeter of the square 12 hides of Glastonbury is 31,680 feet; the significance of 31,680 in the canon of cosmology is illustrated in Fig.11, and we shall also find this number set round the border of Plato's mystical city, described in Laws.
Obviously the number 3168 had an important symbolic meaning, the Christian interpretation of which is provided in New Testament
gematria. The most sacred name of Christianity is (Greek text omitted);
(Greek text omitted), Lord Jesus Christ, and the number of these three words together is 3168. (Greek text omitted) is an astrological term meaning the ruler or dominant influence.
Another sacred phrase from the New Testament, (Greek text omitted) the Power of Christ (2 Corinthians 12.9) has the value 3168 if the alternative spelling of Christos, (Greek text omitted) is adopted.

Page 78

The perimeter of the temple is 3168, Lord Jesus Christ, when the temple is measured by the foot, the most sacred unit of ancient metrology. In terms of the megalithic yard (2.72 feet), however, the perimeter measures 1164, because 3168 feet = 1164 MY. Yet this makes no difference to the symbolic interpretation by gematria, for 1164 is the number of another name of Christ, (Greek text omitted) Son of God.

As a geodetic or earth-measuring number, 3168 also demonstrates the antiquity and sacred origin of British metrology, for
31,680 inches = half a mile

31,680 ft. = 6 miles.

31,680 furlongs = 3960 miles = radius of the earth.

31,680 miles = perimeter of square containing the terrestrial sphere.

31,680 miles = circumference of circle drawn on the combined diameters of the earth and moon (10,080 miles)

Other cosmological correspondences of 3168 are given on page 109.

The Stonehenge sarsen circle with circumference of 316.8 feet
contains an area of 888 square yards, 888 being the number of Jesus, which is equal to 1080 square MY. The circle contained within a square of perimeter 316.8 feet, corresponding to the bluestone circle at Stonehenge, has an area of 666 square MY. Thus the two stone circles at Stonehenge have areas of 1080 and 666 square MY, these two numbers representing the opposite poles of lunar and solar or negative and positive energy.
The number 144 or 122 is characteristic of the New Jerusalem scheme, and 3168 demonstrates the value of (pi symbol 22/7 omitted) in terms of this number, for 144 x 7 = 1008 and 144 x 22 = 3168.

3168 in Plato's city
A remarkable use of the number 3168 occurs in Plato's account in Book V of.Laws of the mystical dimensions of the perfect city. Throughout his work Plato makes guarded reference to a secret canon of numbers that applies universally to every aspect of human life and activity, including government, astronomy, acoustics, kinetics, plane and solid geometry and divination. Linear measurements, areas and volumes are obviously incommensurable, but Plato declares that there are certain numbers that link these with each other and with all phenomena capable of being measured. As an example of these numbers, the study of which Plato recommends as the most sanctifying of all pursuits, he gives 5040. This is the ideal number of citizens in the state and serves other purposes in con­/ Page 79 / nection with the framing of laws and standards. The reason why it is most suitable for all matters of division is that for its size it has the greatest number of divisors, 60 in all, including the entire decad, the numbers 1 - 10. Another property of the number 5040 is that it is the radius of a circle with circumference 31,680. Further examination of the numerical foundations of Plato's state shows that the scheme to which he refers is the ancient plan of the cosmic temple.
The lawgivers in Plato's state are reminded that the perfect human society would be one in which all possessions, wives, children, land and chattels were held in common, where all the citizens were of one mind and acted together so harmoniously that it were as if eyes, ears and hands were also common property. To keep this ideal alive is the function of the prophet. Human nature and conditioning, however, demand a more practical alternative, 'very near to the first in immortality and second to it in merit'. This is provided in Laws V.
Plato's state is arranged in a manner that can scarcely be under­stood literally, and is obviously intended, like the New Jerusalem, as a geometer's allegory. The land is all divided into twelve parts, each dedicated to one of the twelve gods and populated by one of the twelve tribes of the 5040 households. The city is similarly divided, forming a microcosm of the state as a whole. In the centre of the city is the acropolis and 'from this centre he must divide up the city itself and the whole country into twelve parts. The twelve parts must be equalised by making those of good land small and those of inferior land greater. He must mark off 5040 allotments, and each of these he must cut in two and join two pieces to form the allotments, so that each contains a near piece and a distant piece - joining the piece next to the city with the piece furthest off, the second nearest with the second furthest, and so on with the rest.'
The only way in which this division can be represented is by a circle of radius 5040, a hundred times larger than that of Stonehenge measured in feet; the perimeter of this circle is 31 ,680. In Fig. 24 (Figure 24 omitted) the radius of the circle should be divided equally into 5040 parts to produce 5040 concentric circles. These are bisected into 10,080 semi­circles by the diameter and positioned out in Plato's manner into 5040 double allotments, each of equal area.
In this scheme 31,680 is not only the circumference of the circular state, but also the area of each of its 2520 pairs of rings, proving Plato's assertion that linear and area pleasurements can be made / Page 80 / (Figure 24 omitted) commensurable by number. The entire circle is divided into two halves, each containing 39,916,800 square units of land. These numbers, which are inherent in the New Jerusalem scheme, have the following significance:
31,680 is divisible by all the numbers1-12 with the exception of 7

5040 = 1 x 2 x 3 x 4 x 5 x 6 x 7

39,916,800 = 1 x 2 x 3 x 4 x 5x 6 x 7 x 8 x 9 x.10 x 11

5040, the radius of the circular city, is the product of the numbers1 - 7; 7920, the side of the square city, is the product of numbers 8 - 11. In each case the perimeter of the city is 31,680. In Plato's Republic is the famous, cryptic reference to the 'marriage number', which should be consulted by the guardians of the state in all matters relating to the seasonal union of male and female. There appear to be two numbers involved, adding up to a third, but the riddle is so obscure that no firm solution has been reached despite the vast literature on the subject. For various reasons the number 12,960,000 or 36002 is most commonly proposed, and this would seem appropriate, for 12,960 = 5040 + 7920. 12,960 therefore represents the union of square and circle, symbol of the sacred marriage, and the gematria is also appropriate, for 1296 = (Greek text omitted) Mary mother of Jesus.

FIGURE 24 (Figure omitted) Plato's city divided into 5040 rings, Perimeter = 31,680, Areas: A + a = B + b = C + c = 31,680.

 

 

-
FIFTY FOUR
-
-
-
1
F
6
6
6
1
I
9
9
9
1
F
6
6
6
2
TY
45
9
9
1
F
6
6
6
2
OUR
54
18
9
9
FIFTY FOUR
126
54
45
-
-
1+2+6
5+4
4+5
9
FIFTY FOUR
9
9
9

 

 

-
FIFTY FOUR
-
-
-
1
F
6
6
6
1
I
9
9
9
1
F
6
6
6
2
TY
45
9
9
1
F
6
6
6
2
OU
36
9
9
1
R
18
9
9
9
FIFTY FOUR
126
54
45
-
-
1+2+6
5+4
4+5
9
FIFTY FOUR
9
9
9

 

 

-
9
F
I
F
T
Y
-
F
O
U
R
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
6
-
-
+
=
15
1+5
=
6
=
6
=
6
-
-
-
9
-
-
-
-
-
15
-
-
+
=
24
2+4
=
6
=
6
=
6
-
9
F
I
F
T
Y
-
F
O
U
R
-
-
-
-
-
-
-
-
-
-
-
-
6
-
6
2
7
-
6
-
3
9
+
=
39
3+9
=
12
1+2
3
=
3
-
-
6
-
6
20
25
-
6
-
21
18
+
=
102
1+0+2
=
3
=
3
=
3
-
9
F
I
F
T
Y
-
F
O
U
R
-
-
-
-
-
-
-
-
-
-
-
-
6
9
6
20
25
-
6
15
21
18
+
=
126
1+2+6
=
9
=
9
=
9
-
-
6
9
6
2
7
-
6
6
3
9
+
=
54
5+4
=
9
=
9
=
9
-
9
F
I
F
T
Y
-
F
O
U
R
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
--
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
6
-
6
-
-
-
6
6
-
-
-
-
6
occurs
x
4
=
24
2+4
6
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
18
9
F
I
F
T
Y
-
F
O
U
R
-
-
27
-
-
9
-
54
-
27
1+8
-
6
-
6
-
-
-
6
6
-
-
-
-
2+7
-
-
-
-
5+4
-
2+7
9
9
F
I
F
T
Y
-
F
O
U
R
-
-
9
-
-
9
-
9
-
9
-
-
6
9
6
2
7
-
6
6
3
9
-
-
-
-
-
-
-
-
-
-
9
9
F
I
F
T
Y
-
F
O
U
R
-
-
9
-
-
9
-
9
-
9

 

 

9
F
I
F
T
Y
-
F
O
U
R
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
6
-
-
+
=
15
1+5
=
6
=
6
=
6
-
-
9
-
-
-
-
-
15
-
-
+
=
24
2+4
=
6
=
6
=
6
9
F
I
F
T
Y
-
F
O
U
R
-
-
-
-
-
-
-
-
-
-
-
6
-
6
2
7
-
6
-
3
9
+
=
39
3+9
=
12
1+2
3
=
3
-
6
-
6
20
25
-
6
-
21
18
+
=
102
1+0+2
=
3
=
3
=
3
9
F
I
F
T
Y
-
F
O
U
R
-
-
-
-
-
-
-
-
-
-
-
6
9
6
20
25
-
6
15
21
18
+
=
126
1+2+6
=
9
=
9
=
9
-
6
9
6
2
7
-
6
6
3
9
+
=
54
5+4
=
9
=
9
=
9
9
F
I
F
T
Y
-
F
O
U
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
-
6
-
6
-
-
-
6
6
-
-
-
-
6
occurs
x
4
=
24
2+4
6
-
-
-
-
-
7
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
9
-
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
9
F
I
F
T
Y
-
F
O
U
R
-
-
27
-
-
9
-
54
-
27
-
6
-
6
-
-
-
6
6
-
-
-
-
2+7
-
-
-
-
5+4
-
2+7
9
F
I
F
T
Y
-
F
O
U
R
-
-
9
-
-
9
-
9
-
9
-
6
9
6
2
7
-
6
6
3
9
-
-
-
-
-
-
-
-
-
-
9
F
I
F
T
Y
-
F
O
U
R
-
-
9
-
-
9
-
9
-
9

 

 

-
FIFTY FOUR
-
-
-
1
F
6
6
6
1
I
9
9
9
1
F
6
6
6
2
TY
45
9
9
1
F
6
6
6
2
OU
36
9
9
1
R
18
9
9
9
FIFTY FOUR
126
54
45
-
-
1+2+6
5+4
4+5
9
FIFTY FOUR
9
9
9

 

 

2
FIFTY FOUR
126
54
9
1
FORTYFIVE
126
54
9

 

 

1
I
9
9
9
4
PLAY
54
18
9
5
PLATO
64
19
1
5
CHESS
54
18
9

 

 

9
FIFTY FOUR
126
54
9

 

 

9
FIFTY FOUR
126
54
9

 

"Plato declares that there are certain numbers that link these with each other and with all phenomena capable of being measured. As an example of these numbers, the study of which Plato recommends as the most sanctifying of all pursuits, he gives 5040."

 

 

CITY OF REVELATION

John Michell 1972

CHAPTER SEVEN

3168, The Perimeter of the Temple

 

3168 in Plato's city

A remarkable use of the number 3168 occurs in Plato's account in Book V of.Laws of the mystical dimensions of the perfect city. Throughout his work Plato makes guarded reference to a secret canon of numbers that applies universally to every aspect of human life and activity, including government, astronomy, acoustics, kinetics, plane and solid geometry and divination. Linear measurements, areas and volumes are obviously incommensurable, but Plato declares that there are certain numbers that link these with each other and with all phenomena capable of being measured. As an example of these numbers, the study of which Plato recommends as the most sanctifying of all pursuits, he gives 5040. This is the ideal number of citizens in the state and serves other purposes in con­/Page 79/nection with the framing of laws and standards. The reason why it is most suitable for all matters of division is that for its size it has the greatest number of divisors, 60 in all, including the entire decad, the numbers 1-10. Another property of the number 5040 is that it is the radius of a circle with circumference 31,680.

 

FIFTYFOUR = 126 = FIFTYFOUR

FIFTYFOUR = 54 = FIFTYFOUR

FIFTYFOUR = 9 = FIFTYFOUR

 

9
FIFTY FOUR
126
54
9

 

 

-
FIFTY FOUR
-
-
-
1
F
6
6
6
1
I
9
9
9
1
F
6
6
6
2
TY
45
9
9
1
F
6
6
6
2
OU
36
9
9
1
R
18
9
9
9
FIFTY FOUR
126
54
45
-
-
1+2+6
5+4
4+5
9
FIFTY FOUR
9
9
9

 

 

-
FIVE FOUR NINE
-
-
-
1
F
6
6
6
1
I
9
9
9
2
VE
9
9
9
1
F
6
6
6
2
OU
36
9
9
1
R
18
9
9
1
N
14
5
5
1
I
9
9
9
1
N
14
5
5
1
E
5
5
5
12
FIVE FOUR NINE
144
72
72
1+2
-
1+4+4
7+2
7+2
3
FIVE FOUR NINE
9
9
9

 

 

-
FIVE FOUR NINE
-
-
-
1
F
6
6
6
1
I
9
9
9
2
VE
9
9
9
1
F
6
6
6
2
OU
36
9
9
1
R
18
9
9
1
NINE
42
24
6
12
FIVE FOUR NINE
144
72
72
1+2
-
1+4+4
7+2
7+2
3
FIVE FOUR NINE
9
9
9

 

 

-
FIVE FOUR NINE
-
-
-
1
F
6
6
6
2
IVE
36
18
9
1
F
6
6
6
2
OUR
54
18
9
1
NINE
42
24
6
12
FIVE FOUR NINE
144
72
72
1+2
-
1+4+4
7+2
7+2
3
FIVE FOUR NINE
9
9
9

 

 

-
FIVE FOUR NINE
-
-
-
1
FIVE
42
24
6
1
FOUR
60
24
6
1
NINE
42
24
6
12
FIVE FOUR NINE
144
72
72
1+2
-
1+4+4
7+2
7+2
3
FIVE FOUR NINE
9
9
9

 

 

-
12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
6
-
-
-
5
9
5
-
+
=
34
3+4
=
7
=
7
=
7
-
-
-
9
-
-
-
-
15
-
-
-
14
9
14
-
+
=
61
6+1
=
7
=
7
=
7
-
12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
6
-
4
5
-
6
-
3
9
-
-
-
-
5
+
=
38
3+8
=
11
1+1
2
=
2
-
-`
6
-
22
5
-
6
-
21
18
-
-
-
-
5
+
=
83
8+3
=
11
1+1
2
=
2
-
12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
6
9
22
5
-
6
6
21
18
-
14
9
14
5
+
=
144
1+4+4
=
9
=
9
=
9
-
-
6
9
4
5
-
6
6
3
9
-
5
9
5
5
+
=
72
7+2
=
9
=
9
=
9
-
12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
--
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
5
-
-
-
-
-
-
5
-
5
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
6
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
18
12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
27
-
-
12
-
72
-
27
1+8
1+2
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
2+7
-
-
1+2
-
7+2
-
2+7
9
3
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
9
-
-
3
-
9
-
9
-
-
6
9
4
5
-
6
6
3
9
-
5
9
5
5
-
-
-
-
-
--
-
-
-
-
9
3
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
9
-
-
3
-
9
-
9

 

 

12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
6
-
-
-
5
9
5
-
+
=
34
3+4
=
7
=
7
=
7
-
-
9
-
-
-
-
15
-
-
-
14
9
14
-
+
=
61
6+1
=
7
=
7
=
7
12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
6
-
4
5
-
6
-
3
9
-
-
-
-
5
+
=
38
3+8
=
11
1+1
2
=
2
-`
6
-
22
5
-
6
-
21
18
-
-
-
-
5
+
=
83
8+3
=
11
1+1
2
=
2
12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
6
9
22
5
-
6
6
21
18
-
14
9
14
5
+
=
144
1+4+4
=
9
=
9
=
9
-
6
9
4
5
-
6
6
3
9
-
5
9
5
5
+
=
72
7+2
=
9
=
9
=
9
12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
--
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
5
-
-
-
-
-
-
5
-
5
5
-
-
5
occurs
x
4
=
20
2+0
2
-
6
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
12
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
27
-
-
12
-
72
-
27
1+2
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
2+7
-
-
1+2
-
7+2
-
2+7
3
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
9
-
-
3
-
9
-
9
-
6
9
4
5
-
6
6
3
9
-
5
9
5
5
-
-
-
-
-
--
-
-
-
-
3
F
I
V
E
-
F
O
U
R
-
N
I
N
E
-
-
9
-
-
3
-
9
-
9

 

 

-
12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
9
-
-
-
5
9
5
-
+
=
34
3+4
=
7
=
7
=
7
-
-
-
15
-
-
-
-
9
-
-
-
14
9
14
-
+
=
61
6+1
=
7
=
7
=
7
-
12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
6
-
3
9
-
6
-
4
5
-
-
-
-
5
+
=
38
3+8
=
11
1+1
2
=
2
-
-`
6
-
21
18
-
6
-
22
5
-
-
-
-
5
+
=
83
8+3
=
11
1+1
2
=
2
-
12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
6
6
21
18
-
6
9
22
5
-
14
9
14
5
+
=
144
1+4+4
=
9
=
9
=
9
-
-
6
6
3
9
-
6
9
4
5
-
5
9
5
5
+
=
72
7+2
=
9
=
9
=
9
-
12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
--
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
-
-
5
-
5
-
5
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
6
6
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
18
12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
27
-
-
12
-
72
-
27
1+8
1+2
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
2+7
-
-
1+2
-
7+2
-
2+7
9
3
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
9
-
-
3
-
9
-
9

 

 

12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
9
-
-
-
5
9
5
-
+
=
34
3+4
=
7
=
7
=
7
-
-
15
-
-
-
-
9
-
-
-
14
9
14
-
+
=
61
6+1
=
7
=
7
=
7
12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
6
-
3
9
-
6
-
4
5
-
-
-
-
5
+
=
38
3+8
=
11
1+1
2
=
2
-`
6
-
21
18
-
6
-
22
5
-
-
-
-
5
+
=
83
8+3
=
11
1+1
2
=
2
12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
6
6
21
18
-
6
9
22
5
-
14
9
14
5
+
=
144
1+4+4
=
9
=
9
=
9
-
6
6
3
9
-
6
9
4
5
-
5
9
5
5
+
=
72
7+2
=
9
=
9
=
9
12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
--
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
-
5
-
5
-
5
5
-
-
5
occurs
x
4
=
20
2+0
2
-
6
6
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
12
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
27
-
-
12
-
72
-
27
1+2
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
2+7
-
-
1+2
-
7+2
-
2+7
3
F
O
U
R
-
F
I
V
E
-
N
I
N
E
-
-
9
-
-
3
-
9
-
9

 

 

-
PLATO
-
-
-
2
PL
28
10
1
3
ATO
36
9
9
5
PLATO
64
19
10
-
--
6+4
1+9
1+0
5
PLATO
10
10
1
-
-
1+0
1+0
-
5
PLATO
1
1
1

 

 

5
P
L
A
T
O
-
-
-
-
-
-
-
-
-
-
-
-
-
6
+
=
6
-
=
6
=
6
-
-
-
-
-
15
+
=
15
1+5
=
6
=
6
5
P
L
A
T
O
-
-
-
-
-
-
-
-
-
7
3
1
2
-
+
=
13
1+3
=
4
=
4
-
16
12
1
20
-
+
=
49
4+9
=
13
1+3
4
5
P
L
A
T
O
-
-
-
-
-
-
-
-
-
16
12
1
20
15
+
+
64
6+4
=
10
1+0
1
-
7
3
1
2
6
+
+
19
1+9
=
10
1+0
1
5
P
L
A
T
O
-
-
-
-
-
--
-
-
-
-
-
1
-
-
+
=
1
-
=
1
-
1
-
-
-
A
-
-
-
-
-
-
-
-
-
-
-
-
3
-
2
-
+
=
5
-
=
5
-
5
-
-
L
-
T
-
-
-
-
-
-
-
-
-
-
7
-
-
-
6
+
=
13
1+3
=
4
-
4
-
P
-
-
-
O
-
-
-
-
-
-
-
-
5
P
L
A
T
O
-
-
-
-
-
--
-
-

 

 

-
5
P
L
A
T
O
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
+
=
6
-
=
6
=
6
-
-
-
-
-
-
15
+
=
15
1+5
=
6
=
6
-
5
P
L
A
T
O
-
-
-
-
-
-
-
-
-
-
7
3
1
2
-
+
=
13
1+3
=
4
=
4
-
-
16
12
1
20
-
+
=
49
4+9
=
13
1+3
4
-
5
P
L
A
T
O
-
-
-
-
-
-
-
-
-
-
16
12
1
20
15
+
+
64
6+4
=
10
1+0
1
-
-
7
3
1
2
6
+
+
19
1+9
=
10
1+0
1
-
5
P
L
A
T
O
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
3
occurs
x
1
=
3
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
occurs
x
1
=
6
-
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
26
5
P
L
A
T
O
-
-
19
-
-
5
-
19
2+6
-
-
-
-
-
-
-
-
1+9
-
-
-
-
1+9
8
5
P
L
A
T
O
-
-
10
-
-
5
-
10
-
-
7
3
1
2
6
-
-
1+0
-
-
-
-
1+0
8
5
P
L
A
T
O
-
-
1
-
-
5
-
1

 

 

5
P
L
A
T
O
-
-
-
-
-
--
-
-
-
-
-
-
-
6
+
=
6
-
=
6
=
6
-
-
-
  
-
15
+
=
15
1+5
=
6
=
6
5
P
L
A
T
O
-
-
-
-
-
--
-
-
-
7
3
1
2
-
+
=
13
1+3
=
4
=
4
-
16
12
1
20
-
+
=
49
4+9
=
13
1+3
4
5
P
L
A
T
O
-
-
-
-
-
--
-
-
-
16
12
1
20
15
+
+
64
6+4
=
10
1+0
1
-
7
3
1
2
6
+
+
19
1+9
=
10
1+0
1
5
P
L
A
T
O
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
-
-
3
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
6
-
-
6
occurs
x
1
=
6
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
5
P
L
A
T
O
-
-
19
-
-
5
-
19
-
-
-
-
-
-
-
-
1+9
-
-
-
-
1+9
5
P
L
A
T
O
-
-
10
-
-
5
-
10
-
7
3
1
2
6
-
-
1+0
-
-
-
-
1+0
5
P
L
A
T
O
-
-
1
-
-
5
-
1

 

 

 

3
THE
33
15
6
4
FIVE
42
24
6
8
PLATONIC
90
36
9
6
SOLIDS
78
24
6
21
First Total
243
99
27
2+1
Add to Reduce
2+4+3
9+9
2+7
3
Second Total
9
18
9
-
Reduce to Deduce
-
1+8
-
3
Essence of Number
9
9
9

 

 

21
T
H
E
-
F
I
V
E
-
P
L
A
T
O
N
I
C
-
S
O
L
I
D
S
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
9
-
-
-
-
-
-
-
6
5
9
-
-
1
6
-
9
-
1
+
=
54
5+4
=
9
-
9
-
9
-
-
8
-
-
-
9
-
-
-
-
-
-
-
15
14
9
-
-
19
15
-
9
-
19
+
=
117
1+1+7
=
9
-
9
-
9
21
T
H
E
-
F
I
V
E
-
P
L
A
T
O
N
I
C
-
S
O
L
I
D
S
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
6
-
4
5
-
7
3
1
2
-
-
-
3
-
-
-
3
-
4
-
+
=
45
4+5
=
9
-
9
-
9
-
20
-
5
-
6
-
22
5
-
16
12
1
20
-
-
-
3
-
-
-
12
-
4
-
+
=
126
1+2+6
=
9
-
9
-
9
21
T
H
E
-
F
I
V
E
-
P
L
A
T
O
N
I
C
-
S
O
L
I
D
S
-
-
-
-
-
-
-
-
-
-
-
20
8
5
-
6
9
22
5
-
16
12
1
20
15
14
9
3
-
19
15
12
9
4
19
+
=
243
2+4+3
=
9
=
9
-
9
-
2
8
5
-
6
9
4
5
-
7
3
1
2
6
5
9
3
-
1
6
3
9
4
1
+
=
99
9+9
=
18
1+8
9
-
9
21
T
H
E
-
F
I
V
E
-
P
L
A
T
O
N
I
C
-
S
O
L
I
D
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
``-
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
3
-
-
-
3
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
2
=
8
-
8
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
6
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
21
T
H
E
-
F
I
V
E
-
P
L
A
T
O
N
I
C
-
S
O
L
I
D
S
-
-
45
-
-
21
-
99
-
36
2+1
-
-
-
-
-
9
-
-
-
-
-
-
-
-``
-
9
-
-
-
-``
-
9
-
-
-
-
4+5
-
-
2+1
-
9+9
-
3+6
3
T
H
E
-
F
I
V
E
-
P
L
A
T
O
N
I
C
-
S
O
L
I
D
S
-
-
9
-
-
3
-
18
-
9
-
2
8
5
-
6
9
4
5
-
7
3
1
2
6
5
9
3
-
1
6
3
9
4
1
-
-
-
-
-
-
-
1+8
-
-
3
T
H
E
-
F
I
V
E
-
P
L
A
T
O
N
I
C
-
S
O
L
I
D
S
-
-
9
-
-
3
-
9
-
9

 

 

11
TETRAHEDRON
128
56
2
10
OCTAHEDRON
103
49
4
11
ICOSAHEDRON
111
57
3
10
HEXAHEDRON
102
57
3
12
DODECAHEDRON
96
60
6
54
-
540
279
18

Page 78

"Plato declares that there are certain numbers that link these with each other and with all phenomena capable of being measured. As an example of these numbers, the study of which Plato recommends as the most sanctifying of all pursuits, he gives 5040."

 

11
TETRAHEDRON
128
56
2
10
OCTAHEDRON
103
49
4
11
ICOSAHEDRON
111
57
3
10
HEXAHEDRON
102
57
3
12
DODECAHEDRON
96
60
6
54
First Total
540
279
18
5+4
Add to Reduce
5+4+0
2+7+9
1+8
9
Second Total
9
18
9
-
Reduce to Deduce
-
1+8
-
9
Essence of Number
9
9
9

Page78

"As an example of these numbers, the study of which Plato recommends as the most sanctifying of all pursuits, he gives 5040."

 

9
FIFTY FOUR
126
54
9

 

 

11
TETRAHEDRON
128
56
2
10
OCTAHEDRON
103
49
4
11
ICOSAHEDRON
111
57
3
10
HEXAHEDRON
102
57
3
12
DODECAHEDRON
96
60
6
54
-
540
-
-

 

 

THE DIVINE PROPORTION

A STUDY IN MATHEMATICAL BEAUTY

H. E. Huntley 1970

Page 31

THE FIVE PLATONIC SOLIDS

 

3
THE
33
15
6
4
FIVE
42
24
6
8
PLATONIC
90
36
9
6
SOLIDS
78
24
6
21
First Total
243
99
27
2+1
Add to Reduce
2+4+3
9+9
2+7
3
Second Total
9
18
9
-
Reduce to Deduce
-
1+8
-
3
Essence of Number
9
9
9

 

The Greeks took a mystical view of the five regular solids. This is hardly surprising for the forms are beautiful in themselves. No mathematical sophistication is needed for the appreciation of the charm of their outward appearance: that is the given element of their beauty. By contrast, that which is acquired by training and education demands considerable mental effort.
The five regular solids were treated by Euclid in Book XIII of the Elements but are associated with the name of Plato because of his efforts to relate them to the important entities of which he supposed the world to be made. The aura of mysticism with which the Greek geometers surrounded them persisted until the dawn in the sixteenth century of the scientific era. But the aesthetic appeal of what are still known as the Platonic solids is undiminished. Next to the writer's chair, for some years past, there has stood a small dodecahedron of white china!
We are now confronted with our first example of beauty in mathematics. The facts have been common knowledge among mathematicians for 2000 years. Much evidence of their aesthetic appeal in the past is on record
The first point to note about the regular Platonic solids shown in figure 2.6 (Page 32 Figures omitted) is that they are precisely five in number. A little thought shows that, while an infinite number of polygons may be
drawn on a plane surface, it is not possible to construct more than five regular polyhedra in three-dimensional space. The surface of a regular polyhedron is bounded by congruent regular polygons. The simplest polygons. that can form the surface are the equilateral triangle, the square and the pentagon. It is clear from figure 2.6 ( Page 32 Figures omitted) that we cannot form the corner of a polyhedron with fewer than three faces and that a corner may be formed by joining three, four, or five equilateral triangles. With six such triangles, the corner flattens into a plane. The same will happen if four squares are united. Similarly, three regular pentagons at a corner is a maximum. But hexagons, and polygons with more than five sides are all ruled out. This argument for the limitation to five regular / Page 33 / solids is the source of Euler's formula V + F = E + 2, where the letters stand for the number of vertices, faces, and edges respectively.
The second point of interest is that two pairs of the Platonic solids are reciprocal and the fifth is self reciprocating in this sense: if the face centers of the cube are joined, an octahedron is formed, while the joins of the centroids of the octahedron surfaces form a cube. Similar relationship holds between the icosahedron and the dodecahedron., The join of the four centroids of the tetrahedron's faces makes another tetrahedron.
Fig. 2.7. Icosahedron (Figures omitted)
The third noteworthy feature is the relationship of the two pairs of reciprocal polyhedra to the golden rectangle (Figs. 2.7 and 2.8).(Figures omitted)
The Icosahedron. The twelve vertices of a regular icosahedron are divisible into three coplanar groups of four. These lie at the corners of three golden rectangles which are symmetrically situated with respect to each other, being mutually perpendicular, their one common point being the centroid of the icosahedron (Fig. 2.7).
The Octahedron. An icosahedron can be inscribed in an octahedron so that each vertex of the former divides an edge of the latter in the golden section.
The Dodecahedron. The centroids of the twelve pentagonal faces of a dodecahedron are divisible into three coplanar groups of / Page 34 / four. These quadrads lie at the corners of three mutually perpendicular, symmetrically placed golden rectangles, their one common point being the centroid of the dodecahedron (Fig. 2.8).
The aesthetic appeal of the topics of this chapter cannot be doubted. Whether we can explain it or not, the fact that they have been enjoyed by sixty generations of men is good evidence.

Fig. 2.8. Dodecahedron (Figures omitted)

It should be realized that we have only examined the surface of the subject. One of the ingredients of beauty in mathematics is its depth. Even in these familiar waters, "full many a gem of purest ray serene" awaits discovery by the explorer. There appears to be no limit, as Sir Edwin Arnold affirms in The Light of Asia:

Shall any gazer see with mortal eyes
Or any searcher know with mortal mind­
Veil after veil will lift-but there must be
Veil after veil behind.

Page 32 (Figures omitted)

 

 

-
TETRAHEDRON
-
-
-
3
T+E+T
45
9
9
1
R
18
9
9
2
A+H
9
9
9
2
E+D
9
9
9
1
R
18
9
9
2
O+N
29
11
2
11
TETRAHEDRON
128
56
47
1+1
-`
1+2+8
5+6
4+7
2
TETRAHEDRON
11
11
11
-
-`
1+1
1+1
1+1
2
TETRAHEDRON
2
2
2

 

 

-
11
T
E
T
R
A
H
E
D
R
O
N
n
-
-
-
-
-
-
-
-
=
-
-
-
-
-
-
-
8
-
-
-
6
5
+
=
19
1+9
=
10
1+0
1
=
1
-
`-
-
-
-
-
-
8
-
-
-
15
14
+
=
37
3+7
=
10
1+0
1
=
1
-
11
T
E
T
R
A
H
E
D
R
O
N
n
-
-
-
-
-
-
-
-
=
-
-
2
5
2
9
1
-
5
4
9
-
-
+
=
37
3+7
=
10
1+0
1
=
1
-
`-
20
5
20
18
1
-
5
4
18
-
-
+
=
91
9+1
=
10
1+0
1
=
1
-
11
T
E
T
R
A
H
E
D
R
O
N
n
-
-
-
-
-
-
-
-
=
-
`-
20
5
20
18
1
8
5
4
18
15
14
+
=
128
1+2+8
=
11
1+1
2
=
2
-
-
2
5
2
9
1
8
5
4
9
6
5
+
=
56
5+6
=
11
1+1
2
=
2
-
11
T
E
T
R
A
H
E
D
R
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
2
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
-
-
5
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
--
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
10
11
T
E
T
R
A
H
E
D
R
O
N
-
-
35
-
-
11
-
56
-
38
1+0
1+1
-
-
-
9
-
-
-
-
9
-
-
-
-
3+5
-
-
1+1
-
5+6
-
3+8
1
2
T
E
T
R
A
H
E
D
R
O
N
-
-
8
-
-
2
-
11
-
11
-
-
2
5
2
9
1
8
5
4
9
6
5
-
-
-
-
-
-
-
1+1
-
1+1
1
2
T
E
T
R
A
H
E
D
R
O
N
-
-
8
-
-
2
-
2
-
2

 

 

-
11
T
E
T
R
A
H
E
D
R
O
N
n
-
-
-
-
-
-
-
-
=
-
-
-
-
-
-
-
8
-
-
-
6
5
+
=
19
1+9
=
10
1+0
1
=
1
-
`-
-
-
-
-
-
8
-
-
-
15
14
+
=
37
3+7
=
10
1+0
1
=
1
-
11
T
E
T
R
A
H
E
D
R
O
N
n
-
-
-
-
-
-
-
-
=
-
-
2
5
2
9
1
-
5
4
9
-
-
+
=
37
3+7
=
10
1+0
1
=
1
-
`-
20
5
20
18
1
-
5
4
18
-
-
+
=
91
9+1
=
10
1+0
1
=
1
-
11
T
E
T
R
A
H
E
D
R
O
N
n
-
-
-
-
-
-
-
-
=
-
`-
20
5
20
18
1
8
5
4
18
15
14
+
=
128
1+2+8
=
11
1+1
2
=
2
-
-
2
5
2
9
1
8
5
4
9
6
5
+
=
56
5+6
=
11
1+1
2
=
2
-
11
T
E
T
R
A
H
E
D
R
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
2
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
-
-
5
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
=
6
-
--
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
10
11
T
E
T
R
A
H
E
D
R
O
N
-
-
35
-
-
11
-
56
-
38
1+0
1+1
-
-
-
9
-
-
-
-
9
-
-
-
-
3+5
-
-
1+1
-
5+6
-
3+8
1
2
T
E
T
R
A
H
E
D
R
O
N
-
-
8
-
-
2
-
11
-
11
-
-
2
5
2
9
1
8
5
4
9
6
5
-
-
-
-
-
-
-
1+1
-
1+1
1
2
T
E
T
R
A
H
E
D
R
O
N
-
-
8
-
-
2
-
2
-
2

 

 

 

-
OCTAHEDRON
-
-
-
3
OC
18
9
9
1
T
20
2
2
2
A+H
9
9
9
2
E+D
9
9
9
1
R
18
9
9
2
O+N
29
11
2
11
OCTAHEDRON
103
49
47
1+1
-`
1+0+3
4+9
4+7
2
OCTAHEDRON
4
13
11
-
-`
4
1+3
1+1
2
OCTAHEDRON
4
4
4

 

 

-
10
O
C
T
A
H
E
D
R
O
N
n
-
-
-
-
-
-
-
-
=
-
-
6
-
-
-
8
-
-
-
6
5
+
=
25
2+5
=
7
=
7
=
7
-
`-
15
-
-
-
8
-
-
-
15
14
+
=
52
5+2
=
7
=
7
=
7
-
10
O
C
T
A
H
E
D
R
O
N
n
-
-
-
-
-
-
-
-
=
-
-
-
3
2
1
-
5
4
9
-
-
+
=
24
2+4
=
6
=
6
=
6
-
`-
-
3
20
1
-
5
4
18
-
-
+
=
51
5+1
=
10
=
6
=
6
-
10
O
C
T
A
H
E
D
R
O
N
n
-
-
-
-
-
-
-
-
=
-
`-
15
3
20
1
8
5
4
18
15
14
+
=
103
1+0+3
=
4
=
4
=
4
-
-
6
3
2
1
8
5
4
9
6
5
+
=
49
4+9
=
13
1+3
4
=
4
-
10
O
C
T
A
H
E
D
R
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
5
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
6
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
--
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
7
10
O
C
T
A
H
E
D
R
O
N
-
-
38
-
-
10
-
49
-
31
-
1+0
-
-
-
-
-
-
-
9
-
-
-
-
3+8
-
-
1+0
-
4+9
-
3+1
7
1
O
C
T
A
H
E
D
R
O
N
-
-
11
-
-
1
-
13
-
4
-
-
6
3
2
1
8
5
4
9
6
5
-
-
1+1
-
-
-
-
1+3
-
-
7
1
O
C
T
A
H
E
D
R
O
N
-
-
8
-
-
1
-
4
-
4

 

 

 

-
ICOSAHEDRON
-
-
-
1
I
9
9
9
2
C+O
18
9
9
1
S
19
1
1
2
A+H
9
9
9
2
E+D
9
9
9
1
R
18
9
9
2
O+N
29
11
2
11
ICOSAHEDRON
111
57
48
1+1
-`
1+1+1
5+7
4+8
2
ICOSAHEDRON
3
12
12
-
-`
-
1+2
1+2
2
ICOSAHEDRON
3
3
3

 

 

-
11
I
C
O
S
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
-
9
-
6
1
-
8
-
-
-
6
5
+
=
35
3+5
=
8
=
8
=
8
-
`-
9
-
15
19
-
8
-
-
-
15
14
+
=
80
8+0
=
8
=
8
=
8
-
11
I
C
O
S
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
-
-
3
-
-
1
-
5
4
9
-
-
+
=
22
3+5
=
4
=
4
=
4
-``
`-
-
3
-
-
1
-
5
4
18
-
-
+
=
31
3+1
=
4
=
4
=
4
-
11
I
C
O
S
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
`-
9
3
15
19
1
8
5
4
18
15
14
+
=
111
1+1+1
=
3
=
3
=
3
-
-
9
3
6
1
1
8
5
4
9
6
5
+
=
57
5+7
=
12
1+2
3
=
3
-
11
I
C
O
S
A
H
E
D
R
O
N
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
--
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
-
-
-
5
--
-
5
occurs
x
2
=
10
1+0
1
-
--
-
-
6
-
-
-
-
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
--
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
9
=
9
9
11
I
C
O
S
A
H
E
D
R
O
N
-
-
36
-
-
11
-
48
-
30
-
1+1
9
-
-
-
-
-
-
-
9
-
-
-
-
3+6
-
-
1+1
-
4+8
-
3+0
9
2
I
C
O
S
A
H
E
D
R
O
N
-
-
9
-
-
2
-
12
-
3
-
-
9
3
6
1
1
8
5
4
9
6
5
-T
-
-
-
-
-
-
-
-
-
9
2
I
C
O
S
A
H
E
D
R
O
N
-
-
9
-
-
2
-
3
-
3

 

 

-
11
I
C
O
S
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
-
9
-
6
1
-
8
-
-
-
6
5
+
=
35
3+5
=
8
=
8
=
8
-
`-
9
-
15
19
-
8
-
-
-
15
14
+
=
80
8+0
=
8
=
8
=
8
-
11
I
C
O
S
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
-
-
3
-
-
1
-
5
4
9
-
-
+
=
22
3+5
=
4
=
4
=
4
-``
`-
-
3
-
-
1
-
5
4
18
-
-
+
=
31
3+1
=
4
=
4
=
4
-
11
I
C
O
S
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
`-
9
3
15
19
1
8
5
4
18
15
14
+
=
111
1+1+1
=
3
=
3
=
3
-
-
9
3
6
1
1
8
5
4
9
6
5
+
=
57
5+7
=
12
1+2
3
=
3
-
11
I
C
O
S
A
H
E
D
R
O
N
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
--
-
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
-
-
-
5
--
-
5
occurs
x
2
=
10
1+0
1
-
--
-
-
6
-
-
-
-
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
-
--
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
9
=
9
9
11
I
C
O
S
A
H
E
D
R
O
N
-
-
36
-
-
11
-
48
-
30
-
1+1
9
-
-
-
-
-
-
-
9
-
-
-
-
3+6
-
-
1+1
-
4+8
-
3+0
9
2
I
C
O
S
A
H
E
D
R
O
N
-
-
9
-
-
2
-
12
-
3
-
-
9
3
6
1
1
8
5
4
9
6
5
-T
-
-
-
-
-
-
-
-
-
9
2
I
C
O
S
A
H
E
D
R
O
N
-
-
9
-
-
2
-
3
-
3

 

 

 

-
HEXAHEDRON
-
-
-
3
H+E+X
37
19
1
2
A+H
9
9
9
2
E+D
9
9
9
1
R
18
9
9
2
O+N
29
11
2
10
HEXAHEDRON
102
57
30
1+0
-`
1+0+2
5+7
3+0
1
HEXAHEDRON
3
12
3
-
-`
-
1+2
-
1
HEXAHEDRON
3
3
36

 

 

-
12
H
E
X
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
-
8
-
6
-
8
-
-
-
6
5
+
=
33
3+3
=
6
=
6
=
6
-
`-
8
-
24
-
8
-
-
-
15
14
+
=
69
6+9
=
15
1+5
6
=
6
-
12
H
E
X
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
1
-
5
4
9
-
-
+
=
24
2+4
=
8
=
8
=
8
-``
`-
-
5
-
1
-
5
4
18
-
-
+
=
33
3+3
=
8
=
8
=
8
-
12
H
E
X
A
H
E
D
R
O
N
-
-
-
-
-
--
-
-
-
-
-
`-
8
5
24
1
8
5
4
18
15
14
+
=
102
1+0+2
=
3
-
3
=
3
-
-
8
5
6
1
8
5
4
9
6
5
+
=
57
5+7
=
12
1+2
3
=
3
-
12
H
E
X
A
H
E
D
R
O
N
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
--
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
-
5
-
-
-
5
--
-
5
occurs
x
3
=
15
1+5
6
-
--
-
-
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
12
10
H
E
X
A
H
E
D
R
O
N
-
-
33
-
-
10
-
57
-
30
1+2
1+0
-
-
-
-
-
-
-
9
-
-
-
-
3+3
-
-
1+0
-
5+7
-
3+0
3
1
H
E
X
A
H
E
D
R
O
N
-
-
6
-
-
1
-
12
-
3
-
-
8
5
6
1
8
5
4
9
6
5
-
-
-
-
-
-
-
1+2
-
-
3
1
H
E
X
A
H
E
D
R
O
N
-
-
6
-
-
1
-
3
-
3

 

 

12
H
E
X
A
H
E
D
R
O
N
-
-
-
-
-
-
-
-
-
-
-
8
-
6
-
8
-
-
-
6
5
+
=
33
3+3
=
6
=
6
=
6
`-
8
-
24
-
8
-
-
-
15
14
+
=
69
6+9
=
15
1+5
6
=
6
12
H
E
X
A
H
E
D
R
O
N
-
-
-
-
-
-
-
-
-
-
-
-
5
-
1
-
5
4
9
-
-
+
=
24
2+4
=
8
=
8
=
8
`-
-
5
-
1
-
5
4
18
-
-
+
=
33
3+3
=
8
=
8
=
8
12
H
E
X
A
H
E
D
R
O
N
-
-
-
-
-
-
-
-
-
-
`-
8
5
24
1
8
5
4
18
15
14
+
=
102
1+0+2
=
3
=
3
=
3
-
8
5
6
1
8
5
4
9
6
5
+
=
57
5+7
=
12
1+2
3
=
3
12
H
E
X
A
H
E
D
R
O
N
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
--
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
-
5
-
-
-
5
--
-
5
occurs
x
3
=
15
1+5
6
--
-
-
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
10
H
E
X
A
H
E
D
R
O
N
-
-
33
-
-
10
-
57
-
30
1+0
-
-
-
-
-
-
-
9
-
-
-
-
3+3
-
-
1+0
-
5+7
-
3+0
1
H
E
X
A
H
E
D
R
O
N
-
-
6
-
-
1
-
12
-
3
-
8
5
6
1
8
5
4
9
6
5
-
-
-
-
-
-
-
1+2
-
-
1
H
E
X
A
H
E
D
R
O
N
-
-
6
-
-
1
-
3
-
3

 

 

 

-
DODECAHEDRON
-
-
-
1
D
4
4
4
6
O+D+E+C+A+H
36
27
9
2
E+D
9
9
9
1
R
18
9
9
2
O+N
29
11
2
12
DODECAHEDRON
96
60
33
1+2
-`
9+6
6+0
3+3
3
DODECAHEDRON
15
6
6
-
-`
1+5
-
-
3
DODECAHEDRON
6
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
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
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
SEVEN
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
-T
-
-
-
-
-
-
-
-
-
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
-T
-
-
-
-
-
-
-
-
-
3
D
O
D
E
C
A
H
E
D
R
O
N
-
-
9
-
-
3
-
6
-
6

 

 

5
PLATO
64
19
1
6
EUCLID
54
27
9
6
HEDRON
64
37
1

 

 

-
11
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
-
-
-
-
9
-
+
=
9
-
=
9
-
9
-
-
-
-
-
-
9
-
+
=
9
-
=
9
-
9
-
11
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
5
3
3
3
-
4
+
=
18
1+8
=
9
-
9
-
-
5
21
3
12
-
4
+
=
45
4+5
=
9
-
9
-
11
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
5
21
3
12
9
4
+
=
54
5+4
=
9
-
9
-
-
5
3
3
3
9
4
+
=
27
2+7
=
9
-
9
-
11
E
U
C
L
I
D
-
-
-
-
-
-
-
-
1
-
-
--
-
--
-
-
--
--
1
ONE
1
--
--
--
2
-
-
--
-
--
--
-
--
--
2
TWO
2
--
--
--
-
-
-
3
3
3
-
-
-
-
3
occurs
x
3
=
9
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
--
-
--
--
-
--
--
6
SIX
6-
--
--
--
7
-
-
--
-
--
--
-
--
--
7
SEVEN
7
--
--
--
8
-
-
--
-
--
--
-
--
--
8
EIGHT
8
--
--
--
-
-
-
-
-
-
9
-
--
--
9
occurs
x
1
=
9
24
11
E
U
C
L
I
D
-
--
21
-
-
6
-
27
2+4
1+1
-
-
-
-
9
-
-
-
2+1
-
-
-
-
2+7
6
2
E
U
C
L
I
D
-
-
3
-
-
6
-
9
-
-
5
3
3
3
9
4
-
-
-
-
-
-
-
-
6
2
E
U
C
L
I
D
-
-
3
-
-
6
-
9

 

 

11
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
-
-
-
9
-
+
=
9
-
=
9
-
9
-
-
-
-
-
9
-
+
=
9
-
=
9
-
9
11
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
5
3
3
3
-
4
+
=
18
1+8
=
9
-
9
-
5
21
3
12
-
4
+
=
45
4+5
=
9
-
9
11
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
5
21
3
12
9
4
+
=
54
5+4
=
9
-
9
-
5
3
3
3
9
4
+
=
27
2+7
=
9
-
9
11
E
U
C
L
I
D
-
-
-
-
-
-
-
-
-
-
3
3
3
-
-
-
-
3
occurs
x
3
=
9
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
9
-
--
--
9
occurs
x
1
=
9
11
E
U
C
L
I
D
-
--
21
-
-
6
-
27
1+1
-
-
-
-
9
-
-
-
2+1
-
-
-
-
2+7
2
E
U
C
L
I
D
-
-
3
-
-
6
-
9
-
5
3
3
3
9
4
-
-
-
-
-
-
-
-
2
E
U
C
L
I
D
-
-
3
-
-
6
-
9

 

 

-
8
H
E
X
A
G
R
A
M
-
-
-
-
-
--
-
-
-
-
8
-
6
-
-
-
-
-
+
=
14
1+4
=
5
=
5
-
-
8
-
24
-
-
-
-
-
+
=
32
3+2
=
5
=
5
-
8
H
E
X
A
G
R
A
M
-
-
-
-
-
--
-
-
-
-
-
5
-
1
7
9
1
4
+
=
27
2+7
=
9
=
9
-
-
-
5
-
1
7
18
1
13
+
=
45
4+5
=
9
=
9
-
8
H
E
X
A
G
R
A
M
-
-
-
-
-
--
-
-
-
-
8
5
24
1
7
18
1
13
+
=
77
7+7
=
14
1+4
5
-
-
8
5
6
1
7
9
1
4
+
=
41
4+1
=
5
=
5
-
8
H
E
X
A
G
R
A
M
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
1
occurs
x
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
6
-
-
-
-
-
=
-
6
occurs
x
1
=
6
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
-
9
-
-
=
-
9
occurs
x
1
=
9
5
8
H
E
X
A
G
R
A
M
-
-
40
-
-
8
-
41
-
-
-
-
-
-
-
9
-
-
-
-
4+0
-
-
-
-
4+1
5
8
H
E
X
A
G
R
A
M
-
-
4
-
-
8
-
5
-
-
8
5
6
1
7
9
1
4
-
-
-
-
-
-
=
-
5
8
H
E
X
A
G
R
A
M
-
-
4
-
-
8
-
5

 

 

5
XAGON
61
25
7
7
HEXAGON
74
38
2
9
HEXAGONAL
87
42
6

 

 

6
XAGRAM
64
28
1
8
HEXAGRAM
77
41
5

 

 

-
-
-
-
-
THE GOLDEN RATIO
-
-
-
T
=
2
-
3
THE
33
15
6
G
=
7
-
6
GOLDEN
57
30
3
R
=
9
-
5
RATIO
63
27
9
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
-
1+8
-
1+4
-
1+5+3
7+2
1+8
-
-
9
-
5
THE GOLDEN RATIO
9
9
9

 

 

-
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
6
-
-
-
5
-
-
-
-
9
6
+
=
34
3+4
=
7
-
7
=
7
-
-
-
8
-
-
-
15
-
-
-
14
-
-
-
-
9
15
+
=
61
6+1
=
7
-
7
=
7
-
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
7
-
3
4
5
-
-
9
1
2
-
-
+
=
38
3+8
=
11
1+1
2
=
2
-
-
20
-
5
-
7
-
12
4
5
-
-
18
1
20
-
-
+
=
92
9+2
=
11
1+1
2
=
2
-
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
-
-
-
-
-
-
-
-
-
-
20
8
5
-
7
15
12
4
5
14
-
18
1
20
9
15
+
=
153
1+5+3
=
9
=
9
=
9
-
-
2
8
5
-
7
6
3
4
5
5
-
9
1
2
9
6
+
=
72
7+2
=
9
=
9
=
9
-
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
2
-
-
-
-
-
-
--
-
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
=
2
3
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
5
-
-
-
-
--
5
5
-
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
--
-
-
-
9
-
-
9
-
-
-
9
occurs
x
2
=
18
1+8
9
45
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
45
-
-
14
-
72
-
36
4+5
1+4
-
-
5
-
-
-
-
--
5
5
-
-
-
-
-
-
-
-
4+5
-
-
1+4
-
7+2
-
3+6
9
5
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
9
-
-
5
-
9
-
9
-
-
2
8
5
-
7
6
3
4
5
5
-
9
1
2
9
6
-
-
-
-
-
-
-
-
-
-
9
5
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
9
-
-
5
-
9
-
9

 

 

THE GOLDEN RATIO

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
95
41
5
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
7
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
7
-
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
8
-
-
L
=
3
6
1
L
12
3
3
-
-
-
3
-
-
-
7
-
-
D
=
4
7
1
D
4
4
4
-
-
-
-
4
7
-
-
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
7
-
-
-
N
=
5
9
1
N
5
5
5
-
-
-
-
-
5
7
-
-
-
R
=
9
10
1
R
9
9
9
-
-
-
3
-
-
-
-
-
9
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
7
-
-
-
I
=
9
13
1
I
9
9
9
-
-
-
3
-
-
-
-
-
9
O
=
6
14
1
O
15
6
6
-
-
-
-
-
-
6
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
72
4
14
THE GOLDEN RATIO
153
72
72
-
1
4
3
4
15
12
7
8
18
-
-
7+2
-
1+4
-
1+5+3
7+2
7+2
-
-
-
-
-
1+5
1+2
-
-
1+8
Q
-
9
-
5
THE GOLDEN RATIO
9
9
9
-
1
4
3
4
6
3
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
95
41
5
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
7
-
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
7
-
-
-
L
=
3
6
1
L
12
3
3
-
-
-
3
-
-
-
7
-
-
D
=
4
7
1
D
4
4
4
-
-
-
-
4
7
-
-
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
7
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
7
-
-
-
N
=
5
9
1
N
5
5
5
-
-
-
-
-
5
7
-
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
8
-
-
O
=
6
14
1
O
15
6
6
-
-
-
-
-
-
6
8
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
R
=
9
10
1
R
9
9
9
-
-
-
3
-
-
-
-
-
9
I
=
9
13
1
I
9
9
9
-
-
-
3
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
72
4
14
THE GOLDEN RATIO
153
72
72
-
1
4
3
4
15
12
7
8
18
-
-
7+2
-
1+4
-
1+5+3
7+2
7+2
-
-
-
-
-
1+5
1+2
-
-
1+8
Q
-
9
-
5
THE GOLDEN RATIO
9
9
9
-
1
4
3
4
6
3
7
8
9

 

 

Sacred Geometry is the theory of dimensional evolution which assumes the ... door to his academy stating,

"Let no one unacquainted with geometry enter here.". ...
ezinearticles.com/?An-Overview-Of-Sacred-Geometry

 

AN OVERVIEW OF SACRED GEOMETRY

Gregg Hall

Sacred Geometry is the theory of dimensional evolution which assumes the universe is a living system kept together by the existence of a sacred geometry that encompasses the entire cosmos and makes for the blueprint for the mani-festation of what we know as our material universe and in addition organizes the context through which all love evolves.

Our universe was designed to be highly efficient and is capable of performing a wide range of multiple functions at the same time. The very same geometry which provides structure to physical reality also allows for the perceptual environments that people and civilizations must move through as part of a systematic learning process on the path towardsevolution.

Each dimension of this sacred geometry holds a unique place of perceptual space and a context of learning both for personal and social evolution. As each new dimension appears a new set of perceptions and potentials is awakened which we are free to accept and actualize or ignore. It is in understanding the dimensional structure which exists all around us that allows us to be able to understand the path and direction of personal and social evolution.

Even though our modern science generally believes there is nothing of deeper meaning to the dimensional geometry of the universe other than the actual physical aspects, there is a view that is almost diametrically opposed to this that began with the Greek philosopher Pythagoras in 500 B.C. Pythagoras believed and taught the theory or belief that all of the mathematical patterns in the universe were actually expressions of divine intelligence and signified a divine intention.

According to Pythagoras, we are surrounded by organizational intelligence that is shown in its purest from through mathematical formulas and musical harmonies and allowing ourselves to be at the center of our experience; we can know and share the organizing patterns and principles that pervade the universe. This is a thought that was even held by Albert Einstein, who stated that he received his greatest breakthroughs after praying and sleeping. The answers to the questions he was seeking came to him from the Universe while he slept! This is also the way that it can be explained for someone who is blind to be able to sculpt and for a deaf person, such as Beethoven to be able to compose intricate musical scores.

Plato, who taught over a hundred years after Pythagoras, continued in the teachings of Pythagorean thought in espousing that the universe or cosmos as Pythagoras termed it was a place of "harmonious and beautiful order" and placed such a high regard on geometry that he placed a sign above the door to his academy stating, "Let no one unacquainted with geometry enter here."

 

"Let no one unacquainted with geometry enter here."

 

L
=
3
-
3
LET
37
10
1
N
=
5
-
2
NO
29
11
2
O
=
6
-
3
ONE
34
16
7
U
=
3
-
11
UNAQUAINTED
127
46
1
W
=
5
-
3
WITH
60
15
6
G
=
7
-
8
GEOMETRY
108
45
9
E
=
5
-
4
ENTER
62
26
8
H
=
8
-
8
HERE
36
27
9
-
-
42
-
42
First Total
493
196
43
-
-
4+2
-
4+2
Add to Reduce
4+9+3
1+9+6
4+3
-
-
6
-
6
Second Total
16
16
7
-
-
-
-
-
Reduce to Deduce
1+6
1+6
-
-
-
6
-
6
Essence of Number
7
7
7

 

 

On Mathematics and Mathematicians

by

Robert Edoward Moritz

- 2007 - Mathematics - 444 pages that Plato had over the entrance to his school of philosophy: "Let no one who is unacquainted with geometry enter here." WHITE, WF A Scrap-book of ... books.google.co.uk/books

"Let no one who is unacquainted with geometry enter here."

 

 

SACRED GEOMETRY

Miranda Lundy 2006

INTRODUCTION

Sacred Geometry charts the unfolding of number in space and differs from mundane geometry in the sense that its moves, concepts and products are regarded as having symbolic value and meaning. Thus, like good music, the study and practice of geometry can facilitate the evolution of the soul. As we shall see, the basic journey is from the single point, into the line, out to the plane, through to the third dimension and beyond, eventually returning to the point again, watching what happens on the way.
These pages cover the elements of two-dimensional geometry—the unfolding of number on a flat surface. The three-dimensional geometrical story is told in another book in the series. This material has been used for a very long time indeed as one introduction to metaphysics. Like the elements of its sister subject, music, it is an aspect of revelation, a bright indisputable shadow of Reality and a creation myth in itself.
Number, Music, Geometry and the study of patterns in the Heavens are the four great Liberal Arts of the ancient world dealing with quanta, or whole numbers. These simple universal languages are as relevant today as they have always been, and may be found in all known sciences and cultures without disagreement. Just above the entrance to Plato's Academy was a sign: "Let none ignorant of geometry enter here." So let's do some research.

 

"Let none ignorant of geometry enter here."

 

L
=
3
-
3
LET
37
10
1
N
=
5
-
4
NONE
48
21
3
I
=
9
-
8
IGNORANT
34
16
7
O
=
6
-
2
OF
46
19
1
G
=
7
-
8
GEOMETRY
108
45
9
E
=
5
-
5
ENTER
62
26
8
H
=
8
-
4
HERE
36
27
9
-
-
43
-
34
First Total
410
176
41
-
-
4+3
-
3+4
Add to Reduce
4+1+0
1+7+6
4+1
-
-
7
-
7
Second Total
5
14
5
-
-
-
-
-
Reduce to Deduce
-
1+4
-
-
-
7
-
7
Essence of Number
5
5
5

 

 

-
PLATO'S ACADEMY
-
-
-
2
P+L
28
10
1
3
A+T+O
36
9
9
1
S
19
10
1
4
A+C+A+D
9
9
9
2
EM
18
9
9
1
Y
25
7
7
13
PLATO'S ACADEMY
135
54
9
1+3
-
1+3+5
5+4
-
13
PLATO'S ACADEMY
9
9
9

 

 

-
PLATO'S ACADEMY
-
-
-
2
P+L+A+T+O+S
83
20
2
4
A+C+A+D+E+M+Y
52
25
7
13
PLATO'S ACADEMY
135
45
9
1+3
-
1+3+5
4+5
-
4
PLATO'S ACADEMY
9
9
9

 

 

-
-
-
-
-
PLATO'S ACADEMY
-
-
-
P
=
7
-
6
PLATO'S
83
29
2
A
=
1
-
7
ACADEMY
52
25
7
-
-
-
-
13
PLATO'S ACADEMY
135
54
9
-
=
-
-
1+3
-
1+3+5
5+4
-
-
4
8
-
4
PLATO'S ACADEMY
9
9
9

 

 

-
-
-
-
-
PLATO'S ACADEMY
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
-
6
PLATO'S
83
29
2
-
-
-
-
-
-
-
-
-
-
A
=
1
-
7
ACADEMY
52
25
7
-
1
2
3
4
5
6
7
8
9
-
-
-
-
13
PLATO'S ACADEMY
135
54
9
-
-
-
-
-
-
-
-
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
A
=
1
3
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
4
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
29
-
-
-
83
29
20
-
-
-
-
-
-
-
-
-
-
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
C
=
3
8
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
A
=
1
9
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
D
=
4
10
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
12
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
Y
=
7
13
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
-
-
37
-
-
-
83
29
20
-
4
2
6
8
5
6
14
8
9
-
-
-
-
-
PLATO'S ACADEMY
-
-
-
-
-
-
-
-
-
-
1+4
-
-
P
=
7
-
6
PLATO'S
83
29
2
-
4
2
6
8
5
6
5
8
9
A
=
1
-
7
ACADEMY
52
25
7
-
-
-
-
-
-
-
-
-
-
-
=
8
-
13
PLATO'S ACADEMY
135
54
9
-
4
2
6
8
5
6
5
8
8
-
=
-
-
1+3
-
1+3+5
5+4
-
-
-
-
-
-
-
-
-
-
-
-
4
8
-
4
PLATO'S ACADEMY
9
9
9
-
4
2
6
8
5
6
5
8
8

 

 

-
-
-
-
-
PLATO'S ACADEMY
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
-
6
PLATO'S
83
29
2
-
-
-
-
-
-
-
-
-
-
A
=
1
-
7
ACADEMY
52
25
7
-
1
2
3
4
5
6
7
8
9
-
-
-
-
13
PLATO'S ACADEMY
135
54
9
-
-
-
-
-
-
-
-
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
8
9
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
8
9
A
=
1
3
1
A
1
1
1
-
1
-
-
-
-
-
-
8
9
T
=
2
4
1
T
20
2
2
-
-
2
-
-
-
-
-
8
9
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
8
9
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
8
9
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
-
8
9
C
=
3
8
1
C
3
3
3
-
-
-
3
-
-
-
-
8
9
A
=
1
9
1
A
1
1
1
-
1
-
-
-
-
-
-
8
9
D
=
4
10
1
D
4
4
4
-
-
-
-
4
-
-
-
8
9
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
8
9
M
=
4
12
1
M
13
4
4
-
-
-
-
4
-
-
-
8
9
Y
=
7
13
1
Y
25
7
7
-
-
-
-
-
-
-
7
8
9
-
-
-
-
-
-
-
-
-
-
4
2
6
8
5
6
14
8
9
-
-
PLATO'S ACADEMY
-
-
-
-
-
-
-
-
-
-
1+4
-
-
P
=
7
-
6
PLATO'S
83
29
2
-
4
2
6
8
5
6
5
8
9
A
=
1
-
7
ACADEMY
52
25
7
-
-
-
-
-
-
-
-
-
-
-
=
8
-
13
PLATO'S ACADEMY
135
54
9
-
4
2
6
8
5
6
5
8
8
-
=
-
-
1+3
-
1+3+5
5+4
-
-
-
-
-
-
-
-
-
-
-
-
4
8
-
4
PLATO'S ACADEMY
9
9
9
-
4
2
6
8
5
6
5
8
8

 

 

-
-
-
-
-
PLATO'S ACADEMY
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
-
6
PLATO'S
83
29
2
-
-
-
-
-
-
-
-
-
-
A
=
1
-
7
ACADEMY
52
25
7
-
1
2
3
4
5
6
7
8
9
-
-
-
-
13
PLATO'S ACADEMY
135
54
9
-
-
-
-
-
-
-
-
-
-
A
=
1
3
1
A
1
1
1
-
1
-
-
-
-
-
-
8
9
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
8
9
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
-
8
9
A
=
1
9
1
A
1
1
1
-
1
-
-
-
-
-
-
8
9
T
=
2
4
1
T
20
2
2
-
-
2
-
-
-
-
-
8
9
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
8
9
C
=
3
8
1
C
3
3
3
-
-
-
3
-
-
-
-
8
9
D
=
4
10
1
D
4
4
4
-
-
-
-
4
-
-
-
8
9
M
=
4
12
1
M
13
4
4
-
-
-
-
4
-
-
-
8
9
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
8
9
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
8
9
Y
=
7
13
1
Y
25
7
7
-
-
-
-
-
-
-
7
8
9
-
-
-
-
-
-
-
-
-
-
4
2
6
8
5
6
14
8
9
-
PLATO'S ACADEMY
-
-
-
-
-
-
-
-
-
-
1+4
-
-
P
=
7
6
PLATO'S
83
29
2
-
4
2
6
8
5
6
5
8
9
A
=
1
7
ACADEMY
52
25
7
-
-
-
-
-
-
-
-
-
-
=
8
13
PLATO'S ACADEMY
135
54
9
-
4
2
6
8
5
6
5
8
8
-
=
-
-
1+3
-
1+3+5
5+4
-
-
-
-
-
-
-
-
-
-
-
-
4
8
-
4
PLATO'S ACADEMY
9
9
9
-
4
2
6
8
5
6
5
8
8

 

 

PLATO'S ACADEMY 135 ACADEMY PLATO'S

PLATO'S ACADEMY 54 ACADEMY PLATO'S

PLATO'S ACADEMY 9 ACADEMY PLATO'S

 
-
PLATO'S ACADEMY
-
-
-
P
=
7
6
PLATO'S
83
29
2
A
=
1
7
ACADEMY
52
25
7
=
8
13
PLATO'S ACADEMY
135
54
9
-
=
-
-
1+3
-
1+3+5
5+4
-
-
4
8
-
4
PLATO'S ACADEMY
9
9
9

 

PLATO'S ACADEMY 54 ACADEMY PLATO'S

ISLE OF ATLANTIS 54 ATLANTIS OF ISLE

 

 

PLATO'S ACADEMY 135 ACADEMY PLATO'S

PLATO'S ACADEMY 54 ACADEMY PLATO'S

PLATO'S ACADEMY 9 ACADEMY PLATO'S

 
-
PLATO'S ACADEMY
-
-
-
P
=
7
6
PLATO'S
83
29
2
A
=
1
7
ACADEMY
52
25
7
=
8
13
PLATO'S ACADEMY
135
54
9
-
=
-
-
1+3
-
1+3+5
5+4
-
-
4
8
-
4
PLATO'S ACADEMY
9
9
9

 

 

LIFE IS GOD IS LIFE

LIFE IS QUALITIVELY THE SAME CREATIVE INTELLIGENCE IN EACH EVOCATION REGARDLESS OF FORM

ALL IS ONE UNIVERSAL CREATOR MIND ALL AND EVERYTHING

ONE UNIVERSAL PATTERN OF LIFE

 

 

PLATO - NUMBERS ARE THE HIGHEST DEGREE OF KNOWLEDGE

IT IS KNOWLEDGE ITSELF

 
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
7
PLATO
64
19
1
-
1
-
-
-
5
-
7
-
9
-
-
7
-
-
64
19
1
1
-
-
-
-
-
-
-
-
-
N
=
5
2
7
NUMBERS
92
29
2
-
-
2
-
-
5
-
7
-
9
A
=
1
3
3
ARE
24
15
6
-
-
-
-
-
5
6
7
-
9
T
=
2
4
3
THE
33
15
6
-
-
-
-
-
5
6
7
-
9
H
=
8
5
7
HIGHEST
76
40
4
-
-
-
-
4
5
-
7
-
9
D
=
4
6
6
DEGREE
44
35
8
-
-
-
-
-
5
-
7
8
9
O
=
6
7
2
OF
21
12
3
-
-
-
3
-
5
-
7
-
9
K
=
2
8
9
KNOWLEDGE
96
42
6
-
-
-
-
-
5
6
7
-
9
-
-
28
-
-
386
188
35
8
-
-
-
-
-
-
-
-
-
-
-
35
-
42
Add to Reduce
450
207
36
-
1
2
3
8
5
18
7
8
9
-
-
3+5
-
4+2
Reduce to Deduce
4+5+0
2+0+7
3+6
-
-
-
-
-
-
1+8
-
-
-
-
-
8
-
6
Essence of Number
9
9
9
-
1
2
3
8
5
9
7
8
9

 

 
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
7
PLATO
64
19
1
-
1
-
-
-
5
-
7
-
9
N
=
5
2
7
NUMBERS
92
29
2
-
-
2
-
-
5
-
7
-
9
A
=
1
3
3
ARE
24
15
6
-
-
-
-
-
5
6
7
-
9
T
=
2
4
3
THE
33
15
6
-
-
-
-
-
5
6
7
-
9
H
=
8
5
7
HIGHEST
76
40
4
-
-
-
-
4
5
-
7
-
9
D
=
4
6
6
DEGREE
44
35
8
-
-
-
-
-
5
-
7
8
9
O
=
6
7
2
OF
21
12
3
-
-
-
3
-
5
-
7
-
9
K
=
2
8
9
KNOWLEDGE
96
42
6
-
-
-
-
-
5
6
7
-
9
-
-
35
-
42
Add to Reduce
450
207
36
-
1
2
3
8
5
18
7
8
9
-
-
3+5
-
4+2
Reduce to Deduce
4+5+0
2+0+7
3+6
-
-
-
-
-
-
1+8
-
-
-
-
-
8
-
6
Essence of Number
9
9
9
-
1
2
3
8
5
9
7
8
9

 

 
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
7
PLATO
64
19
1
-
1
-
-
-
5
-
7
-
9
N
=
5
2
7
NUMBERS
92
29
2
-
-
2
-
-
5
-
7
-
9
O
=
6
7
2
OF
21
12
3
-
-
-
3
-
5
-
7
-
9
H
=
8
5
7
HIGHEST
76
40
4
-
-
-
-
4
5
-
7
-
9
A
=
1
3
3
ARE
24
15
6
-
-
-
-
-
5
6
7
-
9
T
=
2
4
3
THE
33
15
6
-
-
-
-
-
5
6
7
-
9
K
=
2
8
9
KNOWLEDGE
96
42
6
-
-
-
-
-
5
6
7
-
9
D
=
4
6
6
DEGREE
44
35
8
-
-
-
-
-
5
-
7
8
9
-
-
35
-
42
Add to Reduce
450
207
36
-
1
2
3
8
5
18
7
8
9
-
-
3+5
-
4+2
Reduce to Deduce
4+5+0
2+0+7
3+6
-
-
-
-
-
-
1+8
-
-
-
-
-
8
-
6
Essence of Number
9
9
9
-
1
2
3
8
5
9
7
8
9

 

RE 95 RE

REARRANGED NUMERICALLY REARRANGED

RE 95 RE

 
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
9
-
2
IT
29
11
2
-
-
2
3
4
5
-
7
-
9
I
=
9
-
2
IS
28
10
1
-
1
-
3
4
5
-
7
-
9
K
=
2
-
9
KNOWLEDGE
96
42
6
-
-
-
3
4
5
6
7
-
9
I
=
9
-
6
ITSELF
71
26
8
-
-
-
3
4
5
-
7
8
9
-
-
29
-
19
Add to Reduce
224
98
17
-
1
2
3
4
5
6
7
8
9
-
-
2+9
-
1+9
Reduce to Deduce
2+2+4
9+8
1+7
-
-
-
-
-
-
-
-
-
-
-
-
11
-
10
Essence of Number
8
17
8
-
1
2
3
4
5
6
7
8
9
-
-
1+1
-
1+0
Reduce to Deduce
-
1+7
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
1
Essence of Number
8
8
8
-
1
2
3
4
5
6
7
8
9

 

IT IS KNOWLEDGE ITSELF

 
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
9
-
2
IT
29
11
2
-
-
-
-
-
-
-
-
-
-
I
=
9
-
2
IS
28
19
1
-
-
-
-
-
-
-
-
-
-
K
=
2
-
9
KNOWLEDGE
96
42
6
-
-
-
-
-
-
-
-
-
-
I
=
9
-
6
ITSELF
71
26
8
-
-
-
-
-
-
-
-
-
-
-
-
29
-
19
Add to Reduce
224
98
17
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
9
1
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
11
-
-
29
11
11
2
-
-
-
-
-
-
-
-
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
S
=
1
4
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
10
-
-
28
19
10
1
-
-
-
-
-
-
-
-
-
K
=
2
5
1
K
11
2
2
-
-
2
-
-
-
-
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
W
=
5
8
1
W
23
5
5
-
-
-
-
-
5
-
-
8
-
L
=
3
9
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
D
=
4
11
1
D
4
4
4
-
-
-
-
4
-
-
-
8
-
G
=
7
12
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
42
-
-
96
42
42
6
-
-
-
-
-
-
-
-
-
I
=
9
14
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
T
=
2
15
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
S
=
1
16
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
E
=
5
17
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
L
=
3
18
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
F
=
6
19
1
F
6
6
6
-
-
-
-
-
-
6
-
8
-
27
-
-
71
35
26
8
-
-
-
-
-
-
-
-
-
I
=
9
-
2
IT
29
11
2
-
2
6
6
4
25
12
7
8
27
I
=
9
-
2
IS
28
10
1
-
-
-
-
-
2+5
1+2
-
-
2+7
K
=
2
-
9
KNOWLEDGE
96
42
6
-
2
6
6
4
7
3
7
8
9
I
=
9
-
6
ITSELF
71
26
8
-
-
-
-
-
-
-
-
-
-
-
-
29
-
19
First Total
224
98
17
-
2
6
6
4
7
3
7
8
9
-
-
2+9
-
1+9
Add to Reduce
2+2+4
9+8
1+7
-
-
-
-
-
-
-
-
-
-
-
-
11
-
10
Second Total
8
17
8
-
2
6
6
4
7
3
7
8
9
-
-
1+1
-
1+0
Reduce to Deduce
-
1+7
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
1
Essence of Number
8
8
8
-
2
6
6
4
7
3
7
8
9

 

 
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
9
-
2
IT
29
11
2
-
-
-
-
-
-
-
-
-
-
I
=
9
-
2
IS
28
19
1
-
-
-
-
-
-
-
-
-
-
K
=
2
-
9
KNOWLEDGE
96
42
6
-
-
-
-
-
-
-
-
-
-
I
=
9
-
6
ITSELF
71
26
8
-
-
-
-
-
-
-
-
-
-
-
-
29
-
19
Add to Reduce
224
98
17
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
9
1
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
S
=
1
4
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
K
=
2
5
1
K
11
2
2
-
-
2
-
-
-
-
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
W
=
5
8
1
W
23
5
5
-
-
-
-
-
5
-
-
8
-
L
=
3
9
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
D
=
4
11
1
D
4
4
4
-
-
-
-
4
-
-
-
8
-
G
=
7
12
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
I
=
9
14
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
T
=
2
15
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
S
=
1
16
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
E
=
5
17
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
L
=
3
18
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
F
=
6
19
1
F
6
6
6
-
-
-
-
-
-
6
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
-
2
IT
29
11
2
-
2
6
6
4
25
12
7
8
27
I
=
9
-
2
IS
28
10
1
-
-
-
-
-
2+5
1+2
-
-
2+7
K
=
2
-
9
KNOWLEDGE
96
42
6
-
2
6
6
4
7
3
7
8
9
I
=
9
-
6
ITSELF
71
26
8
-
-
-
-
-
-
-
-
-
-
-
-
29
-
19
First Total
224
98
17
-
2
6
6
4
7
3
7
8
9
-
-
2+9
-
1+9
Add to Reduce
2+2+4
9+8
1+7
-
-
-
-
-
-
-
-
-
-
-
-
11
-
10
Second Total
8
17
8
-
2
6
6
4
7
3
7
8
9
-
-
1+1
-
1+0
Reduce to Deduce
-
1+7
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
1
Essence of Number
8
8
8
-
2
6
6
4
7
3
7
8
9

 

RE 95 RE

REARRANGED NUMERICALLY REARRANGED

RE 95 RE

 
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
9
-
2
IT
29
11
2
-
-
-
-
-
-
-
-
-
-
I
=
9
-
2
IS
28
19
1
-
-
-
-
-
-
-
-
-
-
K
=
2
-
9
KNOWLEDGE
96
42
6
-
-
-
-
-
-
-
-
-
-
I
=
9
-
6
ITSELF
71
26
8
-
-
-
-
-
-
-
-
-
-
-
-
29
-
19
Add to Reduce
224
98
17
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
4
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
S
=
1
16
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
K
=
2
5
1
K
11
2
2
-
-
2
-
-
-
-
-
8
-
T
=
2
15
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
9
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
L
=
3
18
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
D
=
4
11
1
D
4
4
4
-
-
-
-
4
-
-
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
W
=
5
8
1
W
23
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
17
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
F
=
6
19
1
F
6
6
6
-
-
-
-
-
-
6
-
8
-
G
=
7
12
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
I
=
9
1
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
I
=
9
14
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
-
2
IT
29
11
2
-
2
6
6
4
25
12
7
8
27
I
=
9
-
2
IS
28
10
1
-
-
-
-
-
2+5
1+2
-
-
2+7
K
=
2
-
9
KNOWLEDGE
96
42
6
-
2
6
6
4
7
3
7
8
9
I
=
9
-
6
ITSELF
71
26
8
-
-
-
-
-
-
-
-
-
-
-
-
29
-
19
First Total
224
98
17
-
2
6
6
4
7
3
7
8
9
-
-
2+9
-
1+9
Add to Reduce
2+2+4
9+8
1+7
-
-
-
-
-
-
-
-
-
-
-
-
11
-
10
Second Total
8
17
8
-
2
6
6
4
7
3
7
8
9
-
-
1+1
-
1+0
Reduce to Deduce
-
1+7
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
1
Essence of Number
8
8
8
-
2
6
6
4
7
3
7
8
9

 

RE 95 RE

REARRANGED NUMERICALLY REARRANGED

RE 95 RE

 

KNOWLEDGE 96 KNOWLEDGE

 
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
IT
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
IS
-
-
-
-
-
-
-
-
-
-
-
-
-
K
=
2
-
9
KNOWLEDGE
96
42
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ITSELF
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
1
1
K
11
2
2
-
-
2
-
-
-
-
-
8
9
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
8
9
D
=
4
7
1
D
4
4
4
-
-
-
-
4
-
-
-
8
9
N
=
5
2
1
N
14
5
5
-
-
-
-
-
5
-
-
8
9
W
=
5
4
1
W
23
5
5
-
-
-
-
-
5
-
-
8
9
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
8
9
E
=
5
9
1
E
5
5
5
-
-
-
-
-
5
-
-
8
9
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
9
G
=
7
8
1
G
7
7
7
-
-
-
-
-
-
-
7
8
9
-
-
42
-
9
KNOWLEDGE
96
42
42
-
1
2
3
4
20
6
7
8
9
-
-
4+2
-
-
-
9+6
4+2
4+2
-
-
-
-
-
2+0
-
-
-
-
-
-
6
-
9
KNOWLEDGE
15
6
6
-
1
2
3
4
2
6
7
8
9
-
-
-
-
-
-
1+5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
9
KNOWLEDGE
5
6
8
-
1
2
3
4
2
6
7
8
9

 

RE 95 RE

REARRANGED NUMERICALLY REARRANGED

RE 95 RE

 

PLATO'S ACADEMY 54 ACADEMY PLATO'S

ISLE OF ATLANTIS 54 ATLANTIS OF ISLE

 

-
-
-
-
-
ISLE OF ATLANTIS
-
-
-
I
=
7
4
ISLE
45
18
9
O
=
6
2
OF
21
12
3
A
=
1
-
8
ATLANTIS
96
24
6
-
=
13
-
14
ISLE OF ATLANTIS
162
54
18
-
=
1+3
-
1+4
-
1+6+2
5+4
1+8
-
4
4
-
5
ISLE OF ATLANTIS
9
9
9

 

 

-
-
-
-
-
ISLE OF ATLANTIS
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
7
4
ISLE
45
18
9
-
-
-
-
-
-
-
-
-
-
O
=
6
2
OF
21
12
3
-
1
2
3
4
5
6
7
8
9
A
=
1
-
8
ATLANTIS
96
24
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
14
ISLE OF ATLANTIS
162
54
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
1
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
2
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
18
-
-
-
21
12
12
-
-
-
-
-
-
-
-
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
F
=
6
6
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
-
-
12
-
-
-
21
12
12
-
-
-
-
-
-
-
-
-
-
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
8
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
9
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
A
=
1
10
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
N
=
5
11
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
I
=
9
13
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
14
1
S
19
1
1
-
1
-
-
-
-
-
-
-
-
-
-
24
-
-
-
96
24
24
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ISLE OF ATLANTIS
-
-
-
-
4
2
6
4
10
14
7
8
18
I
=
7
4
ISLE
45
29
2
-
-
-
-
-
1+0
1+4
-
-
1+8
O
=
6
2
OF
21
25
7
-
4
2
6
4
1
5
7
8
9
A
=
1
-
8
ATLANTIS
96
24
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
14
ISLE OF ATLANTIS
162
54
18
-
4
2
6
4
1
5
7
8
9
-
-
1+3
-
1+4
-
1+6+2
5+4
1+8
-
-
-
-
-
-
-
-
-
-
-
-
4
-
5
ISLE OF ATLANTIS
9
9
9
-
4
2
6
4
1
5
7
8
9

 

 

-
-
-
-
-
ISLE OF ATLANTIS
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
7
4
ISLE
45
18
9
-
-
-
-
-
-
-
-
-
-
O
=
6
2
OF
21
12
3
-
1
2
3
4
5
6
7
8
9
A
=
1
-
8
ATLANTIS
96
24
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
14
ISLE OF ATLANTIS
162
54
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
1
1
I
9
9
9
-
-
-
-
4
-
-
7
8
9
S
=
1
2
1
S
19
10
1
-
1
-
-
4
-
-
7
8
-
L
=
3
3
1
L
12
3
3
-
-
-
3
4
-
-
7
8
-
E
=
5
4
1
E
5
5
5
-
-
-
-
4
5
-
7
8
-
O
=
6
5
1
O
15
6
6
-
-
-
-
4
-
6
7
8
-
F
=
6
6
1
F
6
6
6
-
-
-
-
4
-
6
7
8
-
A
=
1
7
1
A
1
1
1
-
1
-
-
4
-
-
7
8
-
T
=
2
8
1
T
20
2
2
-
-
2
-
4
-
-
7
8
-
L
=
3
9
1
L
12
3
3
-
-
-
3
4
-
-
7
8
-
A
=
1
10
1
A
1
1
1
-
1
-
-
4
-
-
7
8
-
N
=
5
11
1
N
14
5
5
-
-
-
-
4
5
-
7
8
-
T
=
2
12
1
T
20
2
2
-
-
2
-
4
-
-
7
8
-
I
=
9
13
1
I
9
9
9
-
-
-
-
4
-
-
7
8
9
S
=
1
14
1
S
19
1
1
-
1
-
-
4
-
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ISLE OF ATLANTIS
-
-
-
-
4
4
6
4
10
12
7
8
18
I
=
7
4
ISLE
45
29
2
-
-
-
-
-
1+0
1+2
-
-
1+8
O
=
6
2
OF
21
25
7
-
4
4
6
4
1
3
7
8
9
A
=
1
-
8
ATLANTIS
96
24
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
14
ISLE OF ATLANTIS
162
54
18
-
4
4
6
4
1
3
7
8
9
-
-
1+3
-
1+4
-
1+6+2
5+4
1+8
-
-
-
-
-
-
-
-
-
-
-
-
4
-
5
ISLE OF ATLANTIS
9
9
9
-
4
4
6
4
1
3
7
8
9

 

 

-
-
-
-
-
ISLE OF ATLANTIS
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
7
4
ISLE
45
18
9
-
-
-
-
-
-
-
-
-
-
O
=
6
2
OF
21
12
3
-
1
2
3
4
5
6
7
8
9
A
=
1
-
8
ATLANTIS
96
24
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
14
ISLE OF ATLANTIS
162
54
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
7
1
A
1
1
1
-
1
-
-
4
-
-
7
8
-
A
=
1
10
1
A
1
1
1
-
1
-
-
4
-
-
7
8
-
S
=
1
14
1
S
19
1
1
-
1
-
-
4
-
-
7
8
-
T
=
2
8
1
T
20
2
2
-
-
2
-
4
-
-
7
8
-
T
=
2
12
1
T
20
2
2
-
-
2
-
4
-
-
7
8
-
L
=
3
3
1
L
12
3
3
-
-
-
3
4
-
-
7
8
-
L
=
3
9
1
L
12
3
3
-
-
-
3
4
-
-
7
8
-
E
=
5
4
1
E
5
5
5
-
-
-
-
4
5
-
7
8
-
N
=
5
11
1
N
14
5
5
-
-
-
-
4
5
-
7
8
-
O
=
6
5
1
O
15
6
6
-
-
-
-
4
-
6
7
8
-
F
=
6
6
1
F
6
6
6
-
-
-
-
4
-
6
7
8
-
I
=
9
1
1
I
9
9
9
-
-
-
-
4
-
-
7
8
9
I
=
9
13
1
I
9
9
9
-
-
-
-
4
-
-
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ISLE OF ATLANTIS
-
-
-
-
4
4
6
4
10
12
7
8
18
I
=
7
4
ISLE
45
29
2
-
-
-
-
-
1+0
1+2
-
-
1+8
O
=
6
2
OF
21
25
7
-
4
4
6
4
1
3
7
8
9
A
=
1
-
8
ATLANTIS
96
24
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
14
ISLE OF ATLANTIS
162
54
18
-
4
4
6
4
1
3
7
8
9
-
-
1+3
-
1+4
-
1+6+2
5+4
1+8
-
-
-
-
-
-
-
-
-
-
-
-
4
-
5
ISLE OF ATLANTIS
9
9
9
-
4
4
6
4
1
3
7
8
9

 

 

-
-
-
-
-
ISLE OF ATLANTIS
-
-
-
-
1
2
3
5
6
9
I
=
7
4
ISLE
45
18
9
-
-
-
-
-
-
-
O
=
6
2
OF
21
12
3
-
1
2
3
5
6
9
A
=
1
-
8
ATLANTIS
96
24
6
-
-
-
-
-
-
-
-
-
13
-
14
ISLE OF ATLANTIS
162
54
18
-
1
2
3
5
6
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
2
1
S
19
10
1
-
1
-
-
-
-
-
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
A
=
1
10
1
A
1
1
1
-
1
-
-
-
-
-
S
=
1
14
1
S
19
1
1
-
1
-
-
-
-
-
T
=
2
8
1
T
20
2
2
-
-
2
-
-
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
L
=
3
9
1
L
12
3
3
-
-
-
3
-
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
5
-
-
N
=
5
11
1
N
14
5
5
-
-
-
-
5
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
6
-
F
=
6
6
1
F
6
6
6
-
-
-
-
-
6
-
I
=
9
1
1
I
9
9
9
-
-
-
-
-
-
9
I
=
9
13
1
I
9
9
9
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ISLE OF ATLANTIS
-
-
-
-
4
4
6
10
12
18
I
=
7
4
ISLE
45
29
2
-
-
-
-
1+0
1+2
1+8
O
=
6
2
OF
21
25
7
-
4
4
6
1
3
9
A
=
1
-
8
ATLANTIS
96
24
6
-
-
-
-
-
-
-
-
-
13
-
14
ISLE OF ATLANTIS
162
54
18
-
4
4
6
1
3
9
-
-
1+3
-
1+4
-
1+6+2
5+4
1+8
-
-
-
-
-
-
-
-
-
4
-
5
ISLE OF ATLANTIS
9
9
9
-
4
4
6
1
3
9

 

 

8
ATLANTIS
-
-
-
-
A
1
1
1
1
T
20
2
2
-
L+A+N
27
9
9
-
T
20
2
2
-
I
9
9
9
1
S
19
10
1
8
ATLANTIS
96
33
24
-
-
9+6
3+3
2+4
8
ATLANTIS
15
6
6
-
-
1+5
-
-
8
ATLANTIS
6
6
6

 

 

THE ATLANTIS SECRET

A COMPLETE DECODING OF PLATOS LOST CONTINENT

Alan F Alford 2001

ATLANTIS DECODE

Chapter 19

"In Timaeus and Critias respectively, Plato emphasises the idea that the Isle of Atlantis was supernaturally large:"

 

A
=
1
-
-
ATLANTIS
-
-
-
-
-
-
-
1
A
1
1
1
-
-
-
-
1
T
20
2
2
-
-
-
-
1
L
12
3
3
-
-
-
-
1
A
1
1
1
Q
Q
-
-
1
N
14
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
S
19
10
1
A
=
1
-
8
ATLANTIS
96
33
24
-
-
-
-
-
-
9+6
3+3
2+4
A
=
1
-
8
ATLANTIS
15
6
6
-
-
-
-
-
-
1+5
-
-
A
=
1
-
8
ATLANTIS
6
6
6

 

 

A
=
1
-
-
ATLANTIC
-
-
-
-
-
-
-
1
A
1
1
1
-
-
-
-
1
T
20
2
2
-
-
-
-
1
L
12
3
3
-
-
-
-
1
A
1
1
1
Q
Q
-
-
1
N
14
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
A
=
1
-
8
ATLANTIC
80
26
26
-
-
-
-
-
-
8+0
2+6
2+6
A
=
1
-
8
ATLANTIC
8
8
8

 

 

A
=
1
-
8
ATLANTIC
80
26
8
A
=
1
-
8
ATLANTIS
96
33
6
-
-
2
-
16
First Total
176
59
14
-
-
-
-
1+6
Add to Reduce
1+7+6
5+9
1+4
-
-
2
-
7
Second Total
14
14
5
-
-
-
-
-
Reduce to Deduce
1+4
1+4
-
-
-
2
-
7
Essence of Number
5
5
5

 

 

G
=
7
3
GEO
27
18
9
G
=
7
5
GEODE
27
18
9
G
=
7
8
GEOMETRY
108
45
9
G
=
7
11
GEOMETRICAL
108
54
9
G
=
7
10
GEOCENTRIC
99
54
9

 

 

-
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
-
15
-
-
-
-
-
+
=
15
1+5
=
6
=
6
=
6
-
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
7
5
-
4
5
2
9
25
+
=
39
3+9
=
12
1+2
3
=
3
-
-
7
5
-
13
5
20
18
7
+
=
93
9+3
=
12
1+2
3
=
3
-
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
7
5
15
13
5
20
18
25
+
=
108
1+0+8
=
9
=
9
=
9
-
-
7
5
6
4
5
2
9
7
+
=
45
4+5
=
9
=
9
=
9
-
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
5
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
-
-
-
7
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
=
9
12
8
G
E
O
M
E
T
R
Y
-
-
33
-
-
8
-
45
-
27
1+2
-
-
-
-
-
-
-
9
-
-
-
3+3
-
-
-
-
4+5
-
2+7
3
8
G
E
O
M
E
T
R
Y
-
-
6
-
-
8
-
9
-
9
-
-
7
5
6
4
5
2
9
7
-
-
-
-
-
-
-
-
-
-
3
8
G
E
O
M
E
T
R
Y
-
-
6
-
-
8
-
9
-
9

 

 

8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
15
-
-
-
-
-
+
=
15
1+5
=
6
=
6
=
6
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
7
5
-
4
5
2
9
25
+
=
39
3+9
=
12
1+2
3
=
3
-
7
5
-
13
5
20
18
7
+
=
93
9+3
=
12
1+2
3
=
3
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
7
5
15
13
5
20
18
25
+
=
108
1+0+8
=
9
=
9
=
9
-
7
5
6
4
5
2
9
7
+
=
45
4+5
=
9
=
9
=
9
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
5
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
7
-
-
-
-
-
-
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
=
9
8
G
E
O
M
E
T
R
Y
-
-
33
-
-
8
-
45
-
27
-
-
-
-
-
-
-
9
-
-
-
3+3
-
-
-
-
4+5
-
2+7
8
G
E
O
M
E
T
R
Y
-
-
6
-
-
8
-
9
-
9
-
7
5
6
4
5
2
9
7
-
-
-
-
-
-
-
-
-
-
8
G
E
O
M
E
T
R
Y
-
-
6
-
-
8
-
9
-
9

 

 

-
-
-
-
-
GEOMETRY
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
1
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
2
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
6
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
R
=
9
7
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
Y
=
7
8
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
-
-
45
-
7
GEOMETRY
108
45
45
-
1
2
3
4
18
6
14
8
18
-
-
4+5
-
-
-
1+0+8
4+5
4+5
-
-
-
-
-
1+8
-
1+4
-
1+8
-
-
9
-
7
GEOMETRY
9
9
9
-
1
2
3
4
9
6
5
8
9

 

 

-
-
-
-
-
GEOMETRY
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
1
1
G
7
7
7
-
1
-
3
-
-
-
7
8
-
E
=
5
2
1
E
5
5
5
-
1
-
3
-
5
-
-
8
-
O
=
6
3
1
O
15
6
6
-
1
-
3
-
-
6
-
8
-
M
=
4
4
1
M
13
4
4
-
1
-
3
4
-
-
-
8
-
E
=
5
5
1
E
5
5
5
-
1
-
3
-
5
-
-
8
-
T
=
2
6
1
T
20
2
2
-
1
2
3
-
-
-
-
8
-
R
=
9
7
1
R
18
9
9
-
1
-
3
-
-
-
-
8
9
Y
=
7
8
1
Y
25
7
7
-
1
-
3
-
-
-
7
8
-
-
-
45
-
7
GEOMETRY
108
45
45
-
1
2
3
4
10
6
14
8
18
-
-
4+5
-
-
-
1+0+8
4+5
4+5
-
-
-
-
-
1+0
-
1+4
-
1+8
-
-
9
-
7
GEOMETRY
9
9
9
-
1
2
3
4
1
6
5
8
9

 

 

-
-
-
-
-
GEOMETRY
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
6
1
T
20
2
2
-
1
2
3
-
-
-
-
8
-
M
=
4
4
1
M
13
4
4
-
1
-
3
4
-
-
-
8
-
E
=
5
2
1
E
5
5
5
-
1
-
3
-
5
-
-
8
-
E
=
5
5
1
E
5
5
5
-
1
-
3
-
5
-
-
8
-
O
=
6
3
1
O
15
6
6
-
1
-
3
-
-
6
-
8
-
G
=
7
1
1
G
7
7
7
-
1
-
3
-
-
-
7
8
-
Y
=
7
8
1
Y
25
7
7
-
1
-
3
-
-
-
7
8
-
R
=
9
7
1
R
18
9
9
-
1
-
3
-
-
-
-
8
9
-
-
45
-
7
GEOMETRY
108
45
45
-
1
2
3
4
10
6
14
8
18
-
-
4+5
-
-
-
1+0+8
4+5
4+5
-
-
-
-
-
1+0
-
1+4
-
1+8
-
-
9
-
7
GEOMETRY
9
9
9
-
1
2
3
4
1
6
5
8
9

 

 

-
-
-
-
-
GEOMETRY
-
-
-
-
2
4
5
6
7
9
T
=
2
6
1
T
20
2
2
-
2
-
-
-
-
-
M
=
4
4
1
M
13
4
4
-
-
4
-
-
-
-
E
=
5
2
1
E
5
5
5
-
-
-
5
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
5
-
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
6
-
-
G
=
7
1
1
G
7
7
7
-
-
-
-
-
7
-
Y
=
7
8
1
Y
25
7
7
-
-
-
-
-
7
-
R
=
9
7
1
R
18
9
9
-
-
-
-
-
-
9
-
-
45
-
7
GEOMETRY
108
45
45
-
2
4
10
6
14
18
-
-
4+5
-
-
-
1+0+8
4+5
4+5
-
-
-
1+0
-
1+4
1+8
-
-
9
-
7
GEOMETRY
9
9
9
-
2
4
1
6
5
9

 

 

-
11
G
E
O
M
E
T
R
I
C
A
L
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
-
-
-
9
-
-
-
+
=
15
1+5
=
6
=
6
=
6
-
-
-
-
15
-
-
-
-
9
-
-
-
+
=
24
2+4
=
6
=
6
=
6
-
11
G
E
O
M
E
T
R
I
C
A
L
-
-
-
-
-
--
-
-
-
-
-
-
7
5
-
4
5
2
9
-
3
1
3
+
=
39
3+9
=
12
1+2
3
=
3
-
-
7
5
-
13
5
20
18
-
3
1
12
+
=
84
8+4
=
12
1+2
3
=
3
-
11
G
E
O
M
E
T
R
I
C
A
L
-
-
-
-
-
--
-
-
-
-
-
-
7
5
15
13
5
20
18
9
3
1
12
+
=
108
1+0+8
=
9
=
9
=
9
-
-
7
5
6
4
5
2
9
9
3
1
3
+
=
54
5+4
=
9
=
9
=
9
-
11
G
E
O
M
E
T
R
I
C
A
L
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
3
-
3
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
5
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
11
G
E
O
M
E
T
R
I
C
A
L
-
-
37
-
-
11
-
54
-
36
-
1+1
-
-
-
-
-
-
9
9
-
-
-
-
-
3+7
-
-
1+1
-
5+4
-
3+6
8
2
G
E
O
M
E
T
R
I
C
A
L
-
-
10
-
-
2
-
9
-
9
-
-
7
5
6
4
5
2
9
9
3
1
3
-
-
-
-
-
--
-
-
-
-
8
2
G
E
O
M
E
T
R
I
C
A
L
-
-
10
-
-
2
-
9
-
9

 

 

-
3
G
E
O
-
-
-
-
-
--
-
-
-
-
-
-
6
+
=
6
-
=
6
=
6
-
-
-
-
15
+
=
15
1+5
=
6
=
6
-
3
G
E
O
-
-
-
-
-
--
-
-
-
-
7
5
-
+
=
12
1+2
=
3
=
3
-
3
G
E
O
-
-
-
-
-
--
-
-
-
-
7
5
15
+
=
27
2+7
=
9
=
9
-
-
7
5
6
+
=
18
1+8
=
9
=
9
-
3
G
E
O
-
-
-
-
-
--
-
-
1
-
-
-
-
-
-
1
ONE
1
-
-
-
2
-
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
-
-
-
-
3
THREE
3
-
-
-
4
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
-
-
-
-
6
-
-
6
occurs
x
1
=
6
-
-
7
-
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
9
NINE
9
-
-
-
27
3
G
E
O
-
-
18
-
-
3
-
18
2+7
-
-
-
-
-
-
1+8
-
-
-
-
1+8
9
3
G
E
O
-
-
9
-
-
3
-
9
-
-
7
5
6
-
-
-
-
-
--
-
-
9
3
G
E
O
-
-
9
-
-
3
-
9

 

 

-
3
G
E
O
-
-
-
-
-
--
-
-
-
-
-
-
6
+
=
6
-
=
6
=
6
-
-
-
-
15
+
=
15
1+5
=
6
=
6
-
3
G
E
O
-
-
-
-
-
--
-
-
-
-
7
5
-
+
=
12
1+2
=
3
=
3
-
3
G
E
O
-
-
-
-
-
--
-
-
-
-
7
5
15
+
=
27
2+7
=
9
=
9
-
-
7
5
6
+
=
18
1+8
=
9
=
9
-
3
G
E
O
-
-
-
-
-
--
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
-
-
-
-
6
-
-
6
occurs
x
1
=
6
-
-
7
-
-
-
-
7
occurs
x
1
=
7
27
3
G
E
O
-
-
18
-
-
3
-
18
2+7
-
-
-
-
-
-
1+8
-
-
-
-
1+8
9
3
G
E
O
-
-
9
-
-
3
-
9
-
-
7
5
6
-
-
-
-
-
--
-
-
9
3
G
E
O
-
-
9
-
-
3
-
9

 

 

-
5
G
E
O
D
E
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
-
15
-
-
+
=
15
1+5
=
6
=
6
=
6
-
5
G
E
O
D
E
-
-
-
-
-
--
-
-
-
-
-
-
7
5
-
4
5
+
=
21
2+1
=
3
=
3
=
3
-
5
G
E
O
D
E
-
-
-
-
-
--
-
-
-
-
-
-
7
5
15
4
5
+
=
36
3+6
=
9
=
9
=
9
-
-
7
5
6
4
5
+
=
27
2+7
=
9
=
9
=
9
-
5
G
E
O
D
E
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
FOUR
4
1
=
4
=
4
-
-
-
5
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
-
-
27
5
G
E
O
D
E
-
-
22
-
-
5
-
27
-
18
2+7
-
-
-
-
-
-
-
-
2+2
-
-
-
-
2+7
-
1+8
9
5
G
E
O
D
E
-
-
4
-
-
5
-
9
-
9
-
-
7
5
6
4
5
-
-
-
-
-
-
-
-
-
-
9
5
G
E
O
D
E
-
-
4
-
-
5
-
9
-
9

 

 

5
G
E
O
D
E
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
15
-
-
+
=
15
1+5
=
6
=
6
=
6
5
G
E
O
D
E
-
-
-
-
-
--
-
-
-
-
-
7
5
-
4
5
+
=
21
2+1
=
3
=
3
=
3
5
G
E
O
D
E
-
-
-
-
-
--
-
-
-
-
-
7
5
15
4
5
+
=
36
3+6
=
9
=
9
=
9
-
7
5
6
4
5
+
=
27
2+7
=
9
=
9
=
9
5
G
E
O
D
E
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
=
6
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
5
G
E
O
D
E
-
-
22
-
-
5
-
27
-
18
-
-
-
-
-
-
-
-
2+2
-
-
-
-
2+7
-
1+8
5
G
E
O
D
E
-
-
4
-
-
5
-
9
-
9
-
7
5
6
4
5
-
-
-
-
-
-
-
-
-
-
5
G
E
O
D
E
-
-
4
-
-
5
-
9
-
9

 

 

-
10
G
E
O
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
-
5
-
-
9
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
-
15
-
-
14
-
-
9
-
+
=
38
3+8
=
11
1+1
2
=
2
-
10
G
E
O
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
7
5
-
3
5
-
2
9
-
3
+
=
34
3+4
=
7
=
7
=
7
-
-
7
5
-
3
5
-
20
18
-
3
+
=
61
6+1
=
7
=
7
=
7
-
10
G
E
O
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
7
5
15
3
5
14
20
18
9
3
+
=
99
9+9
=
18
1+8
9
=
9
-
-
7
5
6
3
5
5
2
9
9
3
+
=
54
5+4
=
9
=
9
=
9
-
10
G
E
O
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
3
-
-
-
-
-
3
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
5
-
-
5
5
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
2
=
18
1+8
9
13
10
G
E
O
C
E
N
T
R
I
C
-
-
32
-
-
10
-
54
-
36
1+3
1+0
-
-
-
-
-
-
-
9
9
-
-
-
3+2
-
-
1+0
-
5+4
-
3+6
4
1
G
E
O
C
E
N
T
R
I
C
-
-
5
-
-
1
-
9
-
9
-
-
7
5
6
3
5
5
2
9
9
3
-
-
-
-
-
--
-
-
-
-
4
1
G
E
O
C
E
N
T
R
I
C
-
-
5
-
-
1
-
9
-
9

 

 

10
G
E
O
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
5
-
-
9
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
15
-
-
14
-
-
9
-
+
=
38
3+8
=
11
1+1
2
=
2
10
G
E
O
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
7
5
-
3
5
-
2
9
-
3
+
=
34
3+4
=
7
=
7
=
7
-
7
5
-
3
5
-
20
18
-
3
+
=
61
6+1
=
7
=
7
=
7
10
G
E
O
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
7
5
15
3
5
14
20
18
9
3
+
=
99
9+9
=
18
1+8
9
=
9
-
7
5
6
3
5
5
2
9
9
3
+
=
54
5+4
=
9
=
9
=
9
10
G
E
O
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
3
-
-
-
-
-
3
-
-
3
occurs
x
2
=
6
=
6
-
-
5
-
-
5
5
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
2
=
18
1+8
9
10
G
E
O
C
E
N
T
R
I
C
-
-
32
-
-
10
-
54
-
36
1+0
-
-
-
-
-
-
-
9
9
-
-
-
3+2
-
-
1+0
-
5+4
-
3+6
1
G
E
O
C
E
N
T
R
I
C
-
-
5
-
-
1
-
9
-
9
-
7
5
6
3
5
5
2
9
9
3
-
-
-
-
-
--
-
-
-
-
1
G
E
O
C
E
N
T
R
I
C
-
-
5
-
-
1
-
9
-
9

 

 

G
=
-
7
3
GEO
27
18
9
G
=
-
7
8
GEOMETRY
108
45
9
G
=
-
7
11
GEOMETRICAL
108
54
9
G
=
-
7
10
GEOCENTRIC
99
54
9

 

 

SACRED GEOMETRY

Miranda Lundy 2006

CIRCLING THE SQUARE

the marriage of heaven and earth

The circle is the shape traditionally assigned to the Heavens, and the square to the Earth. When these two shapes are unified by being made equal in area or perimeter we speak of 'squaring the circle', meaning that Heaven and Earth, or Spirit and Matter, are symbolically combined, or married. Fivefold Man exists between sixfold Heaven and fourfold Earth and Leonardo da Vinci's image opposite also shows how a man's span equals his height, that this measure equals seven of his feet and other important ratios.
Amazingly, the Earth and the Moon square the circle, for if the Moon (diameter 3) is drawn down to the Earth (diameter II) then a heavenly circle through the Moon (dotted below centre) has radius 7, and so circumference 44, the same as the perimeter of the square around the Earth. This works because irjelating the circumference of a circle to its diameter is practically 22/7. In Leonardo's image the Moon would fit above the man's head.
Also shown (below left and right) is a construction for a square using ruler and compasses. Octagons soon follow.

 

 

C
=
3
-
8
CIRCLING
75
48
3
T
=
2
-
3
THE
33
15
6
S
=
1
-
6
SQUARE
81
36
9
-
-
6
-
45
First Total
189
99
18
-
-
-
-
4+5
Add to Reduce
1+8+9
9+9
1+8
-
-
6
-
9
Second Total
18
18
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
6
-
9
Essence of Number
9
9
9

 

 

T
=
2
-
3
THE
33
15
6
M
=
4
-
8
MARRIAGE
72
45
9
O
=
6
-
2
OF
21
12
3
H
=
8
-
6
HEAVEN
55
28
1
A
=
1
-
3
AND
19
10
1
E
=
5
-
5
EARTH
52
25
7
-
-
26
-
27
Add to Reduce
252
135
27
-
-
2+6
-
2+7
Reduce to Deduce
2+5+2
1+3+5
2+7
-
-
8
-
9
Essence of Number
9
9
9

 

 

 
 
Top
  
 
Evokation
  
Previous Page
Index
 Next Page