An
Egyptian Geometry Lesson
Two, the Dyad (Gk.
Duas). The first feminine, even number in the classical numerological
litany of the Cabalists. The division between heaven and earth and
symbolic of movement away from Khaos towards symmetry. Yet it also
produces conflict, it is two sides of the same coin that will never
see each other. It needs an observer to see both sides, that's where
the first real number makes its appearance.
Three, the Triad, the Watcher. The first masculine, odd number and
the natural product of one plus two. It is the beginning of dimension
and represents the second stage of creation, fertility. The mathematical
interactions of the first three numbers produce every subsequent number.
Religious history is littered with the symbolism of three and it's
role in Creation. Egypt was no different to Hindu or Christianity
in its recognition of the power of the Trinity.
Four, the Tetrad, the corner posts of the Earth, the squaring of the
circle, two plus two or two times two. It represents symmetry and
surveying and is the number of foundation and land measure. The square
is man's attempt at rationalizing the irrational circle. It can be
symbolized by the cross. X
Five, the Pentad, symbolic of Man and the Golden Mean, the Divine
Proportion. The pentagram has been invoked as a symbol of protection
throughout history, especially in Egypt. The five pointed star enclosed
in a circle was their symbol for the Duat. The Pythagorean theorem
states that in a right triangle, five is the square root of the sum
of the squares of the two previous numbers, three & four.
3 x 3 (9) + 4 x 4 (16) = 5 x 5 (25)
Six, the Hexad, the perfect divider of the circle. The radius of
a circle divides its circumference into the six points of the hexagon
to form the Seal of Solomon, the hexagram. The mathematical properties
of six are abundant in nature, in crystals, honeycomb and flowers.
Seven, the Heptad, the virgin. The magical heptagram was known as the
seven-horned beast. It is the only number of the first ten that cannot
be divided by any of the others except one, it is the only one of
the decad that will not divide evenly into 360 degrees. Seven is the
number of mystery, it does not integrate with the geometries of the
other numbers, or so it is said. However, there is a way, and Giza
describes the path.
Eight, the Ogdoad, symbolized by the cube and it's eight corners. It
represents stability and the Earth. Eight is the square root of the
enigmatic sixty-four, seen in the chessboard and the I Ching. It is
the natural divisor of the square and the beginning grid of the Giza
geometry.
Nine, the Ennead, the Gods of Heliopolis. Re-Atum, Shu, Geb, Nut, Tefnut,
Osiris, Isis, Seth and Nephthys. Representing completion. The Egyptian
mnemonic to calculate the area of a circle was incredibly simple and
accurate to.06%. It goes like this. Subtract the ninth part of the
diameter of a circle and square the result. If the diameter is nine,
minus one ninth = eight. Eight squared is sixty-four, therefore, the
8 x 8 square represents the "nine" diameter circle of completion.
Ten, the Decad, the cycle of the Monad begins again but now it represents
the all-encompassing Universe. The Decad ties together all the number
systems of the Giza Pyramids, it is the underlying mainframe of the
computer
That
was a very brief outline of the symbolism associated with the first
ten numbers, the Decad. I don't intend to give you a complete history
of sacred geometry but some background is essential to the understanding
of what we are about to explore. Geometry is Man's humble attempt
at describing the universe he exists in. There are immutable laws
that nature follows by instinct, as is demonstrated in the spirals
of the sunflower. They are determined by the numbers of the Fibonacci
sequence, the basis of cellular growth. This series operates by progressively
adding the previous number to the sum, beginning :-
1+1=
2+1= 3+2=5+3 = 8+5=13+8 =21+13= 34+21=55+34= 89+55 =144
Before
we proceed to the next level of the Giza geometry it is probably wise
to insert a quote from a man I admire very much. He has written several
eloquent books on the subject of sacred geometry and number symbolism,
John Michell. In his book "The Dimensions of Paradise" (p.65)
is a passage that sums up everything that the pyramid builders were
trying to say in stone.
" Sacred geometry is the essence of the geometers art. Its use is for making descriptions of the universe by combining together in one geometric composition the basic figures which represent the different orders of number and underlie the manifestations of nature. It is the art of synthesizing diverse elements. The tradition survives today in the patterns of Islamic art, but it is no longer thought worthy of scientific study on the grounds that it is irrational. That epithet is applied to mathematical ratios which cannot be defined in terms of whole/number integers.
Ever since the legendary renegade Pythagorean let the cat out of the bag by revealing that ‘pi’, like many of the other important ratios in geometry, is irrational - and that one can not therefore define the relationship between the diameter and circumference of a circle or between the side and diagonal of a square by any pair of integers - the procedures of the old geometers, in squaring the circle and uniting the different orders of geometry in one scheme, have been considered a mare's nest. Rationally indeed, the problem of constructing a circle of the same perimeter or area as that of a given square, or of fitting together geometric shapes developed from the numbers 5, 6 and 7, is insoluble because these shapes are not commensurate.
Yet there is one system in which all numbers and proportions and every disparate or incommensurable element co-exist and function together in perfect harmony, and that system is the universe. It can never be represented perfectly and literally by any artist's or geometer's model because, although it is said ultimately to be a creation of reason, the paradox in its nature is too deep rooted for human resolution. The geometer's aim, therefore, is to imitate the universe symbolically, depicting its central paradox by bringing together shapes of different orders, uniting them as simply and accurately as possible and thus creating a cosmic image which, as Plato claimed for his Magnesia plan,'most nearly resembles the original' and is 'second only to the ideal'. And within that limit of ambition it will be found possible, to all practical intents and purposes, to square the circle, unite the different orders of geometry and depict adequately that most perfect expression of the Monad which the Greeks called Cosmos."
John Michell copyright 1988 Thames & Hudson Ltd. London.
I
wish I'd said that!
The
Giza pyramids are the Ancient Egyptians version of that model of the
Cosmos. All the numbers and principles of sacred geometry are marked
out by the dimensions and positions of the three major pyramids and
their subsidiary satellites, temples and walls. The outcome of their
model is the uniting of all the number geometries of the Decad, including
seven. But that is just the beginning. Their system also includes
the numerical values of music, the courses of the Great Pyramid follow
a sequence in their differing thickness that relate directly to musical
notation, the Great Pyramid actually plays an entire song. The length
in cubits of one side of the G.P is 440. The note A440 is the tuning
note for an orchestra and has a frequency of 440 cycles per second.
The
thickness in inches of the base course, divided into the speed of
sound in inches per second, gives a result of 220, its resonant frequency.
220 is also the half base dimension of the G.P. and one octave below
middle "A”. That A220 becomes the starting note of the Great
Pyramid's tune, and the predominant 35th course (5x7), which is two
sacred cubits deep (50p"), resounds to middle "C “, 261.63
cycles per second. The diagonal of the G.P. is 622.22, corresponding
to an E flat. Because we are dealing with an integral number system
it soon becomes obvious that there is a pattern emerging. The pyramid
builders tried, and possibly succeeded, to incorporate all their numerology
in one grand design.
There
are 20.6066 p.inches in the Royal cubit and 25" in the Sacred
cubit. (See notes) Therefore, all measures quoted are in Royal cubits
(r.c.) unless stated otherwise. The royal cubit equals 20.625 British
inches or.52375 meters, but we will not be needing either of those
systems from now on, though once in a while a comparison will be given
to make you aware of the scale of things and their accuracy.
This
musical aspect is also apparent in the second and third pyramids.
The side of Khafre's measures 411 r.c. .
The length of two sides, 822, divided by Pi also results in 261.63,
middle "C ". Menkaure's pyramid, the third and smallest
measures 196 x 200. 196 corresponds to a "G ".
200
x 200 is the size of the squares which comprise the fundamental framework
of the Giza geometry. As we will see later, each deviation from the
norm has a purpose. For example, the third pyramid measures 200 x
196r.c., where it should be 200 x 200, but 200 x 196 produces a diagonal
of 280r.c., the precise height of the G.P., and it makes its volume
one eleventh of the Great Pyramid. Do you see how it all interrelates
on many different levels? Somebody had a great time working out this
incredible picture.
Just
as we sent our mathematical message into space carried by our ultimate
achievement in technology, so did the Pyramid Builders send theirs
into history on board the Giza pyramids. All it takes to understand
their epitaph is to give them the credit for the knowledge they obviously
had.
When
Petrie measured the Pyramids he concluded that the centres of
the three pyramids bore no mathematical relationship to each other,
and yet he measured them with fantastic accuracy considering that
in 1881 the sand was covering a lot of the detail of what we now see,
especially the corners of Menkaure's pyramid.
Ironically,
if he, or anyone else in history, had placed more importance on the
satellite pyramids of Menkaure, the so-called Queens pyramids, then
the problem could have been solved long ago. Remember how in the Overture
we talked about decoys? We have to give
the designer credit for having a sense of humour. Whatever insight he had into the future you can be
sure he could second-guess how long it would take until someone looked
at the relatively insignificant detail of his masterpiece, the Coda.
This
is the perfect place for a musician's joke, it goes like this: -
A
famous, but notoriously bad, orchestra leader died and went directly
to hell. There he was welcomed warmly and enthusiastically by Old
Nick himself, who expressed his happiness at finally getting the leader
he had always wanted for his orchestra. On expressing his lack of
understanding of the situation, the musician was led into a magnificent
concert hall and there, on the stage, was the largest orchestra he
had ever seen. Old Nick led him to the podium and handed him the baton.
"You shall be my conductor," said the Devil. The musician
broke down in tears of joy and said, " How can I possibly deserve
this? This is my idea of Heaven, not Hell " Old Nick smiled."
You haven't seen the music yet! " The musician opened the pad
on his music stand and read the title " Strangers in the Night
", he laughed, "One of my favourite pieces, I don't suppose you've got 'Ole Blue Eyes
' himself down here as well?" The Devil smiled again, "
You had better check the arrangement, the show begins in two minutes.
" The musician began furiously scanning through the music and
soon a look of panic appeared in his eye and he turned to see Old
Nick wearing a tuxedo and top hat. " I can't find the Coda!!
", he screamed. The Devil's eyes lit up as he swaggered to the
microphone. " There isn't one you fool, this is hell remember.
Now are you ready? After four.....,
a one ana two ana three ana........"
I
hope you see the analogy. Trying to solve the Giza geometry can be
a nightmare if you don't find an exit. The Pyramid builders played
number games just as we play word games today, it was the pastime
of the intellectual ruling class, the priests, and possibly of the
whole society at some time, if Plato's Magnesia ever existed.
The
geometrical paths that can be constructed in the "Rostau"
circle are inconclusive; they can lead you round and round the circle
for an eternity and only replay the same piece over and over. It will
still, eventually, teach you the tune of its sacred geometry, but
the Coda has to be found and played to give the composition a Finale,
so that in the silence, at the end, one can savour the entirety of the piece and possibly, for a moment,
see God.
For
a musician, repetition leads to memorization, as it should. Why read
the music on paper when it can be translated directly from the mind
to the fingers and limbs? After a while the pathways become engrained
in the matrix of the mind and can be recalled, intact, upon demand,
assuming of course that the memory is exercised once in a while. As
we all know, there can be many variations on a theme, but all have
to be contained within the original framework or it becomes unrecognizable
as the original. Even jazz has to stay within certain constrictions
of the original metaphor. The Giza symphony has a set of well-stated
metaphors that have to be included in any rendition, but once learned
they can be the key to an infinite variety of expression.
The Great Pyramid
Khufu (G1)
We
must begin with the prime statement itself, the Great Pyramid and
its measurements. We have to use the measure utilized by the Builder
himself, the Royal Cubit and the Pyramid, or Primitive Inch, since
that is the only system in which the encoded equations actually take
on their true numerical values.
The
Great Pyramid is 280rc. high and 440rc.along its base, giving a perimeter
of 1760rc.and diagonal of 622.22.rc. Divided by 40 its proportions
are 7 high by 11 wide. Consequently it represents the approximate
formula for 
(22 (2 sides) -:- 7 = 3.14285714...)
However, there
has been much deliberation over the years as to its true measure,
each side is fractionally different to the others and the consensus
comes down to an intended base measure of 439.823 r.c., which produces
a true Pi of 3.141592654.. The G.P. is an approximate scale model
of the northern hemisphere of the ratio 1 : 43,200.
1,760
r.c. x
43,200 = 76,032,000r.c =
The
height of 280 is the approximate degree of flattening at the poles,
(1/ 280th) caused by the bulge of the equator. The Golden
Mean proportion, phi, is incorporated in
the dimensions of the half base (220) and the apothem (from apex down
centre face to base centre). This works thus,
*
280 -:- 4 = 70 (ht.)* * 220 -:- 4 = 55 (half
base) *
therefore:
-
70
squared + 55 squared = 89 squared ** 89 -:- 55 = 1.61818 =
(phi)
55
and 89 are consecutive Fibonacci numbers.
89
x 4 = 356 ( length of apothem
)
The
Egyptians called that ratio of 70 : 55 or 7 : 5.5 a Seked angle of
5 1/2 palms. A royal cubit
was divided into 7 palms, or 100 n,
so a pyramid structure one cubit high (7 palms) and a half base of
5 1/2 palms generates an angle of
51* 51' and a tangent ratio of 1.2727, as is found in the G.P..
Now we can expand the visualization process a little. The proportions
we have just been working on have some intriguing properties. 70 :
55 : 89 are the measurements of the top quarter of the G.P. if one
imagines it in four horizontal slices
.
This we can call one pyramid unit and it measures 70 high and has
a side of 110; it constitutes 1/64th. of the volume of
the whole pyramid. The formula to find the volume of a pyramid is,
*
Base squared x Height -:-
3 = volume (cubic cubits) *
The
courses that are missing from the apex of the G.P. constitute 1/16th
of its height or one quarter of the top quarter, or 1/64th
of 1/64th of the volume of the whole. It gets better. The
volume of the missing piece is approximately 4400 cubic cubits, a
mnemonic for the whole that runs like this.
*
(4400 x 64 x 64 x 3) -:- 280 = 439.4282 squared
(440 squared)*
or
we can use (4411 x 64 x 64 x 3) -:- 280 = 439.9771 squared.
It
has to be emphasized that this is a teaching system that was to be
memorized by initiates, so small discrepancies will occur from the
mnemonic numbers, but they are always within.05% of the real value.
Back
to the missing 1/64th and the next mnemonic trick. This
"One Pyramid Unit" we have established which comprises the
top seventy cubits of the G.P, when multiplied by 7, results in the
volume of the next lower quarter slice, which is 7/64th
of the whole. Add that to the top unit and it becomes 1/8th,
therefore the top half, 140 r.c. high, equals 1/8th of
the whole, again multiplied by 7 and we have the volume of the lower
half, 7/8th. And just to take things to extremes, the specific
gravity of the limestone of the G.P. is 2.75, multiply that by seven
and the result is 19.25, the specific gravity of pure gold.
This
coincidence could well be intentional because there is another strange
correlation to gold that appears from this "pyramid unit"
sequence, not only in the pyramid proportions but also indicating
the "Gateway”, as we will see later. For now, to complete this
visualization process, we will mentally construct the whole G.P. from
One Pyramid Units (1/64th). We will find that we can use
44 whole units by beginning with 16 units (4 x 4) as the base and
then inverting others to fill the spaces, like an egg carton. When
we have stacked the 44 whole units we are left with 40 tetrahedral
spaces, which equal the remaining 20/64th (5/16th)
of the whole. Therefore each tetrahedral space = 1/128th.
That crystalline tetrahedron is identical to a gold
crystal.
Also
to be noticed here are the numbers 44 and 40, both additional mnemonics.
If you look at the drawing 
you will also see a series of equations based on the sq.rt. of
41,100,000, this aspect of the Great Pyramids measure has never
before been examined by Pyramid numerologists and is based on the
measure of the second pyramid of Khafre, 411. Before leaving
the G.P. there is one more dimension to be noted and one that is essential
to the final picture. If the side of the G.P. is 439.82 then its diagonal
is precisely 622 cubits. If it is 439.9771 the diagonal is 622.222.
*
622.222 -:- 2.2222 = 280 (G.P.ht.). *
*
4,411 x 64 x 64 x 3 -:- 280 = 439.9771 squared.*
Mnemonics
!
Khafre ( G2
)
The Movable Feast
The
second pyramid of Giza is 274 rc. high and 411 rc. along each
face, making its volume 6/7ths. of the Great Pyramid. It encapsulates
the other Seked ratio favoured by the Egyptians, 5 1/4 palms. For every 7 palms (1rc.) of height the base steps out
5 1/4 palms, this generates
a 3-4-5 triangle.
E.g.:
- (274 -:- 7) x
5.25 = 205.5 x 2 = 411 (see ratio sketch)
(
4 -:- 7 ) x 5.25 =
3 x 2 = 6.
*
205.5 -:- 3 = 68.5 *
274 -:- 4 = 68.5 * 68.5 x
5 = 342.5 (apothem ) *
Petrie's
measurement of the pyramids places the centre of "G2” 929.6rc
south west of the G.P., or 675.63rc south and 638.5 west. Therefore,
by adding or subtracting the half base measure of 205.5, we can determine
the positions of the four sides of G2 relative to the G.P.. The southern
edge is 881.13rc south of the G.P.'s east/west
meridian and the western edge is 844rc west of the G.P.'s north/south
meridian. 881 equals two widths of the G.P. + 1, ( 440 x 2 +1 ), it
seems to have been placed one cubit beyond its logical symmetry, 880.
As it happens, 881.4760348 is the square root of 777,000, so it looks
possible we have been given another mnemonic, and the likelihood is
more apparent when we look deeper into the "illogical "placement
of G2.
The
eastern edge of G2 is 433rc west of the G.P.'s N/S meridian, 440 would
have been the logical position so as to fit the overall geometrical
picture we are working toward, but we are still in the early stage
of understanding the designer's purpose. For now it is easier to accept
that G2 is 7 cubits east and 1 cubit south of where it should be for
geometrical precision and then we can look at why it is so. Consider
that on a scale drawing, seven cubits is only the width of a pencil
line and you will see that it makes very little difference to the
overall picture, put it down to artistic license by the master architect.
There are several other mnemonics encoded into G2 that make it obvious
that its offset is intentional and not a mistake by the builder. If
you consider that the designer or any other geometer were already
aware of the grand geometrical design then the problem becomes one
of solving the hidden equation. We'll get there soon enough.
As
mentioned earlier, two sides of G2 total 822, divided by Pi that results
in middle "C "(261.6). The length of its diagonal, 581.3rc,
multiplied by Pi is equal to the length of the year x 5 (1826.21).
The measure of its centre, south of the G.P., is the square root of
456456 (675.615). 881 minus 844 is 37, this is the horizontal displacement of G2 away
from the diagonal meridian of the G.P., therefore the diagonal displacement
(N.W to S.E) is precisely 26.163, middle "C " divided by
10. The south side of G2 is 661 r.c. south of the south side of G1.The
prime number 37 also contains some other nice relationships, including
a mnemonic for the volume of G2 involving the two displacements, 37
and 661: -
*
3766.1 x 64 x 64 x 3 -:- 274 (G2 ht.) = 411 squared (base)*..
and then-
*
37 x 3 =
111 * *37 x 6 = 222 * * 37 x 9 = 333 * * 37 x 12 = 444 *
*
37 x 15 = 555 * * 37 x 18 = 666 * *37 x 21 = 777 * * 37 x 24 = 888
*
*
37 x 27 = 999 * * 37 x 33
=1221*
Before
we move G2 to its " real " theoretical position there is one more measure to examine, its
perimeter of 1644rc. The western edge is 844 west of the G.P. north/south
meridian. The eastern wall of the Sphinx temple and Khafre's valley
temple is 800rc east of the G.P meridian. 844 + 800 = 1644. Get it? One complete rotation of G2 eastwards aligns
it with the north/south meridian of the Gateway. 
It
is an apparently deliberate clue to lead to the other meridian and
the gateway, if the novitiate has not already found, or been made
aware of, its existence. But since we started with the solution we
have to backtrack to the question. Here's a good one. Did the solving
of the Giza geometry constitute some kind of exam for entry into the
upper echelons of the priesthood? It seems like an adequate problem
to pose to evaluate the mental capabilities of the aspirant. We have
enough modern evidence of this type of education in the oriental religions
like Buddhism, where initiates are pointed in the right direction
and then left to find the Truth for themselves, to construct their
Mandala. That is how I began this search, so I could finish my Mandala and know where I should
be standing to see a 90-degree angle at the apex of the Great Pyramid
and draw the final line for completion.
And
is it not possible that much of what became ritual and ceremony in
Dynastic Egypt was a corrupted version of a science, which had its
origins in a society where a disciplined education in mathematics
and astronomy was the standard? It is quite possible that Pharaonic
Egypt had lost the original premise of its founders and had simply
become a power structure to control the population, an all too familiar
scenario in recent history. Enough digression, it's time to move a
pyramid seven cubits west.
You're
asking why we have to move this pyramid and the explanation is that
we have to put it back to where it started. We have enough numerical
clues to say where it should be geometrically, and we will soon explore
them, but first let's examine symmetry. If G2 were the same size as
G1 (440 x 440), then the position of its southeast corner in the geometry,
relative to the centre of G1, would be 880 south by 440 west, a double
square proportion. This will be explained in detail later, for now
we will see what happens with the equations when we examine the fundamental
geometrical picture.
As
we can see on the drawing, G2, in its new symmetrical position, does
not fill its "box " completely, it becomes 29 cubits short
of the north and west sides of its "box ", since it is only
411 x 411 and not 440 x 440. The true position of the southern edge
of G2 is 881 cubits south, not 880, however, that additional cubit
can be added or subtracted, as in the rules of gematria, to complete
the equations that tie the entire mathematical picture together.
By
moving G2 seven cubits west we have now located its western edge 851
rc. west of the G.P. meridian (440 + 411 = 851). This now reveals
several important number clues. 851w. + 881s. = 1732. Move the decimal
point three places and we have the square root of 3,
(1.732050807). That means that the s.w. corner of G2 measured
from the G.P.centre would be 1224.89rc. 1224.744871 is the square
root of 1,500,000, so it seems we have another mnemonic, but this
one runs right into the cabalistic "magic" numbers 1224
and 1225.
1224,"
the number of Paradise, and the 153 fishes in the net " (Michell)
This number has a great significance in Christian lore and its "gematria
" (see note). One rearrangement in Greek translates as "divine
circle", which is, mathematically, very interesting. Remember
that the aim of the geometer is to merely represent perfection, then
follow this next mnemonic equation: -
(Square
root 6 -:-2) = 1.224744871 x 1000 = 1224.744871
(1224.744871
x 2) squared = 6,000,000
Six
is the number of perfection, the hexad.
There
are many more correlations to be made to confirm the movement of Khafre's
pyramid from the norm, but they involve measurements within the composite
geometries and from the "gateway “. Suffice to say, for now,
that G2's n.w. corner is 2000rc from the gateway and the s.w. corner
is 1800rc from the gateway. The centre of G2 is also now 933.33rc
from the G.P.'s centre, or three times the half diagonal of G1, 311.111,
or musically a E flat. One more number before we move on. The s.w.
corner of the 'box' that G2 does not quite fill is 1244.4 r.c.
from G1's centre, twice the diagonal of G1.. It all becomes clearer,
believe me, but it does take patience. We must not move too quickly
through the process or none of it makes sense. This is a jigsaw puzzle
after all, and even though the solution has been shown already, all
we have are the pieces, and they have to be assembled in a logical
way so as to appreciate the final composition. So now we must interrogate
the next witness.
Menkaure's Pyramid ( G3 )
The
smallest and southernmost of the three pyramids. Its height is 124.444
rc. and its base is 196 x 200. (Lehner). The diagonal measures
precisely 280rc, equal to the height of the G.P..
(196
squared + 200 squared = 280 squared).
124.444..
is 4/9th of the height of the G.P., 280, and 1/5th
of its diagonal. The level of the granite layer of G3 is 31.11..r.c.,
one quarter of the height of G3 and 1/9th the height of
G1 and 1/20th of its diagonal.
196
corresponds to a musical "G" and 200 is one tenth of the
radius of the 2000rc radius circle that encloses the entire geometry,
around which is the 4000 x 4000 square that forms the underlying framework.
By the designer reducing two sides to 196 and leaving two at 200 he
created many intriguing possibilities, one being that it represents
almost precisely one eleventh of the volume of the G.P,
-
*
(4411 x 64 x 64 x 3) -:- 11 = 4927488 -:- 124.444 ( G3ht.)
However
the true ratio is far more interesting, it involves the sq. root of
123.45678987654321, which is 11.11111 or 11 & 1/9th.
*(4411
x 64 x 64 x 3) -:- 11.11111... *
=
(196 x 200 x 124.444) Exactly!
*
196 + 200 x Pi = 1244 *
Musically,
1244.4 is the frequency of E flat.
Once
again, Menkaure has been moved a little from its true theoretical
location and again that movement opens up a vista. We'll begin with
its position relative to the G.P. centre. It is 1,411rc south
and 1,096.765 west, (1244 squared -:- 1411 = 1096.765., also 1411
minus 1096.765 = 314.235..or Pi x 100)
The
direct hypotenuse to G1 measures 1787.125 and that is the first to
draw attention. The diagonal of
a double square rectangle 1600 x 800 measures 1788.8543.. (Sq.rt.
3,200,000), very close considering how many more clues are to be found,
and even more relevant when that is also the distance from the G.P.
centre to the Gateway, 1788.8543 rc.
The centre of G3 falls on the root 5 circle, as does the "gateway”.
Sq.
root of 5 (2.236067977) x 800
= 1,788.8543 * (1,789)*
In
gematria,
Alpha
has a numerical value of 1.
Omega
has a numerical value of 800.
It
doesn't take a great stretch of the imagination to suggest that there
may be a link in that number symbolism to the Great Pyramid and the
"Gateway”.
So
far we have used only the measurements made by Petrie in the 1880's
for reference. There have been several surveys of the Giza plateau
since then, all of which came up with different measurements and with
huge discrepancies between them. For example, the measurement from
the east side of G1 to the east side of G3 ranges from 1180 rc. up
to 1270 rc., that's 90 cubits (150 ft./47 m.) difference, an enormous
gap. The ideal measurement should be 1220, putting the eastern edge
of Menkaure's 1000rc west of the G.P. meridian. 1220 is also the uncorrected
measure from the G.P. centre to the s.w. corner of G2. In Petrie's
day the base of G3 had not been cleared of sand and it was difficult
to establish the corner points, as a result he gave the side values
as 201.5 x 201.5. We now know that it is 196 x 200, which compromises
Petrie's measure at this point, however his reading for the centre
of G3 is accurate in the grand design. It places the eastern edge
of Menkaure's 997.8rc west of G1's meridian, 1000 would have been
better, but as usual the designer had something up his sleeve. The
measure from the northern edge of G1 to the southern edge of G3 is
once again 1732 (sq.root of 3,000,000).
If
we now calculate the distance from G1's centre to the S.W corner of
G3 it comes out as 1930.873997. This is the radius of a circle with
a circumference of exactly 250,000 p.” or 10,000 Sacred Cubits of
25 p.” or one thousandth of the polar radius. The S.W. corner of G3
is 2000 rc. from the gateway. The horizontal displacement of G3's
centre from the G.P. diagonal is 1411 minus 1096.765 = 314.235 as
mentioned previously. This makes the diagonal displacement, NW - SE, 222.22 r.c..
There
are more equations to describe, but once again they involve the higher-level
geometries, and include the causeway and enclosure walls of the Menkaure
complex. They will be included as they become relevant to the construction.
One last important piece of the puzzle before we begin the building stage. The centres of G3's satellites are the only three that fall on a straight line, as it happens, that line runs east-west and is precisely 1600 rc. south of the G.P. meridian, and that's where we take our first step into Rostau, by composing the Bassline.
Putting
the Band Together.
Excuse
the analogy but it's drums 'n' bass, rhythm and fundamental harmony.
It is the basis of the composition and the framework. The time signature
is 4/4 and the first four notes are A, C, E flat and G, an A dim.
7 chord. See it like this; -
The
Great Pyramid is the inspiration for the melody, it encodes in its
numbers the possibility of a beautiful tune, yet it is alone, a one-man
band. A great composer without an orchestra or audience, the Monad,
One. The central thought. Alpha. A.
Geometrically
we have the logical starting point, a centre to place the compass
point, but we must determine a radius for the circle that can be constructed
from that centre. The composer / geometer would see those other musicians
standing nearby and, always thinking he may be able to get a gig together,
would invite them to join in, as long as they didn't want paying.
He would need to see if they could play in tune first, and the first
to be auditioned, since the composer plays the drums, is the bass
player. He has to keep perfect time and always stay within his octave.
He
stands between time and melody, holding both together, parallel, forever
separated but intrinsic to the structure of the harmony. The three
satellites of Menkaure, from east to west; -- G3A, G3B and G3C, look
like a base/bass section, they're small, unnoticed and they have a
lot to say about who joins the band. 
The
"base-players" are pointing straight at the "pianist”,
the "gateway", metaphorically speaking. The gateway is the
final member of the "rhythm section ", the one member who
can substitute for the whole band, except possibly for the drummer,
but that's a personal problem and we won't go there. He has at his
command seven octaves on which he can create an infinity of melodies,
he completes the trio. So we shall heed the wisdom of the bass-players
and listen closely as they play their baseline. We draw the line parallel
to the E-W meridian of the G.P. and through the centres of G3A, G3B
and G3C. As we know already, these two parallel lines are exactly
1600 cubits apart, and even without that knowledge of measure this
is still the first step to take. We are drawing the stave on a blank
sheet of paper.
Using
the baseline as our radius from the G.P. centre we draw the first
circle, it does not enclose Menkaure's complex, but the square drawn
around that circle, utilising the baseline, does. By drawing in the diagonals of
the square we can begin "decomposing the square”. The first division
creates 4 squares, the second creates 16, and the third division produces
64 squares. * ( 8 x 8 )*. Mathematically we now have a 3,200 x 3,200rc
grid containing 64 smaller squares, each divided diagonally and measuring
400 x 400rc. All of these measures are 4 based; a square does not
divide naturally into fifths.
Looking
closely at what we now have it is apparent that what at first seemed
to be random placing is suddenly a little more cohesive. The Sphinx
temple and the adjacent Khafre's valley temple form the edge and corner
of one square. Campbell’s Tomb pinpoints the corner of a double square,
800 south and 400 east of the G.P. The eastern wall of the Sphinx
temple is 800 rc. east of the G.P. n-s meridian and its north wall
is 800 rc. south of the G.P. e-w meridian. The western edge of G3
is defined by the vertical line 1200 rc. west of the G.P. and that
same line runs down the length of the wall of the "Royal Workshops",
west of G2, and then intersects the baseline at the second step on
the west side of G3B, the stepped, middle satellite of Menkaure.
The
second step on the Western side of G3B is 1600 south by 1200 west,
4 down, 3 across and, yes, you've guessed, five (2000) back to the
centre. 
Since our first circle did not enclose Menkaure's complex it is irresistible
not to draw that "5" radius and so get everyone into the
"Magic Circle ". All except poor little G3C, who sits alone,
outside. But G3C is in the bass percussion section of the orchestra,
he plays the Tympani in the Finale. He's sitting outside smoking,
waiting to creep in at the back of the band at the end and play the
enormous crescendo that ends the symphony. We should leave him to
finish his cigarette in peace for now, he'll have enough to say later.
Having drawn that irresistible " 5 " circle with its intriguing
2000 r.c. radius, we must construct the square around it, which measures
4000 x 4000 rc., or 10 x 10. By simple extension of the framework
of the first 8 x 8 square, we can subdivide and absorb the larger
square, around the 4000 diameter circle, into our final framework.
We
could leave our grid at this stage of development with its "pixel
" size of 400 x 400rc, however Menkaure seems to be saying "
Come down to my size ", which is 200 x 200. One more subdivision
of the master framework achieves just that, and makes our grid 20
x 20 and consisting of 400 squares the same size as G3, 200 x 200rc.
This also divides the circle radius into ten. The original "4"
radius becomes "8" and the "5" becomes "10
".
Looking
again at the map we find that several more "hits" have been
made. G3 is now boxed on two sides and the northern edge of its causeway
defines the "7" (1400rc) line south of the G.P.. The southern
wall of Khafre's valley temple sits on the "5 " (1000rc)
line south. It is becoming obvious that there is an underlying pattern of placement
of the structures of Giza within this fundamental framework, but musically
it is rather "stiff ", it is still only drums 'n' bass.
So far Khafre is still not in harmony with the tune. Back to the analogy,
we need the pianist.
G3B
is the predominant player on the circumference of the outer circle,
it marks the corner of the 3, 4, 5 triangle and is 2000rc from the
G.P. centre. It is " the bass player " and it becomes involved
in the movement through several complex "time signatures”, as we will see. Imagine
the circle around the G.P. as the drummer - composer's sound wave,
expanding out from the central point. We also need to draw the "sound
wave" circle for the bass player, with the same radius, so that
he can be heard. The intersection at the western second step of G3B
is the centre where we place the point of the compasses and the GP.
centre is our radius for the "bass groove”. It's a simple statement
that stays in the background but it gets the whole band dancing together.
As the "bass circle" is drawn 
its
circumference passes through only two significant structures, one
"hit" is the centre of the Great Pyramid, obviously, but
the only other hit is the head of the Sphinx. Wow! What a bass line!
Suddenly
harmony appears, out of nowhere, as the "bass groove" circle
intersects with the "baseline”, its natural "repeat sign”.
The pianist joins in with a powerful theme, " Sokar, the Maestro”,
has appeared. This is the "Gateway”; here is where all the equations
coincide. We now have the prime reference point from which to measure
and confirm all the intricacy of the composition. We have to draw
its sound wave to be sure the "piano" is in tune. Drawing
the "Sokar circle" with the same 2000 radius back to G3B
we see an amazing thing happen. First it passes through the S.W. corner
of G3, then the N.E. corner of G2, Khafre is now becoming involved.
The next hit is the eastern meridian point of the G.P.'s 2000 radius
circle. We have just constructed the Eye of Horus, or Vesica Piscis,"
the fish shaped vessel ". 
To
confirm that we have found the right rhythm section, the Architect
has left a wonderful note to be played right here.
Sitting
in the middle of the "eye" is Campbell's Tomb.
Campbell's Tomb is a shaft about 60 cubits (100 ft.) deep and
about 20 cubits square. It is made of massive construction blocks
that could, virtually, never be removed. The shaft could be filled
in but its footprint would never be erased. In the middle of the framing
stonework on the top of the western side of the shaft is an obelisk
pit, the obelisk long since gone, but its melody lingers on. This
is the precise geometrical centre of the eye. It marks the halfway
point on the line that leads from the Great Pyramid to the Gateway.
Halfway between Alpha and Omega.
Join the Dots
The
physical location of the "gateway" is a major clue in itself
as to what may be found beneath the sand. The hill, Gebel Ghibli,
is not very impressive, until you stand at the "gateway”. At
that point, on its northern side, it rises as a vertical cliff behind
you as you align yourself north to the eastern wall of the Sphinx
temple. It rises stubbornly from the debris that has accumulated over
the centuries and poses a question. Why was it not quarried away during
the construction period, why was it left with its vertical face?
All
the pyramids of Giza have their entrance on the northern side. The
entrance passageway of the G.P. is at a
1 : 2 ratio angle, the same angle as the Grand Gallery and
the same angle as from the G.P. centre to the "gateway ".
All the Kings were buried in underground tombs, and it has already
been demonstrated (note) that Akhnaten, grandson of Thutmosis IV,
built his capital, Amarna, geometrically aligned to focus on his tomb
in the hills to the east, where the sun rises. We have every reason
to believe that his grandfather, Thutmosis IV, taught him the principles
of Atenism, the monotheistic belief in which God is represented by
the disc of the Sun. But that belief has to incorporate geometry as
its fundamental model, as Akhnaten so definitely acknowledged, but
possibly he got a little too carried away with it.
Fundamentalism
needs to be appealing to, and be understood by, the masses. As in
Plato's Republic, educating the population into conformity has to
be a gentle process; it must comprehend the reasoning behind the "Philosopher
King" and understand the societal benefits of a truly wise leader.
It appears that Akhnaten lost the plot. The model he was using was
the right one, historically, but he tried to implement it in his own
lifetime, he tried to run before he could walk, and take the glory
for himself.
After two millennia, Akhnaten tried to reinstate a belief system that had been dormant in Egypt all that time. Why? Where did the inspiration come from? We think his grandfather found Sokar and, if he did find Sokar, what did it tell him? More interesting is who told Thutmosis IV where it was. He was the one who cleared the sand from around the Sphinx and erected the "Dream Stelae" between its paws. He is the one recorded as having found, as a youth, a stone in the shape of a divine hawk, which sounds suspiciously like an Omphalos stone. Was he shown an ancient archive and instructed in the Truth, World Order, Maat? Did the Knowledge get passed on to his offspring? Simon will elaborate on that theme, but we can decide later, after we see Thutmosis IV assume the God Form.....
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