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Exact
Reconstruction
of the Layout of the Great Giza Pyramids
Introduction
The geodetic survey of
the three great Giza Pyramids, published in 1883, by Sir William
Flinders Petrie, has since been a perennial subject of analysis for
geometric and numeric significance. To show that
the three great pyramids are parts of one and the same plan, there
have
been attempts
to generate their layout from pure
ideas, from a clean slate.
Indeed, Giza abounds with inspiring
relationships of varied accuracy, but, to safely pin ideas on
the original designers, those have to prove their
mettle by integrating into a system and being compellingly
accurate. Why should we believe the builders had been capable of such accuracy to begin
with? -- There is a
number of
well known examples showcasing the uncanny ability of Giza builders for
accuracy, thus setting a standard for evaluating our own
reconstructions as would-be copies of the
original Egyptian plans.
Often, the
problem is that various elegant elements of design discovered
at
Giza by various researchers seem a
foot or so out. Nevertheless, on a drawing board
35x35 inches, such ideas shall look absolutely accurate because their
inaccuracy is diminished thousandfold. The Giza designers had been well
aware of this convergence of multiple ideas. It adds greatly to the
puzzle quality, but has misled
many authors into wishful thinking.
John
Legon's reconstruction of the Giza plan, however,
stands out among other works in the field, for it works remarkably well along the east-west
axis. It is impressively accurate for three major
lines, the east and west sides of the
Second Pyramid, and the west side of the Third Pyramid. In contrast, the south side of the Second Pyramidc is 1.5
inches, or 40 millimeters off Petrie's plan, the north is 1 inch, or 25 millimeters off.
So, why did
the
builders not achieve comparable accuracy in keeping with their
plan along this N-S axis? We have seen that it was in their
power. Perhaps, Legon's theory is too simple.
Be it as it may, Legon is right
when
stating that axial spacing between the pyramids
approximates square
roots of the first five prime numbers, 1, 2, 3,5, 7, given in 250 x,
and 1000 x multiples of cubits. In fact, his reconstruction comes
fairly close to Petrie's plan, and had I been
familiar with it earlier, I might have accepted it, albeit
reluctantly. Legon's interpretation puts the ancient builders on a
level everybody expects - they had a simple plan,
and executed it really well, although somewhat
unevenly.
For this reason, I find it hard to grasp, why Legon's
theory
does not command more attention and approval from
the academy. His only sin is making plain the Egyptian knowledge
of square roots and Pi to several digits. That's it..
If Legon's theory smacks of heresy to the mainstream, what
reception is in store for a theory like mine? It presents a sophisticated exact construction
which duplicates Petrie's
measurements (stated by Petrie to the tenth of an inch) with superb exactitude. The Golden Section, square roots, and Pi
imbue it with a most appropriate spirit for sacred
ground.
I see our
scepsis prone academy not treating this design kindly, because it is,
frankly, too deeply thought out. After all, the Fourth
Dynasty arrived almost on the heels of hunter-gatherers. How could
its architects reach such sophistication so quickly? In short, if
pressured, the academy might embrace Legon's theory, as the simpler (although incaccurate)
explanation.
As a body, the academy scoffs at
the
very idea of a unified plan for all three pyramids. With so much smoke
around, this
is the burning question:
Was there a GRAND PLAN for the three great Giza pyramids?
A diehard skeptic will try to nip the idea in the bud and say that
even if Giza's groundplan could
be cleanly translated into a meaningful
geometric design, it
would mean nothing. Such prejudice seems to actually dominate
the academic establishment. Anybody, who dares
think otherwise is automatically discounted under various
unflattering labels.
Yet, the true
solution, due to being both accurate and meaningful, will tower above others. Accordingly, the faults of my reconstruction are so miniscule and so insignificant that rather than being faults, they are manifestations of precision.
The maximum
north-south distance between the pyramids is 35,713.10 inches.
To show a fault of 1/100"
in the plan on
a computer screen, we need to assign at least a pixel to it. Then the
total resolution has to be 3,571,310 by 3,571,310 pixels to
show the fault.
A computer
screen 36" square, would sport about 100,000 pixels per
inch in both directions. The fault
of 1/100" would show on this screen as 1/100,000".
Faults in my reconstruction range between 0.0014" to 0.035". Considering the size of Giza,
it can safely be said that the reconstruction is microscopically precise. Consequently, it is in all likelihood the same as the
original plan. By the same token, solutions, which draw over Google's satellite view of Giza need not apply.
A tribute
is due to Petrie's
brilliant achievement:
Although academic opinions suggest that
measurements of the Great Pyramid by Cole supersede those by Petrie, I
am confident in the belief that Petrie did a better job than
Cole
by the virtue of choosing the better method for the
circumstances. I believe in what looks so perfect through the
prism of my own reconstruction; My results
are an order of magnitude nearer to Petrie's actual numbers than to his
margins of error. It is, as if he were given the original plan to convert distances to the nearest tenth of an inch. In any
case, Petrie's Giza work had to be a superbly accurate mensuration of
superbly accurate objects. That would explain why there is
massive agreement between my
reconstruction and Petrie's plan, and none with
Cole's.
*
Length
of the Royal Cubit
This reconstruction
more or less accepts Petrie's cubit.
John Legon makes a strong case that
the
North-South distance between the pyramids was meant by the builders
to equal in cubits 1,000 times the
square root of 3 (1.732...).
As 1,732.05
cubits,
it would belie extremely accurate planning by the Egyptians, as it
extends to five decimals of the square root of
3. (1,732.05 squared = 2,999,997.2025, practically perfect three million, a reason to draw the square) The reconstruction works best at this scale -- flawlessly. Some sensational value
readings pop up, looking
definitely non-random as a
group. Exact numbers rather than geometric procedures are used in three
instances in the reconstruction, which had the wind out of its sails for a while until I
gave up using only dimensionless geometry and applied the cubit
scale.
In this manner,
a strong case is made for the exact length
of the cubit used in planning Giza's layout. It is also
clear that the reconstruction would not work on any other scale, for
then it could not convert to the above-mentioned exacting numbers.
Petrie's measurements at Giza,
and inside the Great Pyramid had produced slightly differing values for the cubit. He settled for an average of 20.62" (inch) per cubit. This reconstruction's cubit
is 20.61897", only 1/1000" off Petrie's value.
South
to North between the pyramids = 35,713.1
inches
= 1,732.05
cubits = a side of the Pyramid Square
1
cubit
= 20.61897..
inches = 523.72202880..
millimeters
It
makes good sense for the designer to stop
at the exact
value of 1732.05
cubits (here is an example of a puzzle using
decimals from La Marche,
Stone-Age France ). The digit 5 for
hundredths is
followed by a
zero, which means no thousandths (milli-cubits?)
to deal with (1/1000 cubit
is just over half-a-millimeter). The next digit is already too fine,
so, this is a natural cut-off
point. As a
representation of the square root of 3, this value differs
from the true by eight
ten-millionths of a unit - 1.73205080..
How
to Recreate
Petrie's Giza Ground Plan From Scratch
part
one
The
Foundation - Classic Geometry
The
Giza plan evolves from a solid theoretical foundation - the
Golden
Section, showcasing the quickest, simplest construction of the regular 5-pointed
star. The strange thing is that there is no mention of this technique anywhere on the internet, so it is definitely obscure. As for myself, I learned it
from the Nazca-monkey. That is not to say no one did this construction in modern era
(post Egyptian), but it is mildly surprizing that efficiency of
construction is not a greater concern nowadays.
 Diagram
1
Start
with the above classic procedure. It begins with a horizontal line,
and takes ten
steps. Two of the steps involve help
circles (to draw the axial cross),
and these are not shown for clarity reasons. The eighth step draws the
key Golden-circle ('c' in the diag.), which is
centered in
the bottom tip of the axial cross. On steps nine and
ten, lines
are drawn from the top of the axial cross, as tangents to this
circle (at P1, and P2). These lines create an angle of 36
degrees exactly
(like
on a 5-pointed star).
Before going on, this simple diagram already has two crucial elements:
* The
base length of the golden triangle, given by the 36° angle
intersecting the horizontal axis, is the same as one side of the
initial Great Pyramid in this reconstruction..
* The
circle from step 2 of the construction, is the actual instrument, which adjusts the initial Great Pyramid to
within ¼ millimeter of Petrie's plan. If this sounds
baffling, check it out in action later. I
thought that I should bring this to the reader's attention now to
emphasize the strong bond between this pattern, and the
reconstruction.
Diagram below:
Three
more steps complete the star: On step eleven, a
circle from the point Q (the
Q-circle)
is drawn through the top and bottom of the
axial cross. This creates two points (1,2), which
are then connected by lines to the bottom point of the axial cross (3). And that's it! The star is complete. (In trade terms, the simplicity
of the construction is 13).
Among constructions, the unique element of this construction is the Q-circle, or its
mirror
image on the other side of the vertical axis.
 Diagram
2
The position below is based on diag.1,
but is
rotated 90 degrees counterclockwise. It creates a slew of golden rectangles with beautiful simplicity.
 Diagram 3
* A
line from 'A'
through '1' and on to
'D' has the angle of a diagonal in a vertical golden rectangle, hence
'ABCD' is a golden rectangle.
* A line from 'B'
through '1' is equivalent to a golden diagonal, as well.
* A line
from 'D' perpendicular
to 'AD' is a golden diagonal in 'CDEF'.
* The combined form of the two golden rectangles (ABCD + CDEF), the 'Horizontal
Column', will be our key to setting the pyramid's side to within a quarter millimeter of Petrie's value.
* The Horizontal Column is
next transformed into a square. One of the ways to do it is adding the golden
circle's diameter to the
underside of the column.
* A 45° line drawn from 'H' will
be one of the diagonals of the Great Pyramid. '
* E'
- will be the center of the initial Great Pyramid in our plan.
A line through the points of intersection of the
two golden circles in the diagram is yet another golden diagonal.
The
ratio of the Horizontal Column's height to the height of the
rest of the square is 2(Φ-1).
The
points A-B-I-J-H mark four segments in a row, where each
segment forms the Φ-ratio with the neighboring
segment(s).
Corners of the Initial Great Pyramid
In the diagram
below, on
both the blue 13-Step star, and its derivative -
the
red star, distances
such as 'AB', and
'EF', are equal to 'CD', a side of the
initial Great Pyramid. At
this scale, it is impossible to see any difference between the interim,
and the true-size Great Pyramid.
The diagram shows one way of drawing the pyramid's print in the sand.
A line-segment through 'P' and 'C' is the same as one of the diagonals in a horizontal
golden rectangle; it rises to 'C' to meet the left-rising diagonal of
the initial Great Pyramid at the pyramid's northwest corner. The segment is key to further construction.
The green square's
extended side crosses a line of the red star at 'X'. This point is
only 4 mms distant to the side of the Second Pyramid. .
Diagram 4
Lines
a, b, c, and d, all have the golden diagonal angle.
In the diagram below,
three of the four - 'a', 'b',
and 'd' project the Great
Pyramid
onto the "13-step" construction.
Line 'b'
is tangential to the inscribed circle of the pyramid,
and that also
permits the
reconstruction. This is the only generic way I know of to construct the initial
Great Pyramid. Every other way has to do with the 13-Step star..
Line 'a': As a radius, it will usher this reconstruction
from theoretical analysis to the applied stage. Let's call it the 'transmission-circle', when it is its turn.
Line 'b' measures 1642.00222202
cubits, a typical measurement in cubits for this reconstruction.
 Diagram 5
In the diagram below: One Q-circle intersects sides of the diamond (square), the other one intersects extended lines of those sides. Lines 'e'and 'f' originate from these intersections, and then intersect the initial pyramid's diagonals at its corners Either 'e', or 'f' suffices for
the pyramid's construction.
Altogether, there are five procedures, each projecting the same initial pyramid. Only
one is
generic, the others are firmly connected to the 13-Step construction. The
design's nature is itself a strong hint at the
designers' familiarity with the entire spectrum of
possibilities therein.
If the designers had known only the single generic way to project the
initial pyramid, the incentive for selecting it at all would
be
strongly diminished.
 Diagram 6
If the
NE corner of the initial Great
Pyramid (G1) is exact in this blueprint, the other
corners
are over 6 inches short of Petrie's locations. Yet, that
distance shrinks to an invisible nothing on any drawing board ( 1/100
millimeter, if the board is 35" tall).
Having the initial pyramid sets the stage for its adjustment to the exact specifications given by Petrie.
The North-South division -
Location of the south side of G2
The point marked 'X' in the above, or below diagrams sets a point on the south side of the
initial Second Pyramid (G2).
If the height of the big square is Giza's 907,112.74
millimeters, then 'X'
is 4.2 millimeters above the south side of
G2, as given by Petrie.
A
fact to note here is that without
the '13-Step' star there can be no point 'X'. Without 'X', the best result in the location of the Third
Pyramid's SW corner (to 0.51 mm) would not happen.
On
the same scale, the length of a side of the initial Great
Pyramid, as well as a side of the pentagon within the 13-Step Star, is remarkably precise 439.50009259 cubits. It's a
strong signal that the units used are
correct.
The distance
between adjacent tips of the 13-Step star, or
one side of the smaller star =
1150.626180 cubits. Here, we see five
consecutive digits of Φ squared (2.6180).
Coincidence?
Diag.7
The
Pyramid Square
After the
initial pyramid
(proto-pyramid), the big
square in diag.3
is extended to the pyramid's north-east corner to function
as a containing
square for the pyramids - the Pyramid Square (35,713.1"
or 1,732.05 cubits per side).
How
to Recreate
Petrie's Giza Ground Plan From Scratch
part
two
Building Upon the Geometric Founfation
Two
Look-alike Circle Pairs
Much
like the ubiquitous instruction guides for "dummies", the Giza
designers provide easy to follow guidance in the form
of pictorial
clues. All one has to do is just get on the right
track.
After the initial Great Pyramid, the following catches the
eye: The widely used Giza containing
rectangle, and the Pyramid
Square have a common south-eastern corner. I
wonder, how many people had drawn
an
experimental circle from there to touch the
Great Pyramid's
circumcircle, which then seems to touch the south-eastern corner of G3,
as well. It is a definite visual impression. Moreover, this circle
appears
to be the same size as the 'e-circle' in the diagram below, also called the 'transmission' circle. Its radius is the golden
diagonal from the pyramid's NW corner, Diagram 4.
diag.8
Not viewing things in the context of Pyramid Square eliminates seeing the Great Pyramid's
circumcircle touch G3 from the other side, when copied to the SW
corner
of the
Pyramid Square (diag. above). This is certainly a startling effect,
albeit up close it is not all
that
accurate.
In
the diagram, the pyramid's circumcircle has been expanded
slightly to touch the transmission circle. So, we have
two pairs of alike circles
,
which look the same from
high up above Giza.
Another
special effect is shown below. Line-g drawn between the
intersection (on
the left) of
the two transmission circles , and the SE corner of
the initial
Great Pyramid then duplicates
the angle of the other golden
diagonals to 0.0015º.
This is a very fine
value. And conversely, the true golden diagonal drawn from the
same corner then
comes to within 0.66'' of
the
same intersection.
Very
Special Effects
This latter transmission circle-pair wheels the entire reconstruction fast forward,
as there are more special effects.
One involves the big circles from each
pair (described above).
 diagram 9
The diagram
above is a close-up from diagram 7 of the situation in the south-east corner of G3,
the Menkaure pyramid. Lines 1 and 2
are sides of the pyramid, 'a' and 'b' are the two big concentric circles (the transmission circle,
and the look-alike circle touching the Great Pyramid's circumcircle).
The circles are spectacularly equidistant to the pyramid's corner!
The centerpoint of the distance between the circle lines
is 3.4
millimeters to the
east of Petrie's pyramid corner. The distance itself is 2.001
cubits, a highly precise value.
Since the
Great Pyramid's position is reconstructed with great accuracy later on
in this article, the trick can be repeated. It
then pins the southeast corner
of Menkaure's pyramid 3.1 millimeters to the east.The distance
between the
aforementioned circle-lines then works out to 2.001011 cubits.
Evidently,
measuring 1 exact cubit to the east from the circle 'b' will
come 0.0005 cubit ( 0.25 millimeter ) closer to Menkaure's
corner.
The
same procedure using the circle 'c' (below), which is
tangential to the circumcircle
of the initial Great Pyramid, produces the SE corner of the interim G3 - version # 2.
It is 3.68 inches to the west of the original, and half-way between a and c.
diagram
10
The "Channeling" Method
Given two possible
versions of the original,
which create a channel of parallel, or concentric lines, often, the
solution is simplicity itself - a single step:
"Go
down the Middle of the Channel!"
This is the method
that duplicates the Great
Pyramid in Petrie's plan by yielding the exact location of its NW
corner (the NE corner is known by
default).
It doesn't end there, though.
In one fell-swoop,
the same procedure (substituting Horizontal
Column for the Rising Column) yields a fairly accurate
location for Menkaure's
SE corner - half-an-inch (0.506"), west of its
location in Petrie's plan. Is this not elegant?
To digress a little, the symbolic language of coincidences around
the transmission circle lets us
follow the trail, but as long as I started the reconstruction from a
golden rectangle without the precursory steps, I was left wondering: Why the transmission circle? Then
I saw its radius in context of the '13-Step' construction (diagram below).
Point '2' is a big point in the original 13-Step-star's construction. The radius 3-4 duplicates the golden circle's radius from the same construction. The segment 1-2-3-4 was used in the construction of the initial G1, and now serves as the radius of
'transmission-circle'. Thus, the continuation builds upon classy geometry.
diagram
11
Menkaure's
initial vertical axis ('go down the middle')
The
midpoint of the gap between the circles sandwiching
G3 in diagram 8, is 4.2 inches to
the west of Petrie's vertical axis for the pyramid. Draw a
vertical axis from there for our initial G3.
The Initial Third Pyramid (#1)
a)
* Center a circle at the point, where
this vertical axis crosses the Horizontal
Column.
* Have it touch the far side of the little
circle
in the column's SW corner.
The little circle is already a part of the position,
since its diameter is
a side of the
inner pentagon of the '13-Step star', and it is also
the same as the inscribed circle of the initial G1.
b)
The
section of the vertical axis below the new circle is taken as equal to
one side of G3. On the basis of this assumption, the pyramid
is completed around the axis.
 diag.12
Together, the NW-corner of the interim G1 and the SE corner of
G3 in the above #1 version enable a simple
operation which makes the reconstruction of the Great Pyramid
identical
to Petrie's
plan.
Operation Rising Column
Robin Cook says that if we
enclose the pyramids between 45º lines perpendicular
to their N.W. by S.E.
diagonals, as in the
diagram below, the long axis of the resulting rectangle (the Rising
Column) is almost exactly the same as one of the Second Pyramid's
diagonals.
diag.
13
The axis actually runs 13.82
inches east of
the pyramid's diagonal; however, this relationship does look accurate on computer screens
or paper.
I noted yet another special effect in
this position, one three times more
accurate: the width W-Z of the Rising
Column is just 4.32 inches more than the width A-B of the
Horizontal Column. Comparing, or substituting the reconstructed columns
for each other could therefore
be of interest, and is next on the agenda.
The idea
that the bottom side
of the reconstructed Horizontal Column should also be the
horizontal axis of Khafre's pyramid (G2), is a 10.12
inches miss. Marking
the actual (Petrie's) thickness of the Rising Column straight
down from the top side of the reconstructed Horizontal Column gets
to 0.94 inch south
of the second pyramid's horizontal axis. This special effect will be put to use later.
Exact
Repositioning of the Great Pyramid
Place the Horizontal Column over the Rising Column (axis over
axis).
Then the situation
in the Great Pyramid's NW corner
looks like the following diagram.
 diagram 14
'go down the
middle'
The long axis of the channel between the western sides of the
two
rising columns is identical to the western side of the rising column in
Petrie's
version. Or, a line
drawn vertically down from the northwest corner of the wider Rising
Column becomes one with the western side of the Great Pyramid in
Petrie's version.
On the line of
the north side of the pyramid, both the west side of the wider Rising Column
and the NW corner of the proto-pyramid G1, are
equidistant to this corner
(as given by Petrie). The difference in length between
the reconstructed and the original versions of
one side of G1 is too tiny to be noticed, at 0.0101.. inch (1/97),
or 0.26 millimeter, or 0.0005
cubit. The
pyramid centers are 0.007
inch, or 0.19
millimeter apart. In
other words, the two versions of the Great Pyramid are
identical.
Our value: 439.827 (439.82732..) cubits,
or
9,068.8
inches ( 9,068.79) per side.
439.82732 /
Pi
= 140.001..
The
pyramid
needs to be about 1 millimeter higher than its theoretical height of
280 cubits, to be perfect with respect to Pi, and this reconstruction..
Petrie's
value:
439.828 (439.8278..)
cubits or,
9,068.8
inches
Petrie's plan is under .005 cubit (about 2.5 mm)
longer than the optimal
value of 439.823
for Pi encoding, while this reconstruction is closer by a
hair:
Pi
times half the pyramid's
height = 439.82297150.. or, 439.823 rounded
A Show of Consistency of Method
The very same procedure works in
the diametrically opposite corner of the Rising Column, as it locates the SE corner of G3 to
0.5". It is far better than solutions by others,
bar Legon, who is only twice as far out, as he misses by an inch.
This is # 3 version of
initial G3. (the SW corner remains a constant in all three versions).
diagram 14b
In the diagram, a and b are sides of the #1 initial version
of G3, and
'c' is the reconstruction of the eastern side of G3. The line "c" is an
obvious improvement over "b"..
An Accurate Reconstruction of the SE corner of G3
With the Great Pyramid duplicated, it is
now possible to repeat the
procedure
from diagram 8.
diagram 8
Above
is a close-up from
diagram 8 of the situation in the
SE corner of G3, Menkaure.
Lines '1' and '2' belong to
the pyramid. Lines 'a' and 'b' are the transmission circle, and
the tangential
circle to
the Great Pyramid's circumcircle. Going down the
middle again, the axis of the channel between them is 0.13
inch, or 3.1
millimeters to the east of the pyramid corner as given by Petrie.
This
formidable result is as close as I got to Menkaure's SE corner by pure geometric
construction, without resort to units.
Records are made to be
broken
The radius of the cyan circle spanning the distance
between lines 'a' and 'b', is 1.0005..,
a rather exact cubit, to about 0.25 millimeter. It follows
that measuring one exact cubit eastwards from the tangential
circle, like in the diagram above, comes 0.25 millimeter closer, at 2.85 millimeters east of the SE corner of Menkaure. I
should mention that Legon
could follow this procedure as well, and get almost as good a result.
at 3.15 millimeters from the said corner (I brought this to his
attention, the two letters I had sent him are near the end
of "Notes").
The reconstruction above is by no means obvious, but once found, it could become a perfect false door, as it might distract one from finding the one ingenious solution for the same corner,
which is a whole order of magnitude more accurate. It uses whole units as well and
is meaningful with respect to Pi.
diag.
15
The
location of the reconstructed SW corner of G3, the third
pyramid, yields some remarkable readings in cubits.
a)
The distance from the SE corner
of
G3, as given by Petrie,
to the SW corner of the Pyramid Square is:
516.005, 516 cubits
almost
exactly.
b)
The distance
between the reconstructed south-west, and the Petrie given south-east
corners
of the third pyramid becomes what many authors posit to be its
intended
length:
201.5
0246
or 201.5 cubits (the small change 0.0026 is worth over one
millimeter)
c)
THe distance between the SW corner of reconstructed G3,
and the SW
corner of the Pyramid Square:
314.5
0275
The above 201.50246
and 314.50275 have very similar fractions:
314.50275
-201.50246
= 113.00029
Put into words, if we flip over westwards
the distance between the reconstructed SW
corner of G3 and Petrie's SE corner, it is short of the SW corner
of the Pyramid Square by:
113.0003 cubits , which is just 1/6 millimeter
short of being perfect 113 cubits.
This remarkable arrangement offers an easy
way of
reconstructing the SE corner of the Menkaure pyramid to 0.0003 cubit,
and an absolute zero on the Giza scale.
Reconstruction of
the Third Pyramid's SE corner
* Mark exactly
113 cubits from the SW corner of the
Pyramid Square towards the SW corner of G3.
* The
remaining gap to this SW corner of G3 becomes the radius of a circle centered in the same corner. This circle
then locates the SE corner, as
given by Petrie, to within the above mentioned (1/6) millimeter, or 0.0003
cubit.
In
plain English, the two locations are perfectly
identical.
π
Numbers 113, and a 314 in association evoke a certain Pi approximation.
113 x
π
= 354.9999.. 355
/ π =
113.0000096
The
circumference of a circle with the diameter of 113 is a perfect 355 for
all the practical purposes.
355/113 approximates π close
to perfection:
355/113
= 3.141592.. - The best
approximation of Pi given as a
ratio of two whole numbers.
Accident?
If so, it joins a
plethora of coincidences pertaining to Pi, for which
Giza,
and especially the Great Pyramid are famous. Without it, there would be
no exact reconstruction of the SE corner of G3,
however. In turn, we
need that one for the
exact reconstruction of the Rising Column. The latter then takes part
in the finding of G2's center.
http://www.ronaldbirdsall.com/gizeh/petrie/c10.html
The
Layout of Khafre's Pyramid (G2)
Lumber
in the Yard
Earlier
the reader saw a
way to position the south side to within four millimeters of
the original Second Pyramid. Given the center, the
reconstruction would come out very nicely. Some
usable data
is mentioned in
the
note under diag. 19: 'marking
the actual thickness of the Rising Column (W-Z), downwards
from the top side of the Horizontal Column, gets to
within 0.94 inch
south of the Second Pyramid's horizontal axis'.
We can duplicate
the Rising Column, and in so doing, get to within 0.94 inch of
G's horizontal axis. Yet, although this
is a nice approximation for the horizontal axis, its true
function is to take part in locating the vertical axis.
"Channeling"
the Solution of
the NW by SE Diagonal of Khafre's Pyramid (G2)
G2 - diagonal simulation # 1
Alison's
circle
Jim Alison had read the
position in the diagram
below, as
saying that a
circle centered in G3, whose radius
is the horizontal
distance between the centers of G1, and G3, closely approximates
the Golden Cut in the given line (marked Phi).
Alison's circle actually misses the ideal cut by about ten inches.
In our experiment, a
line segment runs from the center of the #2 interim version
of G3
to an intersection of the interim Great Pyramid with one of its diagonals.
Next, divide this segment by Phi. The longer section of the segment is the radius of the correct Alison-circle.
Consequently, the golden diagonal 'c' in the diagram, and the Alison's
circle intersect at "I. The point "I" lies on a simulation of G2's the diagonal. This relationship
is an order of magnitude more accurate than the one Alison noted. The simulation is 0.66
(0.6592..) inches northeast
off the original.
diagram
16
G2 - diagonal simulation
# 2
Two major lines: 'a' the axis
of the initial Rising Column (RC was the key to Great
Pyramid's duplication)
'b' the bottom line of the Horizontal Column
meet 0.66 (0.664..) inch southwest of
the diagonal 'd' of the pyramid in Petrie's plan.
This creates a
point of insertion
for the diagonal simulation 'c'.
diag. 17
The channel axis between the diagonal simulations #1 (line 'e'
in the diag), and #2 ,
runs 0.0001 cubit,
0.003 inch, or 0.07 mm southwest of the diagonal as given by
Petrie..
It is truly identical to Petrie's diagonal.
The diagonal simulation #2 will serve us again and help complete the solution for G2.
------------------------------------------------------------------------------------
There
is also another way of replicating the same diagonal, which comes
to within 0.006 inch of Petrie's version. It is really
simple . See it in the notes at the end of the article.
The
Vertical Axis of the Second Pyramid & the Center
Earlier,
the
Horizontal
Column, when substituted for the Rising Column, let us
duplicate the Great
Pyramid, as it is in Petrie's plan. But, since this is the
ambitious Giza plan, the same trick works in reverse, and with even
more spectacular results! The
Rising Column in its final form, when suspended from the
the top line of the Horizontal Column, falls 0.94
inch south of the second
pyramid's horizontal axis (cyan in the diagram below). This
line simulates the axis nicely, but its true purpose is
different.
It meets the Diagonal
Simulation #2 0.0014 inch, 0.035 millimeter, or
0.00007 cubit east of Petrie's vertical axis. An unaided human eye cannot see 7/100000 cubit. Considering the size of our workspace, Giza, any hopes at the
outset for this kind of results would be ridiculous.
diag.18
The vertical axis and the channeled diagonal together
locate G2's center a pinpoint away
from Petrie's plan, at 0.005 inch (1/200"), 0.13 millimeter, or
0.00025 cubit.
We know the position of the southern side to 4 millimeters since early on,
therefore G2, the Second Pyramid can be recreated with impressive
accuracy even before the final adjustment.
Final
adjustments - the Third Pyramid
Having the initial Second Pyramid permits testing Legon's
ideas in our
settings.
1:
The
east-west (axial) distance between the west
sides of the Second and Third Pyramids equals
250√2 cubits.
This postulate works nicely for this reconstruction, locating
the west
side of the Third Pyramid 0.006
cubit (3.45 millimeters) west of Petrie's
version. (It
doesn't work too well for Legon's reconstruction.)
2:
The
east-west distance between the center of the Second Pyramid
and the west
side of the Third Pyramid equals 250√2 + 205.5.
This
postulate works even better, locating the west side of the Third
Pyramid 0.003
cubit (1.5 millimeters) west of Petrie's version
3:
Legon
also implies that counting westwards from the east side of the
Second Pyramid, 250√2
+ 411 cubits locates
the west side of the Third Pyramid.
This formula works the best, as it locates the west
side, or the SW corner, 0.0014 cubit (0.75
millimeter) east of Petrie's plan. The other
corner of
the south side is given with an even greater precision, hence
the Third Pyramid
stands recreated exactly as given by Petrie.
This formula also works nicely for Legon's reconstruction - as
it
gets to 0.01 cubit (5.4 millimeters) east of Petrie's
version.
Diagram
19
4: Working with the Pyramid Square confers an
opportunity to note another Legon-style formula for the reconstruction. Draw line d eastwards from
the SW corner of the Pyramid-Square to the length of 250√2. Then subtract 39
cubits from it. The new segment ends
3.32 millimeters
west of the Petrie's SW corner of G3.
Moreover, segments b and d
have a horizontal overlap
of 0.00026
cubit, 0.005 inch, or
0.13 millimeter!
How
much more proof does one need to recognize that Legon really did
discover something? This is also yet another example of the
validity of the Pyramid Square's concept, which Legon never
worked with, unfortunately.
So, far we counted four accurate ways to deploy an exact formula containing 250√2 towards location of the SW corner of the Third Pyramid.
As to its SE corner, earlier it was located by the means of 113, hence another exact number.
Again, there is the consistency
of design!
So which solution should we use? The fact is that there are at once
four accurate solutions for the
west side of the Third Pyramid, or its SW corner. All four
are accurate - since Petrie's points come with a ±
radius, those points are
in reality
small circles, or
dots. Multiple solutions are designed into the Giza
puzzle, a show of sophistication. We shall never learn, which version
was
the one implemented on the ground.
Final
adjustment - the Second Pyramid
Draw a line westwards from the vertical
axis of the reconstructed Great
Pyramid, whose length is
250√3 , or 433.0127..
cubits). Make this the distance
to the east side of the Second Pyramid. The fault from Petrie's version
then is 0.00126 cubit, or 0.66 millimeter.
Since the center of this pyramid is already located with
utmost precision, the adjusted Second Pyramid
is identical to Petrie's version.
Conclusion:
Petrie's layout of the great pyramids of Giza can be
accurately
recreated from scratch on a clean
slate, beginning with the
'13-Step' construction of the regular 5-pointed
star from a line segment, with some involvement of the basic
prime number square root values. The method
has simplicity, total accuracy, beauty, and intellectual
depth. With such attributes, it must be essentially identical to the actual Egyptian procedure of
planning the Giza layout. For instance, working on any other scale would eliminate the phenomenon of whole and round numbers magic. The North to South
distance
between the pyramids must be taken as exactly 1732.05 cubits
(1000√3
given to
six digits)..
It is noteworthy that the method does
not work with the classical tools of geometry, at
least not without a
microscope. Given the scale of Giza, and
the hairline
differences between drawing objects, the plan had to
be worked out mathematically. In that case,
the knowledge of mathematics guarded by the temples was
clearly on a level unattainable in a neolithic
society less than two millenia removed from the
hunter-gatherer stage.
Asked
beforehand, if a microscopically exacting solution to the Giza ground
plan were possible, I would have naysayed it, for even if there
were an overall plan based on exact ideas, and it were
executed
flawlessly, it would not be possible to measure the perfect result
accurately enough. This would have given rise to discrepances.
Yet, we (the readers and I) see
such a solution
here, thanks to Petrie. I still see it as somewhat inexplicable, for no
matter how good Petrie had been, he himself admitted to a much greater
uncertainty in his measurements.
Of course, my theory about the Agency would explain many
things, but a discussion of the Agency is not an object in this
article.
Jiri
Mruzek
Vancouver,
BC
©Jiri Mruzek
April 15, 2007
Notes
Drawing
Inspiration From Ideas of Others
Intial
Observations - The
Pyramid Square & Khafre's Pyramid
The breakthrough to solution of the Giza plan came from
extending the enclosing rectangle (blue) of the three pyramids, which is rather ubiquitous in other studies, into a
square.
This simple step is very helpful, if not absolutely vital to
understanding the position, but incredibly, no one before me had taken
it!
In this framework, some prior
interesting, but seemingly dead-end, observations by others suddenly become meaningful.
You can see that the Second Pyramid's vertical axis is surprizingly
close to the vertical axis of the Pyramid Square. Next, I divided the
square by the golden
section (green lines). Obviously,
the square base of the Second Pyramid mimics the
small square in the center of the green cross.
For a follow-up experiment, the Pyramid
Square is given a
basic Golden Section grid (diag. below).
Lines of the grid create a Golden-cross within the
square.
Next, another Golden-cross is extrapolated from G2's (Khafre's) square. The two
Golden-crosses are then superimposed over each other for
comparison, as below. The similarity in size is striking.
Below - another try at the same thing
At this magnification, the
result is an
almost perfect illusion of identity . Could
this illusion be a hint that
the design of Giza has something to do with the Golden
Section?
In the diagram
below, golden proportions
added to the G2 in its real location seem to find some correlation to
the south side of G1. Here as well, we encounter facsimiles of golden
rectangles.
diagram b
These
results were encouraging. Not wanting to
rediscover the wheel, before doing anything else, a search was in order
on the subject of
Giza layout. There is an informative article over at Jim
Alison's site: http://home.hiwaay.net/~jalison/gpsp.html
It deals with work by John A.R. Legon, Chris
Tedder, Robin Cook, and Jim Alison himself on
various notions of a ground plan of Giza's
major
pyramids. Note how none of the sites refer to this study,
although there was ample time to make changes. Am I a Johnny-come-lately, am I trying to eat someone's
porridge?
http://www.legon.demon.co.uk/gizaplan.htm
-
Legon's site
http://sevenislands.tk/
- Cook's site
http://www.kolumbus.fi/lea.tedder/OKAD/Gizaplan.htm
- Tedder's site
* Perpendicular
distances between the pyramid centers produce
two golden rectangle facsimiles (ABCD, and DEFP).
Chris
Tedder
sees two golden rectangles given by intersections of lines
in cardinal directions emanating from the center of each
pyramid.
He
proceeds to propose a possible plan, the Egyptians might have used.
Unfortunately, these rectangles have poor accuracy,
a fact
that reflects negatively on the level of Egyptian skills, and thus
the Egyptians were once again the scapegoats for the mistakes of modern
scholars.
Tedder's site:
http://www.kolumbus.fi/lea.tedder/OKAD/Gizaplan.htm
The Giza pyramids dictate their containing
rectangle. The rectangle dictates
the square. The square
constitutes the proper context for analysis.
The
Breakthrough
Adding Tedder's idea to the
context of the Pyramid-square gives a
spectacular result!
Both golden
rectangles create two
new golden rectangles with the
Pyramid
Square:
Tedder's
Secondary Rectangle #1 - An
extension of Tedder's horizontal golden
rectangle A-B-C-D to the west side of the Pyramid Square is
itself a facsimile of a golden rectangle, the vertical rectangle C-D-O-K
C-D divided by C-K = 1.627
Tedder's
Secondary Rectangle #2 -
Alison's circle
intersects the extended diagonal of the second pyramid rising north due
west at the I-point. The distances I-J and I -
L form the
golden ratio.
I-J / I-L =
1.6199 less than 2/1000 off
the true Φ value
The horizontal rectangle I-J-K-L is therefore an
excellent facsimile of
a golden rectangle. 
diag.
d
The diagram below shows how the combination of
the old rectangle with the newer one creates a horizontal column, which
is more accurate as such (a combination of two true golden rectangles)
than either of its components.
Pay good attention to what the top rectangle ABCD from the previous diagram does in the context of our
Pyramid Square in the diagram below.
It now extends from CD to OK. This new rectangle, ABOK is vastly superior to ABCD in representing an
exact idea. While ABCD is a downright lousy imitation of a golden rectangle, it is impossible to tell ABOK from a combination of two exact golden rectangles ( ABCD, and CDOK), one vertical, one horizontal. At least on this scale, since the actual exact figure is 10 inches thicker than the original ABOK.
In
other words, the North-South distance between the two big pyramids
provides a perfect visual experience of a sacred figure of two
golden rectangles, in conjuction with the East-West distance from
the Great Pyramid to the western side of the Pyramid Square. The notion
that this figure was sacred to Egyptians, is supported by my
analysis of a door from Hesire's tomb (Hesire was a colleague of
Imhotep). http://www.vejprty.com/hesirefn.jpg The figure of Hesire engraved upon the door is contained in the same rectangular figure ABOK.
Meanwhile,
we can also see in the diagram, how really inaccurate the original ABCD
was as a golden rectangle, because the original CD line is many cubits
to the west of the exact line.
Moreover,
this diagram lets us see, how the
golden diagonal emanating from O is impossible to tell from a line made from the same point to be tangential to the Great Pyramid's
inscribed circle.
All this is important, because armed with these facts, it is already possible to geometrically reconstruct with visual perfection the Great
Pyramid's square base in relation to the Pyramid Square. The pyramid's
side will then be shorter than
it is in reality by on this scale
invisible seven inches. So, if we suppose that this was how the builders had evolved the Giza plan,
their skills suddenly look vastly superior to the level set for them by
Tedder.
It is unusual that as far as I know, no one
had worked with the Pyramid Square concept before. After all, extrapolating a square
from a rectangle
is so simple! It was the first
thing I had done once I'd had
Petrie's ground plan in a
CAD drawing (Computer Aided Design). To me this step was
elementary, because all my
previous case studies in ancient science-art had involved an all important
square.
Starting
from the
Horizontal Column
This
was my original method of reconstruction of the initial Great Pyramid:
Draw
two golden rectangles, a vertical
one on the left,
and a horizontal one on the right, diagonals
radiating from their corners,. The rectangles form a single
column - the Horizontal Column.
C
divides A-K so that if C-K equals
Φ - 1, then A-C equals Φ, and A-B equals 1.
The
length of the combined rectangle (the Horizontal
Column)
then is 2Φ - 1.)
1) The center of the Great Pyramid is at the top
right corner
of the Horizontal Column.
2) The left side of the Horizontal Column is the
western side of the
Pyramid Square.
3) The diagonal 'a' is tangential to the inscribed circle of
the Great Pyramid (diag.6). This circle is then enclosed in a
square, i.e., the pyramid sides.
4) The lines through the north and east pyramid-sides are
corresponding sides of the Pyramid Square. With three sides of the
square known, so is the fourth - the bottom of Square. The
south
side of the third pyramid lies on the square's bottom (diag.7).
diag.7
Naturally,
the golden rectangles above could be preceded by a number of different
starting positions. But, only the '13-Step' stars
produce a
close location for one side of G2.
*
Alison's
circle - The segment F-H
is very close (0.8º) to
holding the 45º angle from the horizontal. A circle, whose
radius is the east-west distance between centers
of Khufu and Menkaure pyramids, is drawn from the center of
Menkaure's pyramid. It then divides F-H at G by the (quasi)
golden
proportion:
22,616
inches / 13,954.114
inches = 1.621
diag.e
Legon's
theory - Competitor or Contributor?
Since my own theory draws some important benefits from
Legon's
work, he has my gratitude and admiration. As for my criticism of his work, a brief peer review
of
sorts, first
of all, our respective approaches differ greatly. While
this reconstruction is done completely
from scratch, Legon starts out by
copying the Great
Pyramid
(G1) directly from Petrie's plan. Then he derives the rest. At
first, he uses basic geometry, but from there it is mostly a
march of
numbers.
My reconstruction is essentially visual, it is a progression
of
pictures, each showing special effects, which
embolden further
inspection. It is a show packed with action, exactly
as intended by the original designers!
It has long
accepted Legon's premise that
the north-to-south span of the pyramids
expresses the square root of 3.
Acceptance of this idea is a basic requirement. However,
Legon assumes that
the span was intended to be 1732 cubits
exactly. But, it is crucial
to the reconstruction that this distance be taken as more
accurate by two decimal points
of 1000√3, i.e., 1732.05
cubits.
Not using this scale would eliminate the
best result for
the location of the Third Pyramid's south-east corner. The next domino
to fall would be the microscopically accurate reconstruction of the
Second Pyramid's vertical axis. Other dominoes would follow. Only the
Great
Pyramid would remain unaffected.
Reconstructing
Legon's reconstruction - Legon
does not
give the actual procedure. Rather he states several possibilities,
which are all supposed to lead to more or less the same
accurate
result. Consequently, I select those that yield the very best
result on his behalf.
Because there are two possible scales of reconstruction (N-S =
1732 , and 1732.05 cubits), I felt
obliged to
carry out a reconstruction for each.
Also, Legon says that he measures from the center of
the Great Pyramid's theoretical version, the one with an even
number of
cubits per side (440), but actually works from the center
of the version given by Petrie.
Differences
in Legon's plan from Petrie's plan -
When N-S
(35,713.1 inches) = 1732 cubits
Second pyramid
Cubits
Inches
Millimeters
South side
0.0606
1.25
31.76
East
side
0.0012
0.025
0.63
North side
0.0482
0.995
25.26
West side
0.0136
0.280
7.13
Center
0.0549
1.132
28.77
Third
pyramid
Cubits
Inches
Millimeters
South side
0
0
0
forced
North side
0.0603
1.24
31.6
West side
0.0082
0.169
4.3
East side
0.0521
1.075
27.29
Center
0.0373
0.682
19.54
Legon's procedure:
Start by drawing the averaged out square base of the Great Pyramid,
as specified by Petrie. Next, extend the east side to the south to equal 1732 cubits
(Ö3), or 35,713.1 inches. This will be the axial north-south
distance from the north side of the Great Pyramid to the south
side of the Third Pyramid.
Mark 1101 cubits from the north southwards on the extended
east
side of G1 (the Great Pyramid). This will be the axial distance to the
south side of the Second Pyramid.
Mark 433 cubits ( 250Ö3 = 433.0127..) from
the center of the Great Pyramid westwards. This is the axial distance
to the east side of the Second Pyramid. Since the positions of two
sides, south and east are already known, and because Legon postulates exactly 411 cubits as the intended length per side,
the square base of the Second Pyramid can be now be drawn in full.
Next, if we
make the axial distance between the western sides of
the Second and Third Pyramids 250√2
given to two decimals (353.55 cubits), it is just 0.008 cubit short of
the western side of G3. This is a very accurate
result. Since
the positions of two sides, south and west are already known, and
because Legon postulates exactly 201.5 cubits, as the intended length per side,
the square base of the Third Pyramid can be now be drawn.
Differences
in Legon's plan from Petrie's plan -
When N-S
(35,713.1 inches) = 1732.05 cubits
This scale goes
to two more
digits in the fractional part of Ö3 than
the one Legon uses. The same procedures used previously on 'his' scale,
achieve nice great accuracy
here as well, but with one important difference - instead of rounded out
values for the square roots, they work with the actual exact
root values!
But, it must be noted that
overall, Legon's method gets better results on the less
accurate scale of 1732.
Importantly,
what works really well for Legon, works even better for
this reconstruction. It gets Legon's reconstruction to 5.4
millimeters of the western side of the Second Pyramid, but it gets us
to just 0.51 millimeter. However, Legon's result for the eastern side
of the Second Pyramid is closer by a fifth of a millimeter than
this reconstruction, but that is because Legon does not reconstruct the
Great Pyramid in this position like me, copying it instead.
Legon's division along the east-west axis is accurate to half a
millimeter for the east-side of G2, and half a centimeter for
the
west-side of G3. For the division along the north-south axis, Legon
uses two procedures, which he deems both accurate.
Each procedure
works better on one scale and worse on the other, but neither
works nearly as well as the procedures along the east-west
axis.
This unfurls cautionary flags..
Second
pyramid
Cubits
Inches
Millimeters
South side
0.0724
1.49
37.96
East
side
0.0010
0.020
0.53
North side
0.0482
0.994
25.26
West side
0.0252
0.521
13.23
Center
0.0618
1.274
32.35
Third
pyramid
Cubits
Inches
Millimeters
South side
0
0
0
North side
0.0545
1.12
28.6
West side
0.0102
0.212
5.4
East side
0.0648
1.336
33.9
Center
0.0464
0.956
24.29
Differences in my plan from Petrie's plan -
N-S
= 35,713.1 inches = 1732.05 cubits
The Great
Pyramid - final version
Cubits
Inches
Millimeters
South side
0.0005
0.010
0.26
East side
0
0
0 forced
North side
0
0
0
forced
West side
0.0005
0.010
0.26
Center
0.0003
0.007
0.19
Initial
Second pyramid
Cubits
Inches
Millimeters
South side
0.0081
0.167
4.24
East side
0.0082
0.170
4.33
North side
0.0085
0.177
4.49
West side
0.0084
0.173
4.40
Center
0.00025 0.005
0.13
NW-SE diagonal 0.0001
0.0025 0.06
vertical axis
0.00007 0.0014
0.04
How close is 14/10000
inch on the Giza scale?
Second
Pyramid after final adjustment
Cubits
Inches Millimeters
South side
0.0017
0.035
0.88
East side
0.0013
0.026
0.66
North side
0.0007
0.014
0.37
West side
0.0011
0.023
0.59
Center
0.00025
0.005
0.13
NW-SE diagonal 0.0001
0.0025 0.06
vertical axis
0.00007 0.0014
0.04
Center and axes remain, as above.
Third Pyramid after final
adjustment
Cubits
Inches
Millimeters
South side
0
0
0 forced
North side
0.0004
0.009
0.23
West side
0.0014
0.029
0.75
East side
0.0003
0.006
0.15
Center
0.0010
0.021
0.53
The
tables given above show that aside from two accurate, and one semi-accurate readings,
Legon's reconstruction does not reflect Petrie's data with consistent accuracy.
Mirrored Illusions Become Reality
The layout of Giza is very rich in geometric
illusions. It makes
decryption tricky.
In a spiritual sense, false
paths obscuring Giza's
recreation (I blundered down a few of those) may
well symbolize the soul's perilous journey
through the labyrinth of life
to fulfilment.
At the risk of sounding
mysterious, reality then
is the axis of symmetry between illusions..
Measuring Success
-
Precise Values
The
first criterium is how close the reconstruction gets to Petrie's
position. The reader saw that the
faithfulness of the reconstruction to Petrie's plan is
in a class of its own, and nothing else comes close.
Another
aspect of the reconstruction, which puts it into a class of its own is
the undeniable extreme proximity of many of the resulting measurements
to whole or half cubits.
.
Distances
given in cubits
439.50009..
a
side of the
initial G1 - less
than 1/10,000 cubit
from an exact half-cubit
622.009..
a diagonal of the reconstructed
G1
411.007
a side of the
initial G2
201.5027
a side of the initial G3 - less than
3/1000 from
being a perfect half cubit
516.0055
from the
reconstructed SE corner of G3 to the SW corner of the Pyramid Square
1787.5005..
distance between the centers of reconstructed G1
and a version of G3
1642.0022
line 'b'
( diagram 5)
314.5027
from the reconstructed SW
corner of G3 to the SW
corner of the Pyramid Square
-201.5025
from the reconstructed SW corner of G3 to
the SE corner of Petrie's G3
= 113.00029
250√2
- 39
from
one version of the SW corner of G3 to the SW corner of the
Pyramid Square
2.001..
difference between the radii of the transmission circle, and
its lookalike (diagram 5)
1150.626180
distance between adjacent tips of the
13-step-star, or one
side of
the smaller star (diag. 6), five consecutive digits of Phi
squared
439.82732
side of the
reconstructed G1, which yields a
very good value for Pi ( 3.1416..) with half-height of the
pyramid
(140 cubits) 439.8273
/ 140 = 3.1416..
Mike Ivsin's original 14-step construction of the regular pentagon
In my internet search, I came across some 15-step operations,
but then in February of
2010, I got a letter from Mike
Ivsin about his original construction of the regular pentagon.
Applying Ivsin's ingenious original idea to
pentagram construction, I found that it is a 14-step process.
In
the below diagram, the first five steps are in black color, the
following five are blue, and the rest are red. Two of the circles
appear as Vesica Pisces in order to clean-up the image.
Magic
happens, when Ivsin goes on to add just one circle ('c') to
the
basic construction, whose radius equals the length 'a' of the star's
arm (diagram below). Now, extend line 'b' so
it meets 'c',
and draw the line 'd'. What we have here is the cross-section of the
Great Pyramid!
The angle between lines 'e' and 'd' is 51.82729237
degrees, which rounds out to 51º50'.
A quote from Petrie: On the whole, we probably
cannot do better than take 51º 52' ± 2' as
the nearest approximation to the mean angle of the Pyramid..
*
Hence, 51º50'
complies to the lower ±
limit set by Petrie.
*
The ratio
of 'd' to 'e/2' is the exact value of φ
(Phi).
This is better than the usual Phi formula for the Great Pyramid: 356 /
220 = 1.6181818
It is true that the
Giza layout, as well as the Great Pyramid's cross-section seem
derived from Golden Section designs,
namely, the construction
of the regular pentagram needing the least number of
steps, 13,
and possibly, the only other such construction needing less
than 15 steps. In any case, the nature of each construction is
consistent with the other, and thus augments its credibility.
Giza
broadens the Nazca - La Marche
connection (between
the
Nazca-monkey, and one
Stone-Age engraving from the rock shelter of La
Marche, near Lussac-les-Châteaux, France,
with 1,500 masterfully engraved stone tablets, now kept in museum
vaults, and largely unknown to the public).
Why do these three ancient works from different ages
and
continents encode the same construction? Is there some
connection here
to the apparent imagery of the so called Abydos Helicopter,
which also encodes a rigorous construction of the Golden
Ratio?
Petrie's value
of 9,068.8 inches,
or 230.348 meters, or 439.82782340 cubits of the
actual average side by the standard of this study, is
about 2.5 mm off the desired perfect value for Pi. An
error of less than 3mm was allowed for by Petrie. Hence he could be off
by that much, and the pyramid may have been built perfect. If we let
the facts prevail, the conclusion must be that
the average pyramid side was designed with the true value of Pi in mind.
John
Legon
writes:
"In
terms of the Giza royal cubit of 0.52375
metres, the actual mean side of 230.364 metres corresponds to 439.8
cubits, with an average variation in the sides of only 6 cm or 0.1
cubit. Petrie suggested that an adjustment may have been effected in
order that the perimeter of the base should express the so-called
'pi-proportion' in relation to the height of 280 cubits, with greater
accuracy than the value for pi of 22/7. In this case, the
theoretically exact mean side-length would be 439.822... cubits. It
seems that the builders achieved this result while retaining the round
number of 440 cubits in the south side."
Naturally,
a measurement dictates both its exact, and rounded
out
values. The need
for a side of 440 rounded out cubits arises from
the perspective of Phi. The apothem divided by half the side, or
356/220 equals Phi to the first three decimals (1.618 181818...)
Mike Ivsin's construction produces perfect Phi, with a slope,
which is in Petrie's ± range. It seems that the builders had
achieved this result, too.
Legon abstracts a cohesive system from the Giza position, one dealing
with
square roots. Although some of these readings are somewhat
approximate,
Legon suggests logical reasons, why there were
adjustments. In my opinion, Legon correctly identifies an
additional
layer of abstraction in the position. Compared to the
"13 Steps" reconstruction, it is considerably less accurate,
yet it
makes perfect sense in a rounded-out way. The more meaning, the more
reason for selection of this plan.
By the way, Legon seems somewhat unhappy over the east-west
distance spanned by G1 and G3, which is 3 cubits too long to express
the square root of 2 as 1414. Did he discount the fact that the
north-south distance between the centers of the same pyramids is three
cubits shorter than the desired distance and thus produces a correct
average?
east of G1 to east of G3
= 1417.4888..
digits of the
Square root of 2
= 1414213..
north to south distance between centers of G1 and G3 = 1411.414161.., the first three digits are correct; the next five are a peculiarity, for 1.414161.. gives the same root to five correct digits when rounded to 1.4142.
These two distances between G1 and G3 also add up to 2828.. the first four digits of the square root of 8,
which equals 2 x the square root of 2. It is interesting,
because in this case the following formula is true:
the square root of "a" cubed equals two-times the square root of "a".
*
A long-time researcher, Robin Cook adopts the
right approach in observing all strong relations as possible
coincidences first, and then asking, which of the mutually exclusive
relations might be the intended ones. Cook is right, because without
the illuminating background of the 'Pyramid Square', such ideas are a
bit like Plato's shadows dancing on a cave wall. In
such situations it
is easy for a theorist to become convinced that his
recreations mirror the Egyptian planners, before the builders
strayed from the plan somewhat, just as expected, or before the plan
got changed for reasons unknown.
It
makes sense that the designers had started
out with regular squares. Adjustments were then made
to create a new layer of meaning.
Petrie versus Cole
-------------------------------------------------------
Excerpts from an old discussion:
>
What irks me, Jiri, is your persistent refusal to address the question of
intent; you have yet to
>
provide any separate evidence that this pattern was created intentionally by
the designers of the
>
Giza pyramids and temples.
Whenever you reconcile the obligatory appreciation for the
eternal
beauty of the geometry involved, and the fact that this is Giza, the
issue of intent may become clearer. . Such a design certainly places
the layout into the highest category of sophistication.
> According to Lehner
(The Complete Pyramids 1997)
>
the base of Menkaure’s pyramid is 335 x 343 feet.
>
Yes, this is contrary to Petrie’s measurements
>
(mean 346.13 x 346.13) but does this mean that
>
Lehner is wrong?
It most certainly does, considering that his figures differ
wildly from
not only Petrie, but also Cole, who after all did come close to Petrie
in measuring the Great Pyramid.
It pits him alone against two widely acknowledged professionals. plus,
Lehner was caught cheating in a drama of his "This Old Man Pyramid", if
I have the title right. A mechanical shovel was used to move some
blocks, but no mention of the fact was made in the flick.
I found that typical of the PyramiPhobia, which so torments
some
academicians they will sell their soul to the devil.
> > from Cole's
survey report:
>>
"These
differences in azimuth are due to the fact that the new
azimuths
are found from the actual directions of the sides determined
from
the excavated pavement,
Cole took series of measurements of the available sections of
the
excavated pavement. The various hypothetical lines were then averaged
out, and extended until they met near the corners. Considering how
close Cole comes to Petrie, he did a great job!
>
.. a hypothetical base obtained by computing
“a square that
shall pass
> through the points of the casing found on each
side, and having also
> its corners lying on the
diagonals of the sockets.”
a)
Yes, the acclaimed and anomalously accurate casing! The quality of
work, said Petrie, equaled modern opticians, but on the scale of acres.
Determination of lines from the plane of the casing blocks, and their
projection down to the pavement to produce an averaged out line in the
pavement should naturally be superior to relying on measurements from a
single line, as Cole had done.
The descending gallery shows similar accuracy in that it deviates from
its axis over the course of 350 feet by a quarter inch (6 millimeters)
side to side, and only one tenth inch (2.5 millimeters) up and down. In
using the anomalously accurate plane of the casing blocks to obtain a
hypothetical line along the pavement, Petrie seems to have achieved
even greater accuracy.
b)
Corners lying on the diagonals of the sockets:
This was another logical decision by Petrie, which gave his method
further advantage over Cole's. Clearly, the sockets were the target the
builders were aiming the sides at.
Petrie was a consummate professional. In my eyes, his drive
to provide
the most accurate data on Giza surpassed Cole's. After all, Cole
could only be bothered to survey one pyramid. Thus he forfeited further
experience with measuring at Giza, which might have given him a little
more insight.
The fact that Petrie's measurements set the position up for an
incredibly
accurate regeneration from a clean slate using the noble Section, can
by no means be discounted. With this reconstruction of the Giza-layout
everything clicks into place. A click
in the reconstruction - a booming cannon shot across the bow of
Egyptology!
Another way of reconstructiong the Second Pyramids NW to SE axis.
The
big red circle is actually double, with centers in both the #2
and #3 interim versions of G3: From the pyramid-center of each
version, draw a circle through the SW corner of the
(golden) Horizontal Column. Each seems tangential to one axis
of
G2. In fact, both come close.
G3 #2
version - the circle
is 0.8286 inch short of the axis, while in
G3 #3
version - the
circle is 0.8163 inch past the axis.
The
axis line of this channel is 0.006 inch, 0.0003
cubit, or 0.16 millimeter to
the
southwest of the original G2
axis given by Petrie.
My letter to John Legon from September 27th, 2011
Hi, John,
A detail of my analysis also
works for your theory. It gives a nice and easy way of duplicating
Petrie's location of the SE corner of G3, within the framework of the
pyramids'containing rectangle: It works with the NS distance set to
1,732, or 1,732.05 cubits.
a) Circumscribe the Great Pyramid.
b) From the SE corner of the containing rectangle draw a big circle touching the circumscribed circle of the Great Pyramid.
c) Go east exactly 1 cubit from the point, where the big circle crosses the SE boundary of the containing rectangle.
The result is a touch over two millimeters to the east of the SE corner of G3, as given by Petrie.
Since
I learned this fact by geometric means,
and since yet another method was a distraction by yielding an even
better result, it took a while to dawn upon me that this fact also helps
your theory. My apologies for the delay in letting you know.
Best regards,
Jiri
p.s. Since 1,732.05 signals a square root, where is the square?
John,
I think that your theory warrants asking that question. To me, the
square is the extended containing rectangle of the three pyramids. Its area is 2,999,997..
square cubits (or 2,999,824 @ 1.732 cubit). Draw the square, and it
will light up the next clue, bringing the solution a step closer.
From: John Legon <>
To: jirimruzek@
Sent: Friday, March 5, 2010 4:04 AM
Subject: Re: Giza layout
Hi Jiri,
Thank you for drawing my attention to your essay on the Giza plan.
I haven't tried to analyse your findings in detail ........ rest omitted
.
A folllow-up letter on the same day:
Hi, John,
sorry, I made a mistake, for a
saving of 0.0005 cubit represents only 0.25 millimeter, and not 1
millimeter. So, the result brings you to 3.15 millimeters east of the
mark set by Petrie. Still, it is better than the discrepancy of 33.9
millimeters, if one follows the procedure in your report.
Best regards,
Jiri
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