Exact recreation of
Petrie's Giza ground plan from Golden Section ideas
Introduction
Any modern architectural complex starts out from the ground level with
the site's layout. Based upon analysis of Sir William Flinders
Petrie's geodetic survey of the three great Giza Pyramids published in
1883, which is generally accepted as very reliable and extremely
accurate, many researchers believe that Giza, the three pyramids
together, was also built accordingly to such an overall plan.
The Giza ground plan is a monumental puzzle with an aboundance of
potential design ideas. A number of plans has already been proposed on
the basis of those, but all, with but one exception of a plan by J.R.
Legon who gets some key things right, suffer from major inaccuracies in
mirroring the actual position of the pyramids. It then becomes
difficult to prove that it was one particular design idea and not some
other, which had governed the building.
A deceptive factor
Even though some elegant design ideas differ from the surveyed
dimensions by a foot, or more, they do look absolutely accurate on even
the biggest display. For example, their inaccuracy is diminished
thousandfold on a display of 35 x 35 inches. A foot then becomes a
thousandth of a foot which is invisible on even a high resolution
screen or a drafting boards, not to mention diagrams drawn in thick
lines.
The Giza design and Psychology
Among other things, the Giza design is also a mirror for the human
nature. Instead of pooling individual observations and theories into a
coherent body, individual researchers promote their ideas and treat
other ideas as mutually exclusive. Yet, all these ideas collectively
form layers of meaning around the core of exact construction.
I suggest that the Giza designers were aware of all that and had chosen
their plan because of this wide range of choices facing those in the
generations to come, who would want to solve the puzzle and reconstruct
the true plan.
To enable the future solution of the puzzle, extraordinary accuracy in
the plan's implementation was a must. Followed up by W. F.
Petrie's accurate survey of the site, a solid basis became
available for credible reconstructions of the planning process.
As for my own involvement with Giza, I was skeptical of my chances to
contribute anything new to the existing body of research, the result of
many years of effort by many scholars. The change of attitude came
after I made a successful foray into the controversial "Abydos Helicopter"
scene from Seti's Abydos temple.
Egyptologists persist in explaining it away as a so called 'palimpsest' _ an
accidental fusion of unrelated glyphs. As such, its nature has to be
chaotic.
Yet, I had supplied a solid proof of the presence of a thoughtful geometric order in
these glyphs; making the official explanation partisan. The Abydos Helicopter glyphs are not a palimpsest;
it's as simple as that.
Abydos Helicopter was the encouragement I needed to eventually
contemplate the Giza ground plan. For starts, I copied the position of
the three pyramids as surveyed by W.F.M. Petrie, and listed in an
article by John Legon, into my CAD program.
Initial Observations  The Pyramid Square
& Khafre's Pyramid
The first step I took was extending the enclosing rectangle (blue
in the diagram) of the three pyramids, ubiquitous in other
studies, into a square. Taking this simple step is absolutely
vital to opening the floodgates.
It is strange that no one had tried it
before me despite well known facts which should encourage it. For instance, Corinna Rossi on page 122
of "Architecture and Mathematics in Ancient Egypt" says: "From the
Middle Kingdom onwards, ancient Egyptian artists used square grids.."
But why not earlier? After all, the pyramids themselves are square.
Golden Section
The Pyramid Square is divided by the Golden Section lines (green) from
each direction. Altogether twelve golden rectangles are in sight. In
the center of the Pyramid Square is a small square extended by a golden
rectangle on each side. This formation represents a goldencross.
The Second Pyramid's vertical axis is very close to the vertical axis
of the Pyramid Square.
The square is divided by the golden section (green lines).
Does the square of the Second Pyramid mimic the one in the center of
the (green) goldencross?
A followup experiment
A goldencross is extrapolated from G2's (Khafre's) square and made concentric with the Pyramid Square. Now there are
two Goldencrosses superimposed over each other for
comparison. The similarity in size is striking.
At this resolution, the two superimposed crosses create an almost
perfect illusion of there being only one cross.
The actual ratio of a side of G2 to its distance from the
Pyramidsquare side is 1.61 _ the first three digits of Φ after
rounding the actual result of
13,619.1 : 8,474.9 = 1.60699..
Is it a mild 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 situ,
seem to find some correlation to the south side of G1. Here as well, we
encounter somewhat rough facsimiles of golden rectangles.
The results were thoughtprovoking, so far.
In this new framework, some prior interesting, but seemingly
deadend, observations by others suddenly become meaningful.
Not wanting to rediscover the wheel, a search was in order on the
subject of Giza layout. Right away, I found 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 Giza
ground plan.
*
Chris Tedder had noted that perpendicular distances between the
pyramid centers produce two golden rectangle facsimiles (ABCD, and
DEFP) and proceeded to propose a plan that the Egyptians had possibly
used. Unfortunately, these rectangles have poor accuracy.
The ratio of BD to BA is 1.623 instead of the desired Golden Ratio of
1.618..
The BD segment is 74.28 inches too long for a correct golden rectangle.
The ratio of DF to DE is 1.605, which is more than twice as bad.
Judging by these facts, the level of builders' skills, or knowledge is
not impressive. But I learned that reality is different; the builders
are scapegoats for our incomplete analysis!
Advanced Spaceage Accuracy in Implementation of an Exact Idea
It may sound bombastic, yet the heading above is simply a realistic
statement. It is a taste of what's to come in this reconstruction. To
see how outofthisworld good the builders were, let's approach the
same position a little differently:
1) Draw horizontal and vertical lines (axes) from the pyramid centers.
2) Draw an exact golden diagonal from the intersection of 'd' and 'e'.
(Any rectangle with its corners on this diagonal is a
true golden rectangle.)
3) Draw lines 'h' and 'g' from the intersection of 'c' and 'f'.
Now, the rectangle formed by lines abch is an extraordinarily accurate golden rectangle:
Instead of 74 inches, the "inaccuracy" of its long side now stands at 0.18 inches over
the distance of 17,833.8 inches ..
The
ratio of sides in this golden rectangle is 1.618021, which is just
0.00001 off the true value of Phi.
The line 'g' is an extraordinarily accurate golden diagonal:
Instead of the correct 31.71747.. degrees, its angle is 31.71767.. mere ^{2}/_{10,000 }(half a second) degree off.
Note that the center of G2 is perfectly positioned in the EastWest
direction, and likewise the center of G3 is perfectly positioned in the
NorthSouth direction for the purpose. The more these points move along
these directions, the less golden the rectangle becomes.
The contemporary academic concensus is that no Egyptians of the IV Dynasty
were familiar with the Golden section. Tedder's analysis is not
considered a proof due to the inaccuracies involved.
Therefore, I'm really glad to be able to show total accuracy implicit
in the plan. Perhaps, the academy will notice. By the way, I myself
realized this aspect of the position long after I had noticed
other important elements of it; those proved quite a distraction.
The image below is a closeup. 
The Golden Column
The
Golden Column, seen below, occurs naturally in the context of the
Pyramid Square after a pentagram construction. It is a composite of two
golden rectangles, one upright, one horizontal.
C divides AK so that if CK equals Φ  1, then AC equals Φ, and
AB equals 1.
The length of the combined rectangle (the Golden Column) then
is 2Φ  1.)
The diagram below shows two diferent Golden
Columns, although we can only see one at this resolution. One is the
Golden Column from the previous diagram, while the other, the Tedder's
one, is set by the NS distance between
the pyramid centerss.
No doubling nor thickening of lines is seen between the columns on the
outside. However, the western vertical side of Tedder's original
rectangle visibly divides the entire column in a wrong place; it's 74
inches too long.
There is something in this diagram that turns on the proverbial
lightbulb _ a line from the point 'O' on the Golden Column, made to be tangential to the Great Pyramid's inscribed circle, is at the same time a golden diagonal.
Evidently, one can use this golden diagonal to
reconstruct, with visual perfection, the Great Pyramid's square base as
it relates to the Pyramid Square.
The pyramid's side will then be shorter by over 6 inches,
invisible on this scale. So, if we suppose that this was how the
builders had evolved the Giza plan, it would be in agreement with Giza's
sacred nature, because the Golden Section was always considered sacred geometry.
The notion that this figure, the Golden Column, was sacred to
Egyptians, is also supported by my analysis of a door from Hesire's
tomb.. http://www.vejprty.com/hesire.htm
The figure of Hesire engraved upon the door fits snugly
into the same goldencolumn.
Φ  the
Second Nature of the Pyramid Square
Yet another horizontal golden rectangle makes an appearance, exclusive
to the context of the Golden Column within the Pyramid Square, as two
pyramid lines seem to converge into a point on the golden diagonal emanating
from '7', the northwestern corner of the Golden Column:
1) 'h'  extended eastern side of the Third Pyramid
2) 'g'  extended diagonal of the Second Pyramid rising
northwest
That the existence of this rectangle is not accidental is confirmed by
its reappearance based on another method.
(see the second diagram below) Although the two are not identical, they
appear to be identical at this scale, with the latter rectangle being
more accurate.
The second rectangle is an unintentional consequence of a
separate idea proposed by Jim Alison:
Quoting Alison:
A 45° diagonal from the apex of the third pyramid intersects the SE
diagonal of the first pyramid very nearly 220 cubits from the apex of
the first pyramid. Thus, a circle inscribed inside the first pyramid
very nearly marks the point on the SE diagonal of the first pyramid
that is at a 45° angle to the apex of the third pyramid. This 45°
diagonal line from the SE diagonal of the first pyramid to the apex of
the third pyramid is divided by the SE diagonal of the second pyramid
in the ratio of 11/10, as shown in the diagram above. The length of
this diagonal is 1773 cubits, a close approximation of the square root
of π x 1000. The EW distance between the apex of the first pyramid and
the apex of the third pyramid divides the 45° diagonal by the golden
ratio. /End of quote
My simplification of the above text:
1) A line (y) is drawn from the center of the Third Pyramid to the
lower intersection of the Great Pyramid's NWSE diagonal with its
inscribed circle.
2) A circle (x) originates from the center of the Third Pyramid and extends to the vertical axis of the Great Pyramid.
This circle divides the line (y) approximately by the Golden Section
ratio.
Practical Exploitation for the Reconstruction
Here is a Gizapuzzle classic, a more prominent but not
so accurate idea inspires one much more accurate:
This time, let the circle (x) divide the same line (y) by the Golden
Section ratio exactly.
Then it will intersect the golden diagonal originating from the
northwestern corner of the Golden Column at the point (I).
The golden rectangle set by (I) is a seeming twin of the other
rectangle we had identified there before. Moreover, (I) is located just
0.66 inch above the second Pyramid's diagonal.
Later, when we discover another procedure which produces a point 0.66
inch off to the other side of the same diagonal, we shall have located
this diagonal with extraordinary accuracy, because parallel 45° lines
through the two points create a channel, whose axis replicates the true
diagonal.
If instead, the point (I) is set by the intersection of the extended
diagonal of the second pyramid rising due west with the same
circle "x", the horizontal rectangle IJKL will still be an excellent
facsimile of a golden rectangle.
distance IJ divided by IL = 1.6199
less than 2/1000 off the true Φ value
Logical Continuation  Pentagram Construction
The initial observations had reminded me of the style of the creators
of Nazca Monkey and the Athena Engraving. Those two masterpieces both
derive from the construction of the 5pointed star and emphasise the
same unique method above the many available (the 13step method).
Therefore, I tried it out, and it worked just great. While all the
other methods can be used to reproduce the initial Great Pyramid square
base in relation to the Pyramid Square, only the 13Step method of the
star design automatically produces a point which accurately locates one
of the sides of the Second Pyramid: to 4 mm or 1/6 inch north of its
southern side. This is a key tidbit of information, as later on it lets
us reconstruct the position of the southwestern corner of the Third
Pyramid exactly as it's given by Petrie.
Geometry First
The reconstruction remains in the domain of pure geometry
until the Great Pyramid stands exactly reconstructed in
proportion to the
total SW span of the ground plan.
The difference to Petrie's value is microscopic on Giza's scale _0.28
millimeter (1/100 inch)! Rounding like Petrie to the nearest
tenth of an inch, the two designs are identical.
A couple of imitation versions of G3, the Third Pyramid, also appears
during this first stage.
The second stage completes the reconstruction of the ground plan of
Giza by combining geometry and measurements in cubits  either whole or
equal to the square roots of 2 and 3 given in multiples of 250 and
1,000.
Petrie's plan and its interpretation by Legon
Petrie gives the plan like this: "The following are the Axial Distances
in inches that separate the centres of the three Pyramids..."
.................................................................N
..............................E .........
Centre of 1st to centre of 2nd Pyramid 13931.6
..............13165.8 .......
Centre of 1st to centre of 3rd Pyramid 29102.0
................22616.0 .........
Centre of 2nd to centre of 3rd Pyramid 15170.4
.................9450.2
.............
Petrie already gave the mean side for each pyramid; so we can
reconstruct his plan from North to South. The total South to North
distance of this plan is 35,713.2 inches.
Now, in comes Legon who needs to duplicate the plan in
an alternate way..
"From the above surveydata, we can now compute the major components
of spacing between the sides of the three Pyramids, as follows:"
Distances between sides
from North to South
Inches
Cubits Design
N 1st to S 2nd
22,703.4 1,101.04
1,101
S 2nd to
S 3rd 13,009.7
630.93 631
N 1st to
S 3rd
35,713.2 1,731.97
1,732
S 1st to
N 2nd
5,159.7 250.23
250
S 2nd to N 3rd
8,856.1
429.49 429.5
The problem with the above is that the top two lines should add up to
equal the third line:
22,703.4 + 13,009.7
= 35,713.1 but the third line =
35,713.2
Like Petrie, Legon is rounding his distances to the nearest tenth
of inch. In Petrie's model, these distances between pyramid sides work
out to being 0.05 inch longer.
22,703.45 + 13,009.75 =
35,713.2
However, it is very well possible that if Petrie were to state the plan in the
same terms as Legon, he would use exactly the same values as Legon
(values rounded to tenths of inches).
Quoting Petrie:
"A line of levelling was run from
the N.E. corner of the Great Pyramid, to
the S.E., to the S.W., up to
the top, and down to the N.E.
again. The difference
on return was only 1/4 inch, or
1"5, on 3,000 feet distance,
and 900 feet height."
By that standard, it is entirely feasible that Petrie was a tenth of an
inch off the real value of the NS span of Giza.
Legon's Axial Distances between sides from East to
West
Inches
Cubits Design
W 1st to W 2nd
12,868.8
624.09
624
W 2nd to W3rd
7,289.5
353.52
353.5
E 1st to W3rd
29,227.2
1,417.42 1417.5
W 1st to E 2nd
4,393.9
213.09
213
W 2nd to E 3rd
3,135.9
152.08
152
The table above has the same problem. The value of the first two lines
plus the 1st pyramid's side should equal the third line, but it does
not.
12,868.8 + 7,289.5 + 9,068.8 = 29,227.1
But the third line is 29,227.2
In Legon's model the shortened gaps between pyramids add 1/10 inch
to the combined length of pyramid sides, if the model equals the span
of Petrie's Giza plan.
The Basic CAD Model of the Ground Plan
The basic model of the ground plan becomes our target which we try to
reconstruct by mathematical procedures. For this target model I
took Petrie's value for the mean side of the First Pyramid, and
added Legon's value for the gap southwards between it and the
Second Pyramid. To that I added Perie's mean side of the Second
Pyramid, added the gap between it and the Third Pyramid, and added
Petrie's mean side for the Third Pyramid. I used the same method
going from east to west.
9,068.8 + 5,159.7 + 8,474.9 +
8,856.1 + 4,153.6 = 35,713.1 inches =
2,976.1 feet
At the time, I was oblivious of the discrepancy of 0.1 inch between
this version and Petrie's version since I had continued by immediately
switching to cubits as units, counting the SN span as 1732.05 cubits
exactly. Thus, the discrepancy in inches was kept out of sight.. It was
the luckiest mistake I ever made! The reconstruction process turned out
perfect, it could not be any better. Then I found out that my target
model was actually a hybrid of Petrie and Legon models.
True, a difference of a tenth of inch seems insignificant over the
distance of almost one kilometer between the north and south edges of
the Giza plan; and moreover, the size of each pyramid remains identical
to the one given by Petrie; nonetheless, the position of the two
smaller pyramids in Petrie's version shifts slightly in relation to the
Great Pyramid  5/100" for the Second Pyramid, and 1/10" for the Third
Pyramid. Somewhat surprizingly, Legon's model at 35,713.1" is the one
in ideal harmony with the inherent mathematical ideas _a harmony
unachievable if translating 1732.05 cubits into Petrie's 35,713.2 inch
value instead.
What is one to think? What are the main arguments for the notion
that the hybrid model is the correct one, if we suppose that
there is indeed one such correct model available for discovery?
Luxury of Being Documented  Global Mirroring of Ideas
The Giza plan does not stand alone; it forms a geometric trilogy with:
a) Approximately 15,000 years old, an engraved tablet from La Marche,
France is handsdown the mathematical champion among all drawings 
this touchstone matters more than any other picture in the world.
b) The Nazca Monkey from Peru duplicates the essence of the engraving
from La Marche  it is the world's most important geoglyph.
The development of both designs begins with the same unique method of
pentagram construction, just like the Giza plan. The first major
product of both the La Marche and Nazca pentagrams is the same uniquely
positioned square. At Giza it is also a square, but a different one 
the containing square for the three pyramids. Each of the three
masterpieces finds ways to emphasise that its umbilical cord
connects to the same pentagram construction.
ToandFro Feeding of Ideas
Since the trilogy stands on a common geometrical basis, we can
experimentally transfer individual parts from one work to another to
see how they fit the original picture underneath the design. In a crude
comparison, it's like using parts from a Ford on another model of the
same make to see if they fit the same standard.
I took the Giza ground plan and integrated it into the other two works.
The result was a series of instances of a taylormade fit between the
plan and the images. Some invaluable insights ensued.
Above all, the Great Pyramid dominates the design of Athena's head. The
pyramid is virtually Athena's
head, and Athena's head is the Great
Pyramid.
What we have here is a valid historical document of an
indisputible direct connection between the two. In the more disputible
category, the document suggests that Giza's chief architect is a
female. More than just a revision, this a virtual upheaval in
history
Up to now, the academia has never accepted a single argument for the
existence of advanced ancient science. Accordingly, a skeptic
will say that even if Giza's ground plan could be cleanly
translated into a sophisticated mathematical design, it
would still be just one's own plan, nothing more than coincidence.
It is established once and for all that our modern civilisation is the
only highly advanced civilisation ever on this planet; and the
Ancient Egyptians just weren't yet at the required level of
mathematical knowledge.
Of course, such reasoning is specious; there are lots of reasons why
one should not consider the issue closed; and the reader is probably
familiar with most. In passing, let me just mention a virtual mountain
of material evidence for the presence of advanced technology in our
remote past  superheavy stone objects impossible to move by lowtech
means. The Serapeum in Abydos is a perfect example of that  the
cramped space in the Serapeum obviously rules out usage of massed
workforce to move the heavy stone blocks involved. There is no room
there to operate with levers or pulleys, and so on.
Generations of professors of physics, engineers, and other experts had
failed to explain how to accomplish such tasks by primitive means. Such
a state of affairs leads to two conclusions: either the intellectual
capacity of humans has substantially degenerated to where our best
brains prove moronic in comparison to those of several millenia ago, or
such tasks can indeed be accomplished only by powerful machinery.
Cognitive Dissonance
In effect, the academia seems to uphold the first option, which
indicates that it is afflicted by collective cognitive dissonance. I
add this opinion to my report as I'm editing it for the umpteenth time.
It is late June, 2020 and times have been changing at a breakneck pace
for some years, enough so as to deflect public attention from a
tremendously important event having a direct confirmative bearing upon
this narrative, because it is as good a proof of unknown technology
superior to anything humanity owns up to, as it is an example of a
truly outrageous case of worldwide cognitive dissonance.
After seventy years of denial the American government discloses that
flying saucers are real!
They aren't called UFO's anymore; they are UAP's now. Instead of
'flying objects', they are 'aerial phenomena'. The former term is
strongly worded; it is alarming and could cause public panic, lest the
authorities are projecting their own fears, of course. One of the
dictionary defnitions of aerial is 'lacking substance'. The latter is
relatively easy to swallow; it lets people cling to their outdated
beliefs.
In 2019, the American Department of Defense admitted that the US Navy
videos leaked in 2017 are real. https://en.wikipedia.org/wiki/Pentagon_UFO_videos
The disclosure was not followed by
panic.
Although the videos are just fragments of the aavailable footage taken,
they are true game changers. Though seventy years overdue for the
countless people pilloried on the 'altars of science' over being
branded in one way or another with the stigma of treating the issue of
flying saucers seriously  this paradigm shifting disclosure opens a
slew of possibilities.
For instance, it validates the numerous reports of Foo Fighters by WWII
pilots. Those describe the same flying saucers and the same
capabilities. Our technology has advanced by leaps since then, but not
the UFO's. They seem to be perfect, at the ultimate stage of their
technological development.
If so, how long have they been at that level? Thousands of years?
Millions? By any measure, they don't fit the general tempo of
evolution on this planet; therefore, they are unlikely to be from this
planet or solar system.
How long have they been here? Without going into details, historical
sources indicate that they could be here as long as the human kind.
Would it not be cognitive dissonance if I didn't recognize that since
the UFOs' existence now stands on firm ground, there emerges a strong
candidate for the explanation of many seemingly inexplicable mysteries
from prehistory and ancient history, including my own discovery?
The American government has incontrovertible documentation of
flying saucers. It is documentation of their presence and nothing more.
In contrast, the three sites of La Marche, Nazca, and Giza document
hightech encoding of sophisticated mathematical themes into supposed
primitive art. Conveyance of thought through geometry is a form of
communication akin to writng. All that was served to us in the form of
solvable puzzles with chained clues.
Who could be playing such a fantastic game with us? The civilisation
behind the flying saucers is the only bonafide qualifier for the
candidacy. It makes my discovery that much more more attentionworthy.
*
Length of the Royal Cubit
Legon makes a strong case that the NorthSouth distance
between the pyramids was meant by the builders to equal 1.732 cubits,
or 1,000 times the square root of 3. But there is good evidence that
they had meant it as 1,732.05 cubits, the square root of 3 given
to an impressive five decimals. (1,732.05 squared =
2,999,997  practically perfect three million.
This is also a clear hint that drawing the
corresponding square might help us understand the position.
.
Petrie's measurements at Giza, and inside the Great Pyramid had
produced slightly differing values for the cubit. In the end he had
settled for an average of 20.62" (inch) per cubit. This
reconstruction's cubit is 20.618977...", only 1/1000"
off Petrie's value.
North to South distance between the pyramids = 35,713.1
inches = 1,732.05 cubits = a side of the Pyramid Square
1 cubit = 20.618977512196530123264339943997 inches =
523.72202880979186513091423457752 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,
StoneAge France ). This is a natural cutoff point as the
digit 5 for hundredths is followed by a zero, which means no
thousandths to deal with (1/1000 cubit is just over halfamillimeter).
The next digit represents an already too fine distance. As a
representation of the square root of 3, our value differs from the
true by eight tenmillionths 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 an elegant method of construction of
the regular 5pointed star. This method was brought to my attention by
the Nazcamonkey glyph.
Diagram 1
Start with the above classic procedure. It begins with a
horizontal line, the future horizontal arm of the star, and takes
ten steps. Two of the steps involve help circles (to draw the axial
cross), and these are not shown. The eighth step draws the
key Goldencircle (yellow), 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 exact angle of 36 degrees (like on
a 5pointed star).
Before going on, this simple diagram already has two crucial elements:
* The base of the golden triangle, given by the 36° angle
intersecting the horizontal axis, is of the same length as one side of
the Great Pyramid's prototype in this reconstruction..
* The circle from step 2 of the construction is the actual instrument
which adjusts the prototype Great Pyramid to within ¼ millimeter of
Petrie's plan. It shows the strong bond between this pattern and the
reconstruction itself.
On step eleven, we have a choice of three points (D, 1,2) from which to
draw a circle. The one shown above centers in "D" and passes through
"1" and "2".
Then the other two (circled) points at which the circle crosses the 36
degree angle lines mark two more tips of a regular 5pointed
star.
Lines from those tips drawn through "D" then complete the
star. In trade terms, the simplicity of the construction is
13. I was able to find one such construction on the web with the
simplicity of only 12, but no other simplicity 13. The rest seem to be
simplicity 14 and more.
The other choices use the same circle drawn from either "1"
or "2" in the above diagram.
Diagram 2
Golden Rectangles Galore
The position below is based on diag.1, but is rotated 90
degrees counterclockwise. It contains a slew of golden rectangles.
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 'Golden Column', will be our key to setting
the Great Pyramid's side to 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 PPPyramid 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 Golden Column's height to the height of the rest of
the square is 2(Φ1).
The points ABIJH mark four segments in a row, where each
segment forms the Φratio with the neighboring segment(s).
Corners of the Great Pyramid's
Prototype
In the diagram below, on both the blue 13Step 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 imitation, and
the truesize Great Pyramid.
The diagram shows one way of drawing the pyramid's print in the sand. A
linesegment through 'P' and 'C' is the same as one of the
diagonals in a horizontal golden rectangle; it rises to 'C'
to meet the leftrising diagonal of the initial Great Pyramid at the
pyramid's northwest corner. The segment is also 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 north off the southern side of the Second
Pyramid.
Diagram 4
___________________________________________________________________________________
Lines a, b, c, 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 "13step"
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 13Step star..
Line 'a': As the radius of a 'transmission circle", it will usher this
reconstruction from theoretical to applied stage when its turn
comes.
Line 'b' measures 1642.00222202 cubits, a calibrated measurement,
typical for this reconstruction.
Diagram 5
In the diagram below: One Qcircle intersects sides of the diamond
(square); the other one intersects extensions 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
connected to the 13Step 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 much diminished.
Diag.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).
The NorthSouth division  Location
of the south side of G2
The point marked 'X' in the above and 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 907,115.28
millimeters, then 'X' is 4.2 millimeters above the south side
of G2, as given by Petrie.
Without the '13Step' star there can
be no point 'X'  key to the best result in locating of
the Third Pyramid's SW corner (to 0.51 mm).
The length of a side of the initial Great Pyramid, as well as a
side of the pentagon within the 13Step Star, is the remarkably
precise 439.5000... cubits. It's a strong signal that
the units used are correct.
The distance between adjacent tips of the 13Step star, or one
side of the smaller star = 1150.626180 cubits.
Remarkably, that's five consecutive digits of Φ squared (2.6180).
Diag.7
The Pyramid Square
After the initial pyramid (protopyramid), the big
square in diag.3 is extended to the pyramid's northeast
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
Follow the Illusions
The Giza layout is uniquely expressive by
being really close to an unprecedented number of elegant ideas which
might have dictated its design. One by one, these turn out to be
inaccurate, mere illusions. Yet, at
least at Giza, illusions
are part of reality. Like in stereoscopic vision, separate illusions
merge into the real thing. All one has to
do is get on the right track, and
Giza designers provide further guidance in the form
of pictorial clues, special effects. It doesn't take long to the first exact reconstruction
of one of the pyramids in the context of the Pyramid Square, and it is the
Great Pyramid.
Illusion #1  Draw a circle from the southeastern (bottom right)
corner of Giza's containing rectangle, such that it touches
the Great Pyramid's circumcircle. This circle creates an illusion in
that it also touches the southeastern
corner of G3 (Menkaure).
Illusion #2  The same circle appears to be of the same size as
the "Transmissioncircle" in the diagram below, whose radius is
the golden diagonal from the pyramid's NW corner, marked as "a" in
diag.5. So, we copy the Transmissioncircle to the bottom right
corner of the containing rectangle, as well. When concentric, the two
circles merge into a single circle on this scale.
diag.8
Illusion #3  From the center of the imitation Great Pyramid, draw
a circle, which touches the Transmissioncircle, then copy it to
the southwestern (bottom left) corner of the Pyramid Square. It gives
the illusion of touching G3 from the other side (diag. above). This is
certainly a startling effect, albeit up close it is not all that
accurate.
So, we have two pairs of alike circles , which look the same
when the entire Giza is in view. Without viewing things in the context
of Pyramid Square, we would be deprived of this interesting
observation.
Illusion #4  Another special effect is shown below. Line 'g'
drawn
from the left intersection of the two Transmission circles to
the SE corner of the initial Great Pyramid then duplicates
the golden diagonal angle to 0.0015º. The true
golden
diagonal drawn from the same point approaches the same SE corner
to
within 0.66''. Should we overlay this diagram with
diagram 5, line 'g' would be indistinguishable from line 'd'
in diagram 5.
Menkaure's initial vertical axis
('go down the middle')
The midpoint of the gap between the Transmission circle and its
complement circle sandwiching G3 is 4.2 inches to the west
of Petrie's vertical axis for the pyramid. The vertical axis of
our first version of reconstructed G3 is drawn from this midpoint.
The First Imitation of the Third
Pyramid (#1)
a)
* Center a circle at the point, where this vertical axis
crosses the bottom line of the Golden Column, also the central axis of the original mother star, the 13Step star.
* Have the circle touch the far side of the original imitation pyramid (marked 1) centered in the
column's SW corner.
b)
The section of G3's vertical axis below the new circle is taken as
equal in length to the sides of G3. That is all the data needed to
complete this pyramid. Its sides are 6.1 inches shorter than the
original. The left side is just 1.2 inch to the west of the
original. All in all, this is a pretty good imitation of the original.
diag.12
Consistency of the Method
The method of creating the first imitation of G3 is consistency itself.
It bases on the same visual effect  a circle touches either a pyramid
side, or the circle described around a pyramid.
The Benefit
The first version of G3 becomes an instrument enabling a simple operation in which we duplicate the Great Pyramid exactly as specified by Petrie.
In other words, it is a gamepiece created to continue the planners' game.
The Second Version in the Imitation of the Third
Pyramid (#2)
In a variation of the above diagram, we have the
Transmissioncircle again, but this time its lookalike circle is tangential to the circle
around the imitation of the Great Pyramid, rather than the true one. In the magnified detail view
below, 'a' is the Transmissioncircle, and 'c' is its
lookalike circle.
Lines '1' and '2' show the southeastern corner of G3 as given by
Petrie.
The midpoint between 'a' and 'c' becomes the southeastern corner of the # 2 imitation of
G3. It is 3.68 inches to the west of the true corner. The southwestern
corner remains the same. A pyramid side in this version is 2.509
inches shorter than the true side  an improvement over the first
version.
It is
the version used together with the #1 version in one of the two totally accurate reconstructions of the Second Pyramid's
diagonal, the one descending to the southeast.
And, together with the #3 version of G3, it is used for the other totally accurate reconstruction of the same diagonal.
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 exact on
computer screens or paper.
I noted something three times more accurate in this position : the width WZ of the Rising
Column is just 4.32 inches more than the width AB of
the Golden Column, so when superimposed, the two look identical in width. Substitution of the reconstructed columns
for each other could therefore be of interest and is next.
In another interaction between the two columns, the illusion that the
bottom side of the reconstructed Golden Column is identical with 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 Golden Column gets
to 0.94 inch south of the second pyramid's horizontal axis.
This special effect will be put to use later.
The "Channeling" Method of Reconstruction
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!"
Exact Repositioning of the Great Pyramid
Place a copy of the Golden Column (previously called Horizontal Column as in the diagram), axis over axis, over
the Rising Column (using the first version of G3). Then the situation in the
Great Pyramid's NW corner looks like the following diagram.
diagram 14
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.
Or, we can use the axis of the channel between the western sides of the two
rising columns which is then identical to the western side of the Rising Column as
given by Petrie.
The
difference in length between the reconstructed and the
original versions of one side of G1 is too tiny to pay
attention to, at 0.01 inch, or 0.28 millimeter. In other
words,
the two versions of the Great Pyramid are identical.
Our value: 439.827 (439.8273..) cubits, or 9,068.8
inches ( 9,068.81501...) per side.
439.82732 / Pi = 140.001..
Petrie's value: 439.828 (439.8278..) cubits or, 9,068.8 inches
Our value is actually a hair closer to the optimal value of
439.823 for Pi encoding.
Pi times half the pyramid's height = 439.8229..
or, 439.823 rounded
Consistency of Method
The very same procedure works in the diametrically
opposite corner of the Rising Column, albeit not as well, as it locates
the SE corner of G3 to 0.5".
This is version #3 of the initial G3.
its sides are now 0.66897634 inch longer than Petrie's original (the SW
corner remains the same). More versions of the same are still ahead.
diagram 14b
In the diagram, 'a' and 'b' are sides of our first imitation of G3,
and 'c' is the new reconstruction of the eastern side of G3. The line
"c" in this third version is an obvious improvement over the first two versions..
___________________________
Very Special Effects
diagram 10
Coincidences
around the Transmissioncircle certainly make it important. The line
segment which we took for its radius also has some big points on it in
the context of the
'13Step' construction (diagram below).
Point '2' is a big point in the original 13Stepstar's
construction, a corner of this star. The segment 34 duplicates the golden
circle's radius from the same construction.
diagram 11
One of the Accurate Reconstructions of the SE corner of G3
With the Great Pyramid duplicated, it is now possible to do the
following:.
diagram 8
Version #4 of G3
Diagram 8 shows the situation in the SE corner of G3, Menkaure
closeup
.
Lines '1' and '2' belong to the pyramid as given by Petrie.
'a'
which looks like a straight line in this magnification is the Transmissioncircle
'b' is the tangential circle to the
reconstituted Great Pyramid's circumcircle.
The circles are spectacularly equidistant to the pyramid's corner!
Going down the middle again, the
axis of the channel between them is 0.13 inch, or 3.1 millimeters
to the east of Petrie's pyramid corner.
The distance between the
aforementioned circlelines then works out to 2.0010 cubits.
This result is the closest I managed to get to Menkaure's SE corner
by pure geometric construction, without resorting to units.
The same process using the true Great Pyramid's circumcircle gets us to 3.4
millimeters to the east of the pyramid's corner..
A number of other ways
(version 5)
The radius of the cyan circle, between lines 'a' and 'b' in diag.8
is 1.0005.. cubit, a very accurate reading at only ¼ millimeter
off perfection.
Measuring one exact cubit eastwards from the tangential circle
reaches 2.85 millimeters east of the SE corner of Menkaure, 0.25
millimeter closer than the above geometric method.
Perhaps, Legon's
report should mention this fact because it goes well with his
observations of measurements in whole cubits.
diag. 15
The location of the imitation SW corner of G3, the third pyramid, yields
some remarkable readings in cubits.
a)
The distance from the SW
corner of the Pyramid Square to the SE corner of G3 is:
516.005, 516 cubits almost exactly.
(version 6)
So, if we measure 516 cubits exactly from the SW corner of the Pyramid
Square, we'll be some 2.5 millimeters short west of Menkaure's east
corner.
Now, recall the result from above: "measuring one exact
cubit eastwards from the tangential circle comes to 2.85 millimeters
east of the SE corner.
The two results average out to 0.175 millimeter (1/6 millimeter) east
of the SE corner of Menkaure.
b)
The first imitation southwestern corner of G3 in combination with the real southeastern corner of G3 is accurate to 0.00246 cubit (a little
over one millimeter) to what many authors posit to be the intended
side of G3 :
201.50246 or 201.5 cubits
(version 7)
Thus, going 201.5 cubits eastwards of the imitation SW corner gets to 1
millimeter west of its SE corner.
c)
The distance between the imitation SW corner of 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
initial 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.
Exact 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 target SE corner of G3 to within the above mentioned (1/6)
millimeter, or 0.0003 cubit east of Menkaure's SE corner. In
plain English, the two locations are perfectly identical.
Noteworthy:
Do you recall this tidbit from above? <the two results average out to 0.175 millimeter (1/6 millimeter) east
of the SE corner of Menkaure>
Hence we have two ways to get the same result for Menkaure's SE corner 
each 1/6 millimeter east of that corner.
Precise location of Menkaure's SW corner
Once we know the exact location of Menkaure's SE corner, we can
also locate its real (as given by Petrie) SW corner, thanks to a brilliant idea of Legon; I
presume it's his because he cites no sources for it.
Draw a square 555 cubits per side, as in the above diagram. Inscribe a
circle with the radius of 250 cubits and draw a tangent line parallel
at 45 degrees. You obtain an exact half base of Menkaure, the Third
Pyramid, which is easily completed into a whole square whose sides then
come to 201.446 cubits, or 4,153.6 inches long, exactly as specified by
Petrie. This is as simple as it is brilliant, and it is yet another
glimpse of Giza plan's theoretical depth.

z
With one corner of
the third Pyramid already located, we apply the same method from there
and reconstruct the pyramid, as in the diagram above. It's as simple as
that. The new design blends into the old nicely, namely, the
green circle and the green square.
There is another way to reestablish the Third Pyramid exactly to
Petrie's specification, but to do it, we have to wait until we
get to pinpointing the center of the Second Pyramid later. It too, is
Legon's idea, but we do it our way and get a closer result than Legon.
Hints at Pi (π) and the Golden ratio (φ)
Once we draw a circle with the radius of 113 cubits from the
Southwestern corner of the Pyramid Square, there is an incidence of
what seems to be hints at π and φ values on both sides of it.
The distance from the circle to the western side of G3 is 314.5 cubits.
Numbers 113, and a 314 together evoke a widely known Pi
approximation.
355/113 = 3.141592..
The distance from the circle to the adjacent corners of the Pyramid
Square is 1619 cubits.
1732  113 = 1619 is just 1 off Golden Ratio's 1618
1618 rounded to three digits is 162 _ φ rounded to three digits.
1619 rounds out to 162 _ φ rounded to three digits.
The speed of light in nautical miles is 162,000
More specifically, it is: 161874.977 which rounds out to four
digits as 1619. 
1732  162 = 1570 ...... 157 x 2 = 314 (π digits)
The Legon's construction dealt with above divides the 555 cubits long
side of a square into 250 x √2 and 201.446..., the latter being the
side length of G3.
250 x √2 = 353.55339...
355 and 339 are in sequence
353.55339  39 = 314.55339
It is 314.50275 from the initial SW point of G3 to the SW point
of the Pyramid Square.
The difference between the two is 0.050... cubits, 2.5
millimeters, or 0.1 inch
355/339 = 1.047197640 = π /3 because:
1.047197640 x 3 = 3.141592 ....... the first seven digits of π
Pi, Golden Ratio, and speed of light are recurring topics in Giza
discussions.
Reconstruction of G2 (Khafre, the Second Pyramid)
First, we need to locate the SE corner of G3 for the exact
reconstruction of the Rising Column.
Then, marking the width of the Rising
Column (WZ), downwards from the top side of the Golden Column, gets to
0.94 inch south of the Second Pyramid's horizontal axis'. Although
this is a nice approximation for the horizontal axis, its
true function is to help us locate the vertical axis of G2.
diag. 13
G2  diagonal simulation # 1  Channeling" the Solution
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
(the green line marked Phi). The line is also very close (0.8º) to
holding the 45º angle with the horizon. The circle divides it rather
nicely in the Golden Proportion:
22,616
inches / 13,954.114 inches = 1.621
However, with the golden diagonal emanating from the Golden
Column's norhwestern corner in the picture, an interesting visual
effect occurs  the golden diagonal, the Second Pyramid's extended
diagonal, and Alison's circle all seem to meet in a point. It is an invitation
to experiment like in the diagram below:
This circle intersects the golden diagonal "c" at "I" in the
diagram. The point "I" simulates a point on G2's diagonal.
This relationship is an order of magnitude more accurate than the one
Alison noted. The simulation falls 0.66 (0.6592..) inches northeast off
the original.
Note: This method works this well only with #2 version of G3.
G2  diagonal simulation # 2 (Diagram 17))
Two major lines:
'a'  the axis of the initial Rising Column
(it was the key to Great Pyramid's duplication) 'b'  the bottom line of the Golden 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 'd' between the two diagonal simulations (lines
'e' and 'c' in the diag), 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 (line 'c') also serves another, and
equally important, function, this time to find the vertical axis
of G2.
The vertical axis together with the recreated diagonal locate the center of G2.

Substituting the Columns Again  Finding the Second Pyramid's Center
Earlier, the Golden Column, when substituted for the Rising
Column, let us duplicate the Great Pyramid, as it is in Petrie's
plan. Now, the same trick
works in reverse!
The Rising Column in its final form, when suspended
from the the top line of the Golden 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. Those
are mighty fine specs. Considering the size of our
workspace, Giza, any hopes at the outset for this kind of
results would be far from realistic.
diag.18
The axis and the 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 G2's southern side to 4 millimeters since
the very beginning; therefore, the Second Pyramid can be
recreated with impressive accuracy even before its final
adjustment.
Another exact way of reconstructing the Second Pyramid's NWSE
diagonal
Having two perfectly adequate ways of accomplishing the task is a pure
luxury.
The big red circle is actually double, with centers in both the #2
and #3 versions of G3: From the pyramidcenter of each
version, draw a circle through the SW corner of the Golden Column. Each circle seems tangential to the same 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.
Final adjustments  the Third Pyramid
Having the initial Second Pyramid permits testing Legon's ideas in our
settings. This means that after the center is located, the size
of the pyramid is determined by using the point (x) from diagrams 4 and
6, which is over 4 millimeters closer to the center.
1: The eastwest (axial) distance between the west sides of the Second
and Third Pyramids equals 250√2 cubits.
This postulate locates the west side of the Third Pyramid 0.006
cubit (3.45 millimeters) west of Petrie's version.
2: The eastwest 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 best, as it locates the west side, or
the SW corner, 0.0014 cubit (0. 5175 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 Legonstyle formula for the reconstruction. Draw line
d eastwards from the SW corner of the PyramidSquare 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!
This is yet another example of the validity of the Pyramid
Square's concept, which Legon never worked with, unfortunately. It
does, however, also support the validity of Legon's observations.
So, far we counted five accurate ways to deploy an exact formula
containing 250√2 towards location of the SW corner of the Third
Pyramid. So which solution should we use? The fact is that there are at
once five accurate solutions for the west side of the Third Pyramid, or
its SW corner. All are accurate  since Petrie's points come with
a ± radius, those points are in reality small circles, and
all five solutions fall inside the circles.
Multiple solutions
designed into the Giza puzzle are a show of sophistication. We
shall never learn, which version was the one actually 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 in size to that given by Petrie.
Conclusion:
Petrie's layout of the great pyramids of Giza can be accurately
recreated from scratch on a clean slate, beginning with the
'13Step' construction of the regular 5pointed star from a line
segment, with some involvement of the basic prime number square root
values. The method has simplicity, total accuracy, elegance, and
intellectual depth. With such attributes, it must
be essentially identical to the actual procedure of planning
the Giza layout.
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 tiny differences between
drawing objects, the plan had to be worked out mathematically
because the tiny differences could not possibly be seen and worked
with on even the largest drawing board. 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 huntergatherer 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 see such a solution here. I still see it as somewhat
inexplicable, for no matter how good Petrie was, he himself assumed a
greater margin of error 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.
©Jiří Mrůzek April 15, 2007
Vancouver, BC
last edited December 27, 2020
Notes
Measuring Success  Precise Values
The reconstruction succeeds in duplicating the Petrie given sizes of
the pyramids. 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.
622.009.. a diagonal
of the exact duplicate of G1
439.500
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 the exact duplicate of 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 13stepstar, 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 halfheight of the pyramid (140 cubits)
439.8273 / 140 = 3.1416..
Mike Ivsin's own
14step construction of the regular pentagon
In my internet search, I came across some 15step 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 14step 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 cleanup 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 crosssection 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
The Giza layout, as well as the Great Pyramid's crosssection seem
derived from Golden Section designs. Giza broadens the Nazca 
La Marche connection (between the Nazcamonkey, and one StoneAge
engraving from the rock shelter of La
Marche, near LussaclesChâteaux, 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 separated by ages and continents encode the same
construction? There is also a connection here to the apparent
imagery of the so called Abydos Helicopter,
which encodes yet another 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 socalled
'piproportion' 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 sidelength 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 roundedout way. The more meaning, the
more reason for selection of this plan.
By the way, Legon seems somewhat unhappy over the eastwest
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
northsouth 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 in my reconstruction =
1,417.. cubits north
to south distance between centers of G1 and G3 =
1,411..
cubits
The average of these two distances between these two pyramids is 1,414
cubits  the first four digits of the square root of 2. It is arrived
at in a straightforward manner _ no numerology involved.
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.
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 twotimes the
square root of "a".
*
A longtime 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.
Considerations of Accuracy
A quote from Petrie:"The probable error is an
amount on each side of the stated
mean, within the limits of which there
is as much chance of the truth
lying, as beyond it; i.e., it
is 1 in 2 that the true result
is not further from the
stated mean than the amount of
the probable error.
If the glass is half empty, 1 in 2 (fiftyfifty) results will exceed
the stated mean error. The excess is still most likely to be very close
to the mean, because the odds against increase exponentially with
distance from the true result (see the quote below). And since the
glass is half full, half of Petrie's measurements should be be 1/10",
or more, off.
Quote: "Outside the
Pyramids all the measures
were ascertained by triangulation from
the measured baseline of the
survey; the first class points
being fixed with a probable
error of ±*o6 inch. The
base of the survey was thrice measured,
and is known with a probable
error of ±"03 inch, or 1 in
200,000. A line of levelling
was run from the N.E. corner of the Great
Pyramid, to the S.E., to the S.W.,
up to the top, and down
to the N.E. again. The
difference on return was only 1/4
inch, or 1"5, on 3,000 feet
distance, and 900 feet height."
The quote describes amazing accuracy in Petrie's mensurations at Giza.
"The mean base being 90688
± 0.5 inches"
For all his modesty, Petrie got the mean base of G1 exactly, as judged
through the prism of our authoritative geometrical model, and his mean
sides for the other two pyramids are in full agreement with the
CAD model, too.
__________________________
Full length minus half the G3 containing box at 45 degrees
long side 42982.68777595'' x short s. 13935.87257566 = 599,001,259.80
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
Golden Section, can by no means be discounted. With this reconstruction of the
Gizalayout everything clicks into place. A click in the reconstruction
 a booming cannon shot across the bow of orthodox Egyptoology!
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 is perfect through the prism of my own experience.
G2
Our CAD model of the G2 pyramid has all its sides to the outside of
Petrie's model:
0.02599292 inch in the east = 0.66 mm
0.02322149 inch in the west = 0.59 mm
0.01970865 inch in the north = 0.50 mm
0.02950576 inch in the south = 0.75 mm
On the average the pyramid sides are 0.625 millimeter apart.
The sides of G2 are 1.25 millimeter longer in CAD than in Petrie's
model, but exactly the same after rounding to the nearest tenth of
inch. Thus, the biggest difference between the two models (1.25
millimeter) is found on G2.
The tiniest difference between the two models is also found on G2:
the vertical axis of G2 in CAD is
0.00138572 inch = 0.035 millimeter (1/28 millimeter!) east
of Petrie's model,
while the horizontal axis of G2 in CAD is
0.00489856 inch = 0.12 millimeter south of Petrie's model.
The center and diagonals of the two models of G2 are also
extraordinarily close:
0.00248395 inch = 0.06 millimeter gap: NWSE
diagonals:
0.00444365 inch = 0.11 millimeter gap: SWNE
diagonals:
0.00509078 inch = 0.13 millimeter gap:
center
G3
G3 pyramid sides are 0.01424592 = 0.3618 millimeter shorter in CAD
0.00595555 inch = 0.15 millimeter in CAD
east of Petrie's east side
0.02020147 inch = 0.51 millimeter
in CAD east of Petrie's west side
0.01424592 inch = 0.3618 millimeter in CAD south of Petrie's
north side
(south side shared by default)
Gaps between diagonals:
0.01428459 inch = 0.36 millimeter NWSE diagonals
0.00421121 inch = 0.11 millimeter SWNE diagonals
J.R. Legon's reconstruction of Giza
It is noteworthy because it gets some things right. Legon starts out
with a key assumption. After observing that the total North to South
distance for the pyramids rounds out to 1732 Royal Cubits
(determined by Petrie from his study of the Great Pyramid), he assumes
that this was the intended value, because it gives the first
four digits of the square root of 3. This correct assumption enables
him to carry on and provide a solid proof for it, as axial spaces between the
pyramids exhibit group behavior in approximating the square roots
of 2, 3,5, and 7.
Legon's study is also impressively accurate for the positions
of the east and west sides of the Second Pyramid, and the west side of
the Third Pyramid. In other cases it cannot be deemed accurate,
however. Its shortcoming is one sided focus on numbers at the
expense of geometry.
Overall, Legon's interpretation puts the ancient builders on
a level most expect  they had a simple plan, and executed it
really well, albeit somewhat unevenly. For this reason, I find it hard
to grasp, why Legon's theory does not command more attention from the
mainstream. His only sin is making plain the Egyptian knowledge of
square roots to several digits, whereas the mainstream scoffs at
any idea of a unified plan whatsoever.
While our reconstruction inherits Legon's basic idea (1732
cubits), it gives the Giza creators a lot more credit for technical
prowess by having them intend this distance as the far more precise 1,732.05
cubits. Due to this upgrade the cubit system then starts shining
brightly.
Overall,
this is a case of reality surpassing SciFi. It is self evident
that the workshop from which the trilogy came could
only belong to a powerful and highly advanced agency with a
global reach. 
