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Exact Reconstruction of the Layout of the Great Giza Pyramids
Foreword
Were the three famous Giza pyramids conceived and laid out in a single grand-plan? The best way to tell is to investigate geometric and numeric relationships in their layout. Subsequently to the provision of reliable measurements from the geodetic survey of Giza by Sir William Flinders Petrie in 1883, more than a century of testing has revealed great many apparent relationships. No one has been able to derive the layout with complete accuracy, and so builders' errors get the blame for all discrepancies from the original. There are many such theories out there, and they often contradict one another. Uncertainty adds weight to a perennial denial tactic in like cases: "You can use mathematics to derive the position of anything... such as the cars parked on my street."
A skeptic will stick to the extreme opinion that in Giza's case even the most ingenious design means absolutely nothing without discovery of the original blueprints, or a signed statement from an architect like Imhotep himself. There is a kernel of truth in these objections. Sure, there must be a number of ways to recreate Petrie's Giza layout.
But, read the small print! Efficiency and sharp focus is an absolute must! If the layout is random to begin with, even the most efficient procedure will more than likely turn out quite complex, and the more so, the closer to the original the recreation wants to get. Other than in special cases, it is impossible to accurately describe a random position by just a few simple geometric constructions, or mathematical formulae.
I believe that these general principles will apply to any three oriented squares like the pyramid bases tossed randomly upon a 2-D plane. By the same token, if there were discovered an elegant and easy method of developing the Giza layout in exact agreement with Petrie's blueprint, it should justify the extrapolation that the architect was also familiar with this method, and had selected it on purpose.
The secret to quick and easy discovery of the method of replicating Giza's pyramidal layout was in the synthesis of existing ideas with an original concept. Such framework was born, the minute I had extended the usual enclosing rectangle of the three pyramids into a square. Against this background, the Golden Section elements so appropriate for the special character of Giza grounds stand out with clarity.
Architecturally, Giza is a forest of coincidences, which can fool one into straying off the true (here geometric) path. It was planned in full awareness of all the possible geometric coincidences, which would render later detection of the true plan somewhat tricky. In one aspect, the plan of Giza may well symbolize the soul's perilous journey through life. Since the geometry of the design helps create sacred nature of the grounds, Giza becomes a geometric allegory of a successful passage through life.
July 31, 2008 - A sensational development!!!
Until that day I never stopped to fill in the preliminary constructions to a golden rectangle, from which the rest of the plan was derived. I just grabbed an available template, and carried on. It was just as well, because I had taken one of the correct paths to the solution. The drawback was being in the dark about key aspects of this profound position.
It would have been correct to start with the below classic construction, which begins with a horizontal line segment, and produces an axial cross, four points of a square, a circle through those points, and also a circle, which actualizes the Golden Section for the given position. For instance, with this 'Golden-circle' centered in the bottom corner of the square, lines drawn from the top point of the square as tangents to this circle hold the exact angle of 36 degrees.
Diagram
1
From there, it takes just three steps to complete the regular 5-pointed star (pentagram) - one circle and two lines. This is the fastest such construction. It has the "simplicity of 13", which means that it takes thirteen elementary steps from start to finish.
Diagram
2
The diagram below includes the position from the first diagram, which has been rotated 90 degrees. It is remarkably simple, but it does a lot.
Circle-1: Its diameter is AB, the half-diagonal of AG of the initial diamond (square).
Circle-2: It is the Golden Section circle for the diamond.
Circle-3: It is the same circle copied to the center of the square by the circle's north-pole.
Just these three circles do the following:
The south-pole of circle-3 gives the exact south extent of G3, and thus the line through HE.
The points A-B-F-G-H mark four segments in a row, where each segment is in Φ-proportion with the neighboring segment, or segments.
A-E-H is therefore half a square, a perfect reason to complete the square ADEH. In this position, the center of the Great Pyramid falls upon the diagonal DH. The south side of G3 falls upon the segment EH.
A line through the two mutual intersections of circles 2 and 3 has the exact angle of a diagonal in a golden rectangle, and points to the exact center of circle-1.
The rectangle ABCD is a combination of two exact golden rectangles, one vertical, one horizontal. Onwards it is called the Horizontal Column. By the way, a facsimile of one of the component golden rectangles of the Horizontal Column has been noticed by Robin Cook. I noticed the other component, and that was enough to start an avalanche of consequences.
Diagram
3
Reconstruction of the Great Pyramid - stage 1Diagram below:
From the left corner of the original diamond draw two (red) lines as tangents to circle-2. This produces points 4 and 5 where the lines cross the diamond's diagonal. A circle is drawn from each point, which passes through both the west and north corners of the diamond. In this position, adding two more lines will complete a 5-pointed star.
Of course, we could do the same construction on on the eastern side (he segment C-D). Then the two latter circles give the four points of the Great Pyramid in conjunction with two extended sides of the original diamond! At least that's how it looks on this scale. In fact, the north-east corner of G1, the Great Pyramid, is in its exact location in this blueprint, however the other corners are over 6 inches short of Petrie's locations. This is the initial stage of the exact reconstruction of G1.
Diagram 4
Diagram below:
Having G1 means that the square ADEH can be expanded to the pyramid's north-east corner. This is the so called Pyramid Square. The diagram also shows the 5-pointed star produced by the 13-step method, and an associated star, whose production took just three lines to existing points.
So, one average side of the Great Pyramid in the first stage of reconstruction is identical in length to one side of the star's inside pentagon - an exciting observation! Here we get to the truly important part. The Egyptians are making this particular star the dominant element of Giza's architectural layout. The star itself is special. There is a bevy of regular pentagrams, which can be derived from the basic construction in diag.1, but the things is, all those take a greater number of basic steps to make. The 13-step star is most efficient and thus elegant.
The catch is that this star, and the method of its making is not to be found anywhere on the Internet. How did it come into my possession? I learned it from playing the game of connecting dots on the world-famous glyph of a monkey on the decorated plain of Nazca, Peru.. There is great significance to the fact that these ancient sites share some very special geometry. Could more be discovered? Given the exact coordinates of everything at Giza, information which is not public at this time, I am quite sure that there would be major revelations.
Diag. 5
A way of locating the south side of G2
In the diagram above an X marks a point on the south side of G2. A line of the smaller star, and an extension of the diamond meet just 4 millimeters from the south side of G2 as given by Petrie. Despite being so accurate, later, there is an even better way to locate the same side, which gives this side to within about 1/250 of an inch, or 1/10 millimeter.
1150.626180cubits - the distance between two adjacent tips of the big star - this is a noteworthy reading since the fraction contains five consecutive digits of Φ squared (in cubits as determined by this study).
My first method of achieving exactly the same result:
Stage 1- Reconstruction of the Great Pyramid and the Pyramid Square,
diag.7
The Horizontal Column
Draw a vertical golden rectangle with diagonals radiating from its corners on the left, and add a horizontal rectangle to its right side. The two rectangles together form 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 goes to the top right corner of the Horizontal Column.
2) The left side of the Horizontal Column will be part of the Pyramid Square.
3) The diagonal 'a' is a tangent to the inscribed circle of the Great Pyramid (diag.7)
The pyramid's inscribed circle lets us have the pyramid sides.
The Horizontal Column and the pyramid then recreate lines through altogether three sides of the Pyramid Square, enough to draw the whole square (diag.8).
diag.8
Establishment of the length of the Royal CubitHaving progressed this far, we get the first chance to relate the reconstruction to Legon's preeminent theory on the layout of Giza. He makes a very strong case that the North-South distance between the pyramids (one side of our Pyramid Square) was meant by the builders to equal in cubits 1,000 times the square root of 3.
Accordingly, I've tried setting the Pyramid Square's side to 1,732 cubits, as well as the exact value of 1,732.0508.. cubits obtained by construction. However, the sensational results pop up when one side of the Pyramid Square represents 1,732.05 cubits, the first six digits of the square root of 3.
The test works like a magic wand, as many key dimensions take on the aspect of fine calibration. By this virtue, we get a good look at the exact length of the cubit used in planning Giza.
The bigger we make things, the easier it is to maintain accuracy. The planners had apparently realized this very well. It was easier for Petrie to measure the entire north-to-south length of Giza to a given error-percentage than the much smaller dimensions of King's Chamber, where each mistake had to take on a proportionally greater role. Therefore, the cubit values obtained in this study deserve consideration alongside Petrie's results from King's chamber.
South-west distance = 1,732.05 cubits = 35,713.1 inches
1 cubit = 20.61897.. inches = 523.722 millimeters
Phi as 2.618 times 1.2 = 3.1416 divided by 6 =. 0.5236 ( often given as the royal cubit in millimeters)
A Slew of Highly Precise Values
As was the case, I never stopped to convert the units from inches to cubits until the end of geometric reconstruction. By that time there were four versions of the reconstructed Great Pyramid, each a little closer to Petrie's version, for as the reconstruction went on, there were opportunities for readjustment.
Initial reconstruction:
439.50009258.. cubits less than 1/10,000 cubit from an exact half-cubit
The other three versions all result from adjusting the Rising Column to the width of the Horizontal Column. One seems uninteresting:
439.7667..
The other two involve a circle I call Alison's circle (diag.17a) after its discoverer Jim Alison:
439.79990284.. less than 1/10,000 cubit from an exact eight-tenths of a cubit.
439.83008582 .. looks calibrated as well .
The last value is just 0.0022 of a cubit, or 1.18 millimeter longer than Petrie's value
439.82782340.. cubits 9,068.8 inches
and so it becomes the final length of the reconstructed side - 9,068.85 inches
Petrie's plan is about 2.618 mm off the perfect side length of 439.823 for Pi encoding.
In two of four cases, the difference from perfect calibration of the numbers was less than 1/10,000th of a cubit. (When we compare the differences themselves, they differ by 0.00000458.. , or about 1/250,000th of a cubit.)
Such fine calibration of values clearly indicates that Giza's planners knew that the plan yields these values, when one side of the Pyramid Square equals the square root of 3, given specifically as 1,732.05. It's that simple.
If it weren't that simple, then precise values would be very unlikely to reoccur on on the very next step, pinpointing the east side of G3 with authority, which strikes awe in my heart, to be frank.
Final Solution to the Third Pyramid's Size and Position
diag. 9
Once I had reconstructed the south-west corner of the third pyramid, there were some strange readings. Firstly, the distance between this corner and the actual south-east corner of the pyramid became what many posit was the intended length:
201.50264414, or 201.5 cubits, if we disregard the 26/10,000 cubit, or about one millimeter. The distance between the south-west corner of the pyramid, and the square is:
314.50257453. The distance from the actual south-east corner of the third pyramid to the nearest corner of the Pyramid Square is:
516.00521867, or 516 cubits precisely. Something is obviously up.The fractional parts in 201.50264414 and 314.50257453 are very similar:
0.50264414 minus
0.50257453 leaves 0.00006961, or about 7/100,000 of a cubit.
Projecting the pyramid's length on the south side towards the south-west corner of the Pyramid Square leaves a gap of 112.99993038 cubits. This value is only 0.00006961 cubit, 1/714 inch, or 1/27 millimeter short of being perfect 113 cubits. So, in whole numbers. there is a 113 next to a 314 here, two thirds of a certain Pi approximation. Accident? Not in view of the following.
Exact Duplication of the South-east Corner of G3
Once the reconstruction produces the south-west corner of G3, mark exactly 113 cubits from the south-west corner of the Pyramid Square towards the south-west corner of G3. The remaining gap becomes the radius of a circle centered in the pyramid's south-west corner. This circle then misses the south-east corner as measured by Petrie, by the above mentioned 0.00006961 of a cubit. This truly is the mother of all coincidences! The final reconstructed line through the east side of G3 will be 0.00007 cubit off Petrie's plan.
Why 113?
Because the circumference of a circle with the diameter of 113 is 354.9999.., a perfect 355 for all the practical purposes. Therefore 355/113 must approximate π close to perfection.
355/π = 113.0000096
355/113 = 3.141592.. - The best approximation of Pi given as a ratio of two whole numbers.back to the reconstruction
In the diagram below, lines which appear to be single lines, are in reality black lines of the original plan, side-by-side with color lines of the reconstruction. Visually, the two Horizontal Columns become one.
On the other hand, the vertical line from the third pyramid's center visibly misses being the line that cuts the Horizontal Column into two golden rectangles. To showcase these facts, the diagram is page-wide.
diag.10
the Rising Column
It was Robin Cook, who first observed the basic idea of the below diagram: if we enclose the pyramids between two lines perpendicular to their diagonals, the central axis of the resulting column is almost exactly the second pyramid's diagonal.
diag. 4
But, this axis actually runs parallel east of the pyramid's diagonal at the distance of 13.82 inches. The relationship only looks accurate on computer screens, or on paper. Meanwhile, I've noticed that the thickness W-Z of the slanted column is just 4.32 inches more than the thickness A-B of the horizontal Column. Since this latter relationship is three times more accurate than the former one, it may also be more important.
The idea of comparing these two dimensions again, as they change during reconstruction, will pay off. Each new value for either dimension will be tried on the other, in the form of a circle of that diameter.
diag.11
Marking the actual thickness of the Rising Column W-Z straight down from the top side of the reconstructed Horizontal Column gets to within 0.94 inch of the Second Pyramid's horizontal axis. This is promising, because otherwise the idea that the bottom side of the reconstructed Horizontal Column is also the horizontal axis of the reconstructed G2, misses the actual axis by 10.12 inches.
Twin-like Circles
Three circles, one with the diameter W-Z (radius in cubits 337.94), one with the diameter A-B (radius 337.83), and one with the diameter of reconstructed A-B (radius 338.16), are completely indistinguishable from each other in the above diagram. Radius-wise, the smallest circle is 99.9 percent of the biggest one. To achieve the separation of 1 millimeter between them on a drawing board, the larger circle would have to be two meters across, and the lines would have to be infinitely thin. The Pyramid Square at this scale would be almost 5 meters per side. Such special effects usually happen for a reason. In this case, a researcher's appetite should be whetted for drawing some exploratory circles in search of progress to teh solution. He should be watchful for further incidence of similar circles in the scheme of things.
Reconstruction of Menkaure's Pyramid (G3)
How can we make further progress from this basic position? There isn't much to go on, but we have the square for G1, and to a geometer a square comes with a set of basic features like diagonals, axes, an inscribed circle, and a circumscribed circle.
So, how about some other implied circles? One such circle (E1 - for exploratory circle 1) centers in the south-east corner of the Pyramid Square, and is tangent to the circumscribed circle of G1.
e-circle
The diagonal 'e' in the diagram below is perpendicular to the diagonal "b", and ramps down from the north-west corner of the pyramid. This distance is taken as the radius of another implied circle, the e-circle.
Size-wise, this circle looks just like E1.
Copy this circle to the south-east corner of the Pyramid Square so it is concentric with E1, and two remarkable things happen:
1)
The two e-circles intersect only 0.66 inches away from the diagonal 'g', and a line from that intersection to the pyramid's south-east corner duplicates the true golden diagonal with the difference of 0.0015º, a very fine value.
diag. 12
The S.E. Corner of G3
2)
(diag.13) The two big circles (E1, and e-circle) centered in the south-west corner of the Pyramid Square seem to touch the east side of the third pyramid. Zooming in, it is then seen that both circles miss the south-east corner of G3 by about the same distance, but one overshoots, and the other one falls short. Their average, the midpoint of the distance between them is just 2.25 inches west of the pyramid corner. This point will serve as the south-east corner of the third pyramid in this reconstruction until the final adjustment.
The circumscribed circle of the Great Pyramid is complementary to circle E1. Next, the e-circle also gets its own complementary circle centered on the Great Pyramid. This is why the circles in the below diagram seem slightly fuzzy. They are dual circles. When the smaller circles are copied to the bottom left corner of the Pyramid Square they seem to touch the west side of the third pyramid. This is an evident and indisputable special effect.
s
diag.13
Menkaure's temporary vertical axis
Diag.14 - the small circle complementary to the e-circle opposes the e-circle from the south-west corner of the Pyramid Square. The axis of the gap between them is 4.2 inches to the left of the vertical axis of Menkaure's pyramid. Although not a close replica, it will serve as a one-time construction line.
diag.14
Left Bottom Corner of G3
Now it is the turn of G1's inscribed circle to be useful again. It is copied diagonally across the Horizontal Column to its south-west corner. Next, go to the first intersection of the new vector line (imitating Menkaure's vertical axis) with the Horizontal Column, and from this point draw a circle tangent to the far side of the transplanted inscribed circle. The point at which this circle intersects the interim vertical axis is experimentally considered the midpoint of the pyramid's north side, and the pyramid is completed on the basis of this assumption.
diag.15
The last action establishes the south-west corner of G3 with nice accuracy - to 1.2 inches, but otherwise this reconstruction fails. Yet, since we already have the south-east corner from the previous construction, that's all we need to complete the third pyramid. This reconstruction proves to be pure magic for the task still ahead.
diag.16
A view of the original and reconstructed bases of the third pyramid in their respective positions
One side of the reconstructed pyramid measures 4152.52 inches. It is 1.08 inches shorter than the original side. No wonder then that in the image above there is no visible separation between the two pyramids.
Two Accurate Reconstructions of the Same Diagonal of G2
The reconstruction continues from the center of new G3. There is some low hanging fruit in this position, which I had missed for quite a while: Draw a circle from the center of G3 through the south-west corner of the (reconstructed) Horizontal Column. This circle is then only 2.31 millimeters, 1/11 inch, or 0.00442 cubit short of touching the G2's diagonal (diagram below). The circle's radius is 442,237 mm, so the miss is aabout 1/191,000 of the radius, 0.002 of a percentage point. I suppose, this is breath-taking accuracy, which comes courtesy of embracing the Pyramid Square-concept.
In contrast, the same idea using the original G3, and the original Horizontal Column misses the same diagonal by more than half a cubit.
diag 17a
Alison's idea of a circle, which gives the Phi point in the diagram below gets a radical twist. The hypothesis is that it intersects the golden diagonal 'c' at the same point, as the second pyramid's extended diagonal. When put to test, the reconstructed diagonal courses just 0.58 inches off the original.
By the way, this Alison's function is an excellent illustration of how we can be lured into making a mistaken assumption about what the builders had really thought, and done. The point is that people automatically ascribe the minor inaccuracies in how their ideas work to the builders. Yet, this study shows time and again that the builders didn't make any mistakes.
diag.17b
The reconstructed Rising Column
Having reconstructed the first and third pyramids, we can also reconstruct the Rising Column idea, as shown below. Lines from pyramid corners P4, and Z extend at 45º from the horizon towards each other's diagonal. The result is an elongated rectangle. It seems that the central long axis of the rectangle duplicates one diagonal of G2, but this is just an illusion, as up close the miss is substantial.
diag. 18
Vesica Pisces over the Second Pyramid's diagonal
In the diagram, a circle of the column's diameter W-Z is projected along the column's central axis, beginning from Menkaure's extended diagonal, so that each new circle passes through the previous circle's center. The third and fourth circles in the column intersect only 0.89 of an inch above the actual diagonal of the Second Pyramid, and only 0.31 of an inch from the reconstructed diagonal d in diag.17.
In comparison, a repeat of the same procedure for Petrie's plan gives a fault of over ten inches, so again the reconstruction is working better than Petrie's plan.
Alison's multi-purpose Golden Section circle
diag. 19
Alison's Golden Section circle, of which we see just an arc is also pure magic. All our remaining steps depend on it. We know what points marked 1, and 3 do. Now point 2, which is at the intersection of Allison's circle with one side of the Rising Column, lets us duplicate the Great Pyramid's layout to 0.05 of an inch.
In diag.11, we copied a circle with the diameter of W-Z across the width of the Horizontal Column. This time we do the opposite. We make the Rising Column as thick as the Horizontal Column, using a circle for the transfer. The column widens by 7.77 inches on each side. It is this wider column, which is involved in point 2.
Exact Reconstruction of the Great Pyramid's layout
A horizontal line through point-2 then looks like a tangent from below to the circle we just placed at the top of Rising Column's axis (point-5). So, we take a hint, and draw a concentric circle, which is a true tangent to the aforementioned line at the point-4. This circle presents an opportunity to reset the pyramid side directly to 439.8 cubits even (see top of page) at the point of its intersection with the north side. But, that is not all:
Switch this circle to center anywhere on the central axis of the Horizontal Column. Let a horizontal line rest on the north-pole of this circle. Then that line is the horizontal (east-west) axis of the Great Pyramid!
The actual difference between the two is completely insignificant, and flies under the radar just 0.023 inch (1/46 inch) below the east-west axis of the Great Pyramid. The center of the newly readjusted pyramid will be at the intersection of this line with the pyramid's diagonal due from the north-east corner.
This works out to one pyramid's side being 9068.846.. inches long, under 1/20 of an inch, or 1.2 mm longer than Petrie's version. But, Petrie would have rounded this distance to 9,068.8 inches, so we have duplicated his result.
Looking back, the creation was all between the Great, and the Menkaure pyramids, G1 and G3. The second pyramid was completely absent.
It turns out to be the offspring of the other two, at least on the construction board. Having produced each other, the two outside pyramids combine again to define the inner one. The recreation of G2 is especially accurate on its south-side. There, the error is zero inches, when the result is rounded to two decimals. This is interesting, because Legon himself locates the same south side, with what he deems high accuracy. Yet, upon checking, the fault is an inch and half.. It is easy to understand Legon's perspective - what is an inch and a half in comparison to half a nautical mile, the south-north distance covered by the pyramids? It may have been the most accurate result ever obtained. In contrast, the one and only fault in this reconstruction is more accurate than the previous world record in accuracy, at less than 1.2 inches.
Reconstruction of Khafre's Pyramid (G2)
The task seems simple, all we need to reconstruct the second pyramid (averaged out sides) is knowing one of its sides, plus the center. To get the center, we need two of the lines crossing there.
There is some lumber in the yard already, three variants of the diagonal (d) - 0.09 inch due south-west, plus 0.58 inch, as well as, 0.89 inch due north-east of the diagonal. And in diagram 9, measuring the width of reconstructed Horizontal Column by the actual width of Rising Column had brought us to within an inch of the Second Pyramid's horizontal axis. The reconstruction can now repeat this trick using the reconstructed element instead. The exact location of the south-east corner of G3 (since diagram 7), and the reconstructed north-west corner of G1 just 0.046 inch from Petrie's ideal permit an exact reconstruction of the Rising Column.
Unfortunately, diagram 7 involved measuring with numbers, which belong to the final adjustment stage, while the reconstruction should stay in the realm of geometry. Again, the solution comes courtesy of the reconstructed Rising Column. The diagram below shows a close-up of the situation at the Great Pyramid's north-west corner after adjustment of the Rising Column to the width of the Horizontal Column. But now its corner is above the horizontal line of the pyramid base, which we know to be correct. Therefore the corner of the Rising column is brought down perpendicularly to the existing base. This gives us another version of the base, which is a little closer to the actual plan, and yet another version of the Rising Column.
diag.20
Reconstruction of Khafre's horizontal axis
Once more, we measure the Horizontal Column by the new Rising Column's width. It stops just 1.54 inches short of second pyramid's horizontal axis. Finally, this imitates the pyramid's horizontal axis with some accuracy.
Reconstruction of Khafre's vertical axis and south side
diag. 21
The position around the center of G2 is pictured above. The vesica-pisces line intersects the reconstructed horizontal axis closer to the center in the east-west direction than Alison's line, so that intersection marks the pyramid's interim vertical axis.
Exact Reconstruction - Southern Baseline of G2
Having the vertical axis allows placing the vertical lines through the sides of G2 from diagram 1 (the inner square of the Pyramid Square's Golden-cross).
The vertical line through the west base of G2 then meets the Rising Column's central axis just 0.0037 of an inch off the south base as given by Petrie. This duplicates the line through the second pyramid's south base with zero error (!!!), rounding to two decimals. At the scale we are operating on, this is undeniably Microscopic Precision!
The Cascading Ball Machine - Final Adjustments
In the same diagram, the action is by the reconstructed center imitating a ball rolling down available pathways. First, Alison's diagonal 'd' intersects the new vertical axis closer to the pyramid's center than the vesica-pisces line. So, the center drops down vertically to that point, and a new horizontal axis is drawn closer to Petrie's original.
Exact Reconstruction - Vertical Axis of G2
The circle from the newest center passing through the older one also crosses the Alison-line just 0.02 inches, about half a millimeter east of Petrie's original vertical axis. This is the final version of the vertical axis.
At this time, the Menkaure Pyramid also gets the final adjustment (diag.9), and changes the width of the Rising Column for the last time. By now, the established tradition is to test either column by the other. This test produces a line about an inch south of Khafre's horizontal axis, almost a mirror image of the 'new horizontal axis' on the other side. The central axis between these two lines runs only 0.03 inch from the actual axis, and therefore becomes the final reconstructed horizontal axis.
Now, we have a vertical and a horizontal line, each within a hair of Petrie's center. This locates the second pyramid's center just 0.03 of an inch, less than a millimeter from the center as given by Petrie. In conjunction with the exactly duplicated south base, this data allows an exact reconstruction of G2, the Khafre's pyramid.
A Fact: Petrie's plan can be recreated with great accuracy and simplicity from a profound idea.
G2 (Khafre) final discrepancies in cubits
West side 0.002
South side 0.00
North side 0.006
East side 0.004
Center 0.003
Rounding to two decimals, the average error for the layout of G2 is zero ..
There was an interesting change in my sentiments, as the study had progressed. Now, I don't marvel anymore at how close the reconstruction is to Petrie's plan, but rather at how close Petrie's plan comes to this set of exact ideas.
Jiri Mruzek
August 8, 2007
Vancouver, BC
©Jiri MruzekNotes
Integration of ideas by Legon, Alison, Cook and Tedder into the Pyramid Square added on April 15, 2008
Reconstruction of the Giza Plan added on April 29, 2008
Framing - the Pyramids
The CAD drawing of Giza's layout on my computer uses Petrie's measurements. I drew it to check out various interesting claims. To me the subject seemed already fully explored by numerous scholars, including Isaac Newton, hence I had no expectations of finding anything new. But extending the rectangle often used by experimenters to enclose the three pyramids into a square (the Pyramid Square), gave me a so far untested approach to Giza, although it has long been in my toolbox. For instance, in an experiment with the Nazca monkey figure, the monkey's feet were first enclosed in a rectangle oriented to the cardinal points (the Foot-frame). Extending it into a square then was the breakthrough move..
The Pyramid Square & Khafre's Pyramid
The below was my first experiment:a) The Pyramid Square is divided by the Golden Section into a basic grid. The lines create a cross within the square. Let's call this type of a cross " Golden-cross".
b) Khafre's pyramid is taken as the center square of its own Golden-cross. Next, the two Golden-crosses are superimposed over each other for comparison.
a sign of things to come
The square of Khafre's pyramid overlaps the Pyramid Square's golden-grid by 22 inches (56 centimeters on each side). So, the Golden-cross based upon Khafre's pyramid is a fairly close facsimile of the Golden-cross created by the other two pyramids together. But it certainly is not accurate enough to warrant being called a reconstruction. It is the 'step one', however, to the eventual exact replica of Khafre's south side. Until then, it is all there is for the reconstruction of G2.
diag.1
In the diagram below, the same golden proportions added to the G2 in its true location seem to find some correlation to the south side of G1.
diag. 2
Ideas of Chris Tedder, Jim Alison, Robin Cook integrate with the Pyramid Square
There is an informative article over at Jim Alison's site
http://home.hiwaay.net/~jalison/gpsp.html
It deals with some interesting work by John A.R. Legon, Chris Tedder, Robin Cook, and the author himself on the notion of a general plan of Giza's major pyramids.
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
a) Coming across Jim Alison's rendition of certain ideas by Chris Tedder was absolutely key:
Perpendicular distances between the pyramid centers produce two golden rectangles ( A-B-C-D, and D-E-F-P).
b) Jim Alison is the author of another key observation:
The segment F-H is then very close (8/100) to a 45º angle to 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 divides F-H at G into the golden proportion.
diag.3
Jim Alison's circle divides the segment F-H at the point G as 22,616 / 13,954.114.. = 1.621)
Integration
Merged into the Pyramid Square, both golden rectangles fashion new golden rectangles:
Tedder's Secondary Rectangle #1 - An extension of Tedder's horizontal golden rectangle A-B-C-D to the western side of the Pyramid Square happens to be another golden rectangle, the vertical rectangle C-D-O-K
C-D divided by C-K = 1.627
Tedder's Secondary Rectangle #2 - Allison's circle intersects the extended diagonal of the second pyramid rising north due west at the I-point. The distances from I to J and L form the golden ratio.
I-J divided by I-L = 1.6199 less than 2/1000 off the true Φ value
The horizontal rectangle I-J-K-L is therefore a golden rectangle.
Allison's circle seems to intersect at the I-point with two other lines - a golden rectangle's diagonal, and a diagonal of the second pyramid, hinting at a way of reconstructing that diagonal.
diag. 5
A view of five golden rectangles without the Pyramid Square.
diag. 6
Consistently accurate results signal correctness of the Pyramid Square-concept. Striving to show the precise clock-work inherent in the Giza layout is important. Consider the following refreshingly candid statement from a discussion on Randi's (the skeptical fortress).
http://forums.randi.org/showthread.php?t=81034&page=7
Jiri, the bottom line is that your idea is unprovable.
You could reconstruct Giza with extreme accuracy and precision, and show that it fits into a golden ratio scheme, and you will still have proven nothing. Except that it's possible to fit it into a golden ratio scheme. It doesn't prove that that was how the Egyptians planned it, let alone that they even knew about the golden ratio.
And this is your biggest problem. What you are doing is, ultimately, a waste of time. The only way to prove that the Egyptians knew about the golden ratio, or that they used it in the planning of Giza, is to find a contemporary account that says that that's what they did. An architect's plan with a notation that says, "Here's where we use the golden ratio". An historian or scribe noting that they measured out distances from one pyramid to the next, carefully noting diagonals as they did it would be a start, but not proof.
There is no proof that you idea is correct. None. And no amount of reproducing floor plans on your part will change that, even if your floor plans were 100% accurate, which they aren't, it would still be mere speculation.
To sum up the above extreme judgment by a fellow nicked Wollery, he declares circumstantial evidence worthless! That is patently untrue. For one, enough of such evidence can cast a deep shadow of doubt over any final conclusions to the contrary, and invalidate them. Any statements to the effect that the Giza pyramids were not part of a unified plan looks totally false next to this analysis.
It is a fact that the Golden Section clock-work is a quality inherent in the Giza architectural arrangement, if Petrie's measurements are correct. By Ockham's rasor rule, if something looks like an elaborate plan, then it is an elaborate plan. In combination with other data on design characteristics we have on Egyptian architecture, and Giza in particular, all that circumstantial evidence has already become so overbearing that the only rational conclusion is that Egyptians had extensive knowledge of the Golden Section, and had utilized it in sacred architecture.
Notes on pyramid dimensions when the N-S distance equals 1732 cubits even
final side of the reconstructed Great Pyramid - 439.81738902 cubits
Pi multiplied by 280/2, half the pyramid's height equals - 439.82297150..
The average reconstructed side is only some 0.00558248 cubit, 0.11 inches, or less than 3 mm shorter than the ideal side.
Divide the reconstructed value by half the pyramid's height of 280 cubits The result is 3.14155.. about 4/100,000 off the true value of Piactual side of the pyramid - 439.81512666
It gives 3.14153 about 7/100,000 off the true value of Pi. So, both the reconstructed and the actual sides express Pi to four decimals correctly.
side of reconstructed G3 - 201.49666 cubit , about 0.0033 off exactly 201.5 cubits, a number a lot of people mistakenly think is the true side-length.
Notes on pyramid dimensions when the N-S distance equals 1732.05 cubits (instead of 1.73205080..)
439.8300.. cubits per side of the final reconstructed pyramid
Pi times half the pyramid's height of 280 cubits equals 439.823cubits, rounded to three decimals. The side is about 7/1,000 cubit, 0.14 inch, or 3.7 mm too long to graphically express Pi.
The errors from both versions of cubits average out to 439.82373742 - less than 1/1,000 cubit, or 1/63 inch, or 0.38 mm from the best value.
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 2.618 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 could be perfect. The conclusion is 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. [10] 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."
Having expressed Pi to a finer value for Pi than 22/7, the need for a side approaching 400 cubits from the perspective of Pi exists mainly to add another dimension to the subject of Pi, and thus enhance and emphasize the pyramid's encyclopedic character (as a depository of knowledge).
The greater need for a side of 400 cubits arises from the perspective of Phi. The apothem divided by half the side, or 356/220 equals Phi to the first three decimals (1.618181818...)What then about the claim that a bit of fiddling around would net a competitive solution of its own? Well, what a perfect occasion this is to expose the claim as false. Just point to the limited success at this very endeavor by a significant number of researchers over a protracted period of hundred and twenty-five years since Petrie's publication. Suffice it to say that the best results from these combined efforts I have seen so far are all less accurate than the only and thus also the biggest fault in my reconstitution of the position This single fault results in two sides of G3 being out 1.2 inches. The best accuracy by others I have seen so far was one relation by John Legon, accurate to 1.5 inches. Giza is so big that when you reduce it to a plan the size of a regular drawing-board, inaccuracies of several inches will be completely invisible. That's why there are many claims of great accuracy, such that it cannot be a coincidence. When looking closer, the amazing accuracy turns into inaccuracy.
Pioneering Theories
Of course, some researchers have done invaluable work on the problem. John Legon abstracted a cohesive system from the Giza position, whereby one must start out from scratch, with just a simple idea, and develop it step by step into a plan closely resembling the Giza layout, or more accurately put, Petrie's plan. Another long-time researcher, Robin Cook displays a correct approach in observing all relations as possible coincidences first, and 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 a situation it is easy for a theorist to come to a conviction that his recreations mirror what the Egyptian planners had done, before the builders strayed from the plan, just as expected for the historical period, and such a titanic task, or before the plan got changed for reasons unknown. In all of these cases, the picture we get of Egypt is that of a brilliant, but early civilisation, adequately low in technology, and nothing to revolutionize Egyptology or History.
I drew inspiration from studies by Legon, Alison, Tedder and Cook, and found myself coordinating some of their elements against the background of my own ideas. The result is a unique procedure for the recreation of Giza's layout. Its greatest problem is at once its greatest strength - it is exact. That was not supposed to happen by the present paradigm. Of the twelve pyramid sides, ten are exactly on the line given by Petrie, because all discrepancies fall completely under the radar scanning for errors.
While the recreated south-east corner of the third pyramid works out to being less than 1/27 millimeter from the position given by Petrie, the south-west corner strays 1.2 inches to the west (the only fault). Since the reconstruction of the third pyramid depends on these two corners, two of the sides then end up being 1.2 inches longer.
Yet, with the exception of one, the north-east, corner of the Great Pyramid, the positions of all the other corners depend on this south-west corner of the third pyramid being right where it is. It is absolutely pivotal to the rest. In my opinion, it is fully justified by performing this role. Moreover, this corner by being in its place, creates another record of Pi to be found in the Giza pyramids, this time a record of the approximation of Pi, as the ratio 355/113. Apparently, Petrie had some difficulties measuring the third pyramid because of ruination on its north-west corner, and that could explain the discrepancy, once the sides were averaged out in his basic plan.
It makes sense that the original plan also started out with regular squares. Adjustments were then made to encode further data, or perhaps, to accomodate some of the seeming coincidences. One of the lessons of this study is that we should expect that data to be extremely accurate as well.
This recreation tells a lot about the Egyptian planners, and builders. Accepting my solution as essentially selfsame with the original plan would do more than just raise high the bar of Egyptian knowledge of mathematics. The dynamic nature of the plan's development all but eliminates the possibility that it could have been drafted. Considering the scale of Giza, and the finesse of the method, some of the work to be done is virtually subliminal, small enough to be invisible on the biggest drawing board, because of interference by nearly identical objects, such as concentric circles. If the plan cannot be worked out by drafting, it had to be worked out by calculation. Thus, the knowledge of mathematics guarded by the temples had to be on a level categorically unreachable for a neolithic civilisation less than two millenia removed from the hunter-gatherer stage.
In this case, speculation about advanced prehistoric science, which had somehow continued to exist in dynastic Egypt in secrecy, simply cannot be avoided. That is unless this solution is taken to represent 'just another one of those series of consistent coincidences' so typical for the Great Pyramid, and Giza. Never mind that this reconstruction satisfies all the criteria, which differentiate it from random. All the same, as long as the sandheap for the proverbial ostriches just keeps on getting smaller, all is well.
Regardless of complications in evaluating the meaning of this discovery, it is now a matter of public record that Petrie's layout of the three pyramids can be produced by a profound exact method, easily, quickly, and with maximum accuracy.
Hesire's Tomb Engraving & The Golden Section
The below arrticle is really interesting in that it contains certain findings I made in 2007, which it seems, might have been made by a Russian architect long time before me. I can't be sure of exactly how the other work goes, because it is onily available if you buy the book. I don't buy such books out of principle just like I don't try to make anyone pay for the information available here.
The Graven Image of Hesire from a wooden door in hs tomb is almost 47 centuries old. Like Imhotep, architect of the first pyramid in Egypt, Hesire belonged to King Djoser's (Zoser's) intellectual elite, and held many titles like Overseer of Scribes, and Chief Dentist.
In this image, Hesire grips two staffs in such a manner that they form a right angle. The upright staff looks approximately twice as long as the level one. If there is a clue that the image is constructed on a geometric basis, this is it. At the same time, Hesire looks not unlike a warrior 'en guarde' with a sword, and a shield . Indeed, these weapons refute those Egyptologists, and historians, who uphold the present consensus that ancient Egyptians had only rudimentary knowledge of mathematics, their value for Pi was about 3.16, and they didn't know and use the Golden Section. Yet, the truth is that Egyptians had considered the Golden Section sacred, and thus perfectly suitable, one could say a must, for the planning of their temples and tombs. We have seen, how extensively the three Giza pyramid temples embody the Golden Section.
To put the clue to a test, the length of the upright shaft serves as the basis for two side by side squares. The diagram below shows that the testing squares fit the figure. For example, the line dividing the rectangle into two squares is also a line of Hesire's belt.
The cut of Hesire's sword
The line along the lower edge of Hesire's 'sword' gives the Golden Proportion with the (parallel) central axes of both squares in the column.
When we fit two golden rectangles below the top of the lower square, as in the image, or when we center a golden rectangle in the middle of the two-square column, in the process, we recreate the lower edge of the engraved clue line.
A natural progression of experiments
Since Hesire's right foot meets the ground at a perfect right-angle, it provides an experimental socket for the two-square column from above. First, we just transport the column to the socket to anchor it there without changing its angle. In this case, the left side of the column shows definite alignment with Hesire's body. The yellow lines in the diagram below are those of the Golden Section. The Golden Section line on the right runs with the edge of the long staff. On the other side the equivalent line runs to Hesire's eye. There are other harmonies. One diagonal of the top square runs with the edge of Hesire's forearm for most of its length. The top side of the column limits hesire's left shoulder at the armpit.
All details considered, this experiment also shows some merits, and yields further insight into the designer's methods.
The obvious next step
Next, the anchored column is aligned to Hesire's foot. The result is more harmony between Hesire's figure and geometry. For example, a different golden-section line now passes through the same corner of Hesire's eye. Notably, the column in this position determines the limits of Hesire's figure on two sides, the bottom, and the right.
This leads to an idea that these two limit lines are like half a bounding box (frame) for the figure, and so, completion of this bounding box is the subject of the (for now) final experiment.
The short staff sets the width of the bounding box (the frame) on the left. The top side is then added automatically. This is a standard experiment, in which three rectangular lines form a socket for corresponding figures. We can insert a square, or golden-rectangles, or their combinations into the socket. If the designer had worked with the Golden Section at all, then these experimental expansions make it possible to fall in tune with the original design.
Hesire's frame can be seen in several ways - a square sandwiched between two horizontal golden rectangles, or two stacked golden rectangles, one vertical, one horizontal. Careful scrutiny of the result reveals much harmonious correlation between the image and the grid. For instance, the vertical limit line on the left of the previous column is also incorporated into the new grid, and so is the top line of Hesire's right foot.
Conclusion: The designer of Hesire's image on the wooden door in his tomb had deliberately integrated the Golden Section in its layout.
©Jiri Mruzek
August 8, 2007
Vancouver, BC
Note: The rasterization module in my vector driven program has a stubborn kink, which elongates the rasterized images vertically by about three and a half percent (1.034 to 1.035). This is a recent problem, dating from "The Giza Pyramid Temples & the Golden Section. I am not willing to readjust raster images, it just does not seem right. But, such adjustment will produce the optically correct ratio.
