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Exact Reconstruction of the Layout of the Great Giza Pyramids
Foreword
Since Sir William Flinders Petrie had provided reliable measurements from his geodetic survey of Giza in 1883, more than a century of public scrutiny of this data has revealed great many apparent geometric and numeric relationships. Many tried to generate the layout of the three great pyramids, from pure ideas inspired by such relationships, but no one could do so with enough compelling accuracy to pin those ideas on the original planners. Some authors mention builder errors to explain why pyramid corners and edges are not where they say they ought to be.. Of course, theories often contradict one another, and the academy, in general, scoffs at the whole idea. With so much smoke around, this is the burning question: Were the three great Giza pyramids laid out as parts of one grand plan?
A diehard skeptic will try to nip the idea in the bud, venturing that even if Giza could be cleanly translated into a meaningful geometric design, it would mean nothing without the discovery of original blueprints, or a statement from the architect. Indeed, there is a speck of truth in every big lie, and in this case, there must be a number of procedures to recreate Petrie's Giza layout.
But, for a procedure to survive more than a glance, accuracy, and swiftness are mandatory. If the layout is random to begin with, even the most efficient procedure, the geometrical how-to of its recreation, will perforce be dragged out, and soulless. As well, the closer a reconstruction wants to get to the original, the more its complexity grows. Other than in rare special cases, there is no way to accurately describe a random position by a few simple geometric operations.
By the same token, if such an elegant method of developing the Giza layout in almost exact agreement with Petrie's blueprint does exist, then it is, in all likelihood, the original method rediscovered. It is my honor, and pleasure to present that method here.
The reader might think my claim presumptuous, and perhaps, a little arrogant. But, cross your heart, would not true arrogance be to think that it was me, who had somehow imbued the layout of Giza with so much spirit, and science? Read on, and see for yourself, how well the method works. The sheer existence of one successful method, due to its fully defined mathematical nature, precludes successful deployment of other methods. So, supposing that I am not brazenly lying about the glorious nature of the Giza layout, but delivering on the promise, who is its creator? Me, Petrie, the ancients, or pure chance? Is it not time to put a stop to all the skeptical nonsense about Giza, about how its creators did not realize what profound concepts they were to bestow upon its sacred grounds? The answer must be that any excellent fit between profundity and Giza is due to Giza's creators, not me. Of course, they knew what they were doing! A tribute is also due to Petrie's brilliant scholarship, and his translation of the pyramid positions on a sloping Giza plane onto a two-dimensional plane, successfully reversing implementation of the original 2-D plan by the builders.
(I hope that now is not too late to caution the reader about developing prejudice against this work on account of my 'literary' English. Suffice it to say that upon landing in Canada, as a young adult, mustering a few sentences in broken English, was all I could do.)
The secret to the success of recreating Petrie's Giza layout from scratch (tabula rasa) was in the synthesis of existing observations on the subject within the framework of an original concept. Put simply, the solution was born the minute I had extended the usual enclosing rectangle of the three pyramids into a square. Simple, yes, but against this background, scattered geometric elements start making sense together.
Mirrored Illusions Become Reality
The layout of Giza is rife in coincidences. Coincidences create illusions. That fact makes decryption tricky, and may have been one of the major reasons for its selection. In a spiritual sense, false paths obscuring Giza's recreation (I blundered down some of those) may well symbolize the soul's perilous journey through life to fulfilment.
Testing illusions geometrically brings an immediate disappointment. Things do not fit nowhere well enough. But, in this first pass of exact geometry over the Giza layout, new illusions appear. Reality then is the axis of symmetry between illusions..
How to Recreate Petrie's Giza Scheme from a Line Segment
July 31, 2008 - A sensational find deepens the position!
Until then, I never paused to fill in the preliminary constructions to the golden rectangle, from which my plan was derived. When needing it, I just grabbed an available template. The price to pay was obliviousness to the fact that Giza plan's creation must begin with one of my most favorite prehistoric constructions, one I had learned from the geometry of Nazca Monkey, and thus, lack of deeper understanding of the Giza position. Not using this method would complicate the effort, and make the reasons for existence of the given layout much less clear.
The method creates a solid foundation of theoretical Geometry for the layout on the first pass, and on the second pass, things fall into place by simple application of the right elements. The correct start uses the below classic construction. It begins with a horizontal line, and takes ten steps to produce. Two of the steps are help circles, and not shown. The eighth step gets the key Golden-circle ('c' in the diag.), which is centered in the bottom of the axial cross. Steps nine and ten are the drawing of lines from the top point of the axial cross, as tangents to this circle. The lines give the exact angle of 36 degrees (like on a 5-pointed star).
Diagram
1
In the diagram below three more steps complete the regular 5-pointed star (pentagram) - first, a circle from Q (the Q-circle) through the top and bottom points of the axial cross, then two lines through just created points (points marked by small circles). This is the fastest such construction in geometry, I believe - thirteen moves from start to finish (its simplicity is 13).
The unique element of this construction is the Q-circle (or its mirror image) since other constructions generally produce differing stars. It passes through four out of ten vertices of the star. After the Q-circle there is a choice of things to do.
Diagram 2
a) two steps to complete the star (as above)
b) three steps to get the four corner points of the Great Pyramid (as below), in the first of two stages.
c) return to diagram-1, and diverge from it on step-8, using a slower method of getting the same four interim corner points of the GP.
The reconstruction directly references every available method, as if to accentuate the creators' familiarity with the entire spectrum of possibilities.
Reconstruction of the Four Corners of the Great Pyramid - stage 1
With lines of the diamond extended, and points 1&2 already in existence, a single step produces the other two points, corners of the Great Pyramid:
a) Either draw in the other Q-circle (diag. 4),
b) or draw a circle centered in the corner of the diamond, whose radius is the distance from the corner to points 1 or 2, where the Q-circle intersects the diamond.
Either circle intersects the extended sides at the other two corners of the interim Great Pyramid. Using the smaller circle is preferable since it is also the pyramid's circumcircle.
The diagram below shows the inscribed diamond instead of the circle-3 from diagram 1.
Diagram 2b
The diagram below is based on the above diagrams, but since I prefer to start on the western side of the square, the construction now moves in the other direction, and the star points the other way in the following diagrams, This actually seems to be the correct procedure order. There is a lot of goings on in the design the reader needs to see.
Diagram 3
a)
The points A-B-F-G-H mark four segments in a row, where each segment is in the Φ-proportion with the neighboring segment.
b)
The south-pole of circle-3 (circles 2 and 3 are golden circles) gives the exact southern reach of G3, the Menkaure pyramid, marked by a line through E-H.
c)
A-E-H is half a square. Complete it as the square A-D-E-H. The center of the Great Pyramid will always be on the diagonal DH.
d)
The rectangle A-B-C-D is a combination of two true golden rectangles, one vertical, one horizontal. Onwards it is called the Horizontal Column.
A facsimile of the larger golden rectangle of the Horizontal Column was discovered in the position by Chris Tedder. With the advantage of viewing Tedder's Golden Rectangle in context of the square (diag.), the remainder of the Horizontal Column stands revealed as another, this time vertical, Golden Rectangle. This was a crucial piece of intelligence, which set off an avalanche of new observations.
e)
A vertical tangent line to the right of each golden circle (2&3) divides the Horizontal Column into two golden rectangles. Tedder's Golden Rectangle is one of these.
Note that the line through the two intersections between circles 2 and 3 has the exact angle of a diagonal in a (vertical) golden rectangle.
Both Q-circles together also produce the pyramid's four corners (diag. below). The distance between points 4 and 5 (side of a pentagon in diag.2) equals one side of the interim Great Pyramid.
Diagram 4
In fact, if the north-east corner of the Great Pyramid (G1) is in its exact location in this blueprint, then the other corners are over 6 inches short of Petrie's locations, but on this scale (about 1 : 7000 on an average monitor screen) that distance shrinks to effectively nothing. This is the interim stage of exact reconstruction of the proto-pyramid of G1.
The Pyramid Square
After the proto-pyramid, the square A-D-E-H (diag.3) can be extended to the pyramid's north-east corner. This creates a containing square for the pyramids - the so called Pyramid Square.
The diagram also shows the 5-pointed star produced by the 13-step method, and a smaller associated star. To make the smaller star, one needs to draw only three lines between points already in existence in this context.
The 13-step-star is the guiding star of Giza's ground plan. Although there are other procedures on record to construct a pentagram, I have seen none faster than 15-steps (despite repeated Internet searches). Fortunately, I had already learned the 13-step method from the giant Nazca Monkey (part of the spectacular Nazca Lines in Peru).
The fact that Giza and Nazca seem to share this special geometry is nothing short of sensational, as it augments the broad geometric connection, inclusive of the 13-step method, between the monkey, and one Stone Age engraving from western France (the rock shelter of La Marche, near Lussac-les-Châteaux, yielded 1,500 masterfully engraved stone tablets). It would be nice to be able to dig into hard to get detailed data on Giza, Nazca, and La Marche. Who knows what major revelations might follow?
Diag. 5
Locating the south side of G2
In the diagram above an X marks a point on the south side of G2. There, a line of the smaller star, and an extension of the original diamond meet 4 millimeters above the south side of G2. Now, if we had the center, we could already reconstruct this pyramid's layout with unsurpassed accuracy..
A remarkable coincidence!
A side of the pentagon drawn between the tips of the 13-step-star = 1150.626180 cubits. This is another indication that the units used are correct, as is the construction (the '13-step'):
We see five consecutive digits of Φ squared (26180).
Altogether, I found three ways of reconstructing the proto-pyramid.
diag.6
Starting from the Horizontal Column
This was my original method of reconstruction:
Draw two golden rectangles, a vertical one on the left, and a horizontal one on the right, diagonals radiating from their corners,. The rectangles form a solid column - the Horizontal Column.
C divides A-K so that if C-K equals Φ - 1, then A-C equals Φ, and A-B equals 1.
The length of the combined rectangle (the Horizontal Column) then is 2Φ - 1.)
1) The center of the Great Pyramid is at the top right corner of the Horizontal Column.
2) The left side of the Horizontal Column is the western side of the Pyramid Square.
3) The diagonal 'a' is tangential to the inscribed circle of the Great Pyramid (diag.6). This circle is then enclosed in a square, i.e., the pyramid sides.
4) The lines through the north and east pyramid-sides are corresponding sides of the Pyramid Square. With three sides of the square known, so is the fourth - the bottom of Square. The south side of the third pyramid lies on the square's bottom (diag.7).
diag.7
Naturally, the golden rectangles above could be preceded by a number of different starting positions. But, only the '13-step' stars produce a close location for one side of G2.
Reconstruction of Menkaure's Pyramid (G3)
Operation Rising Column
Look-alike Circle Pairs
The widely used Giza containing rectangle, and the Pyramid Square share the same south-east corner. I wonder, how many people had drawn an experimental circle from there to touch either the Great Pyramid's circumcircle, or the south-east corner of G3 to find out that it seems to touch the other object, as well.
diag.8
It is doubtful that anyone had seen how the Great Pyramid's circumcircle seems to touch G3 from the other side, when copied to the SW corner of the Pyramid Square (diag. above). This is certainly a startling effect, albeit up close it is not all that accurate. (for another special effect inherent in this position, see diagram f in Notes)
The above diagram is misleading, however, in that the alleged circles are being simulated by a different pair of circles, but, from this high up above Giza it makes no visual difference. As is, the big circles are both 'e-circles'.
The radius of the original e-circle is given by the distance between the NW corner of G1, and the golden diagonal emanating from the NW corner of the Horizontal Column. The smaller e-circle centers in the center of the Great Pyramid, and is tangential to the big e-circle centered in the SE corner.
Very Special Effects
This latter e-circle-pair wheels the entire reconstruction fast forward. First, an amazing effect, involving both the big circles described above, vies for our attention.
Shattering the Giza Record
diagram 9
The diagram above is a close-up from diagram 8 of the situation in the south-east corner of G3, the Menkaure pyramid. Lines 1 and 2 belong to Menkaure. Lines a and b are the e-circle, and its look-alike, the tangential circle to the Great Pyramid's circumcircle (Petrie's version). Both are almost exactly equidistant to the pyramid's corner! The centerpoint of the distance between them is 3.4 millimeters to the east of the pyramid corner as given by Petrie. The distance itself is 1.0005 cubits, a fascinatingly round value.
This means that given the Great Pyramid's position, we could pinpoint the SE (southeast) corner of Menkaure's pyramid with, for Giza unsurpassed, accuracy.
The method used above is truly simple. Given two versions of the original, the solution is their average. In practice, this method will work out over and over to a simple step:
"Go down the Middle!"
For now, we can use the proto-pyramid of G1 for the same operation. It locates the SE corner (green lines) 3.5 inches to the west of the original. This corner plays a part (#2) in the second interim version of G3. Both versions use the same SW corner.
diagram 10
The visual effect (diag.8), in which the e-circle pair hugs the Menkaure pyramid from both east and west, had then led straight to the first interim reconstruction of G3 - the Menkaure Pyramid, permanent establishment of its SW corner, and exact positioning of the Great Pyramid. Duly, I never questioned the means, guided by the symbolic language of coincidences, i.e., clear hints on what to do next. Nevertheless, the geometric statement, which was so strong in the opening phase of the reconstruction was lacking. In the end, a big question was left looming:
Why use the e-circle?
A look at the e-circle in context of the '13-step' construction reveals its integration (via line 'e' - its radius). It also establishes a third method of deriving the proto-pyramid of G1. Time out for two diagrams:
diagram 11
a)
Line 'a' in the diagram above constitutes step 7 in the 13-step construction of a pentagram. After line 'a', the construction can diverge from the routine.
b)
From the point at which line 'a' intersects the circle, a new line can be drawn to the top corner of the diamond, and extended down to the horizontal diagonal. This is line 'b'. Its angle makes it a golden diagonal.
c)
From the intersection of 'b' with the horizontal diagonal, 'c' is drawn vertically upwards. (Line 'c' completes a golden rectangle in the position)
d)
The intersection of 'c' with the upper right side of the diamond is the origin of horizontal line 'd'. (Line 'd' divides the golden rectangle made by 'c' into a square and a smaller golden rectangle)
e)
Finally, the point at which 'd' intersects 'b' is the origin of line 'e' (e-circle radius) - perpendicular to 'b'..
Diagram below:
()
Line 'e' connects directly to the NW corner of G1, and shows the third way to construct the same proto-pyramid - a 135 degree line through the NE coorner of the Horizontal Column meets line-e (or the e-circle, if it it is centered at '1') at the NW corner of G1.
()
The point-2 on 'e' connects to both the west and east corners of the original square (yellow), and the SE corner of the Horizontal Column by lines, which form the stars in the diagram below. In other words, the point is part of the '13-step' star construction.
diagram 12
The e-circle dominates the reconstruction of G3. From the viewpoint of legitimacy, it is significant that this circle has such a classy geometric origin. The first phase of the reconstruction deals with fundamental geometry. In the second phase, some elements of the position are manipulated in several simple operations.
Menkaure's interim vertical axis ('go down the middle')
The midpoint of the gap between the pair of circles sandwiching the third pyramid in diagram 8 is 4.2 inches to the west of the pyramid's vertical axis, as given by Petrie's version. Draw a vertical axis from there.
The Interim G3 (#1)
a)
Draw a circle from the intersection of the vertical axis with the Horizontal Column, such that it touches the far side of the little circle in the column's SW corner. The little circle is already a part of the position, because its diameter forms one side of the inner pentagon of the '13-step star'.
b)
The section of the vertical axis below the new circle is taken as equal to one side of G3. On the basis of this assumption, the pyramid is completed around the axis ('go down the middle' again).
diag.13
The last action permanently establishes the southwest corner of G3 with nice accuracy - to 1.18 inches. Eventually, this will be the biggest, and one could argue the only discrepancy of the reconstruction from Petrie. The SW corner of G3 is now permanent, the SE corner needs work, being seven inches too far west, making the proto-pyramid side six inches shorter, but is just what we need with respect to finding the Great Pyramid's true position.
The interim G1 and G3 (#1) are prerequisites for a simple operation, which renders the proto-Great-Pyramid identical to the Petrie version.
Robin Cook says that if we enclose the pyramids between two 45º lines (perpendicular to their N.W. - S.E. diagonals), as in the diagram below, the long axis of the resulting column is almost exactly the same as one of the second pyramid's diagonals.
diag. 14
But in Petrie's plan, the axis actually runs east of the pyramid's diagonal at the distance of 13.82 inches. This relationship only looks accurate on computer screens, or paper. However, it does create a spectacular illusion, a smoking special effect. But, where is the fire?
Meanwhile, I noticed yet another special effect, the width W-Z of the Rising Column is just 4.32 inches more than the width A-B of the Horizontal Column. Comparing the reconstructed columns could therefore be of interest, and is next on the agenda.
The idea that the bottom side of the reconstructed Horizontal Column should also be the horizontal axis of Khafre's pyramid (G2), is a 10.12 inches miss, and of course, yet another puzzling special effect. But, marking the actual thickness of the Rising Column W-Z straight down from the top side of the reconstructed Horizontal Column gets to 0.94 inch south of the second pyramid's horizontal axis.
Exact Repositioning of the Great Pyramid
Make the Rising Column given by the proto-pyramids G1, and G3, as wide as the Horizontal Column. Then the situation in the Great Pyramid's NW corner looks like the following diagram, just under lifesize (16/17 on my screen).
diagram 15
'go down the middle'
A line drawn vertically down from the northwest corner of the wider Rising Column becomes one with the western side of the Great Pyramid in Petrie's version.
On the line through the north side of the pyramid, both the west side of the wider Rising Column, and the NW corner of the proto-pyramid G1, are equidistant to this corner in Petrie's version. The difference in length between the reconstructed and the original versions of one side of G1 is too tiny to be noticed, at 0.011 inch (1/90), or 0.28 millimeter, or 0.0005 cubit. The pyramid centers are 0.008 inch, or 0.2 millimeter apart. In other words, the two versions of the Great Pyramid are identical. Our final value is: 439.8273 cubits (439.82727850) or 9,068.8 inches (9,068.7887..) per side.
439.82727850 / Pi = 140.001.. The pyramid needs to be about 1 millimeter higher than its theoretical height of 280 cubits, to be perfect with respect to Pi, and this reconstruction..
Petrie's value: 439.8278.. cubits or, 439.828 or 9,068.8 inches
Petrie's plan is under .005 cubit (about 2.5 mm) longer than the ideal 439.823 for Pi encoding, while this reconstruction will be half a thousandth closer:
Pi times half the pyramid's height = 439.82297150.. or, 439.823 rounded
It is interesting to note that the reconstruction of the Great Pyramid does not require Khafre's pyramid at all.
an Accurate Reconstruction of the SE corner of G3
With the Great Pyramid duplicated, it is now possible to repeat the steps from diagram 9, using the duplicate.
diagram 9
Above is a close-up from diagram 8 of the situation in the south-east corner of G3, the Menkaure pyramid. Lines 1 and 2 belong to the pyramid. Lines a and b are the e-circle, and the tangential circle to the Great Pyramid's circumcircle. Going down the middle again, the centerpoint of the distance between them is 0.13 inch, or 3.2 millimeters to the east of the pyramid corner as given by Petrie. This result is a whisker outside of Petrie's range of fault-tolerance (3 millimeters, missing by .2 millimeter).
Of interest is the distance between lines 'a' and 'b'. It is expressed by the cyan circle, whose radius is 1.0001.., a rather exact cubit. How significant is this sharply defined measurement in view of the fact that it marks procedurally the very last operation in reconstructing both G1, and G3?
In my opinion, it is a checkpoint at the two-thirds mark of the journey, a check-sum of one cubit, which tells the explorer of being about 0.0001 degree off the true course, before setting out for the final operation.
Records are made to be broken
There is also an ingenious, and numerically meaningful way to pinpoint Petrie's south-east corner of G3 (Menkaure) with accuracy, which strikes awe in me. The reconstructed SW corner of G3 is the only prerequisite, aside from using our own version of the cubit. Thus this reconstruction becomes possible before the reconstruction just described above, but it is by no means obvious. One could easily be distracted by the other accurate solution for the same corner, and come to think about it, another perfect false door.
Final Solution to the Third Pyramid's Size and Position
diag. 16
The location of the SW corner of G3, the third pyramid, yields some notable readings in cubits.
a)
First, the distance from the SE corner of G3, as given by Petrie, to the reconstructed SW corner of the Pyramid Square is:
516.005, 516 cubits almost exactly.
b)
The distance between the reconstructed south-west, and the Petrie given south-east corners of the third pyramid becomes what many authors posit to be its intended length:
201.5 02644, or 201.5 cubits (the small change 0.0026 is worth over one millimeter)
c)
Distance between the SW corner of G3, and the SW corner of the Pyramid Square:
314.5 02574The fractional parts in the above 201.502644 and 314.502574 are very similar:
0.502644 minus
314.5 02574
0.502574 leaves 0.00007, 7/100,000 of a cubit difference between them
- 201.5 02644
= 112.99993
Put into words, if we flip over westwards the distance between the reconstructed SW corner of G3 and the Petrie given SE corner, it is short of the SW corner of the Pyramid Square by:
112.99993038 cubits
0.00006961 cubit 0.0014 inch, (1/28 millimeter) short of being perfect 113 cubits.
This remarkable arrangement seems to be suggesting an easy way of reconstructing the SE corner of the Menkaure pyramid to 0.00007 cubit, and an absolute zero on the Giza scale.
Exact Duplication of the South-east Corner of G3
For the reconstruction of the SE corner of G3 mark exactly 113 cubits from the SW corner of the Pyramid Square towards the SW corner of G3. The remaining gap becomes the radius of a circle centered in that corner. This circle then locates the SE corner, as given by Petrie, to within the above mentioned 0.0014 inch, 0.036 (1/27) millimeter, or 0.00007 cubit. In plain English, the two are perfectly identical. For Giza reconstructions, such accuracy is superlative, and never seen before.
π
In terms of whole numbers. there is a 113, and a 314 here, two thirds of a certain Pi approximation.
Why 113?
113 x π = 354.9999..
355 / π = 113.0000096
The circumference of a circle with the diameter of 113 is a perfect 355 for all the practical purposes. Therefore, 355/113 must approximate π close to perfection:
355/113 = 3.141592.. - The best approximation of Pi given as a ratio of two whole numbers.
Accident? If so, it joins a plethora of coincidences pertaining to Pi, for which Giza, and especially the Great Pyramid are famous. This 'accident' is indispensable to the exact reconstruction of the SE corner of G3, which we need for the exact reconstruction of the Rising Column. The latter is then instrumental in the reconstruction of G2.
The impression is clearly that Giza's planners knew that their plan yields these values, when one side of the Pyramid Square equals the square root of 3, given specifically as 1,732.05 cubits.
The Layout of Khafre's Pyramid (G2)
This reconstruction has undergone wholesale changes since March 8-9, 2009. I found a small mistake made rather early on. It was one of those small mistakes with far reaching consequences, and I was devastated, at first. The only solace was memories of similar predicaments in the past, which had invariably led to remarkable improvements overall in the subjects involved. And so it was this time around. Correction of the mistake perfected the result for the Great Pyramid, and greatly simplified the layout's reconstruction. Such simplification also holds true for the Khafre Pyramid. The good thing about the mistake is that it can easily be repeated, and with it, some interesting effects, so my time wasn't entirely wasted.
Lumber in the Yard
Earlier we saw a way to position the south side to within four millimeters of the original G2, and so, if given the center, this reconstruction of G2 would show considerably greater accuracy than any other reconstructions (to the best of my knowledge, of course).
Some usable data is mentioned in the note under diag. 19: 'marking the actual thickness of the Rising Column (W-Z ) downwards from the top side of the reconstructed Horizontal Column gets to within 0.94 inch south of the Second Pyramid's horizontal axis'.
Of course, with Rising Column's accurate reconstruction, the 0.94 inch result above stands duplicated. It approximates Khafre's horizontal axis fairly closely, but its function will be to reproduce the vertical axis.
G2 - diagonal simulation #1
Alison had read the position in the diagram below, as saying that a circle (Alison's circle) centered in G3, whose radius is the horizontal distance between the centers of G1, and G3, then closely approximates the Golden Cut in the given line (marked Phi) from the center of G3 to the intersection between the circle inscribed into the Great Pyramid, and a diagonal.
In the Pyramid Square context, this line runs from the center of the second interim version of G2 to the point of intersection between the inscribed circle of the proto-pyramid of G1, and its diagonal. In plain view, the Alison's circle finds the intersection between the golden diagonal 'c', and the second pyramid's extended diagonal (point I). Let the golden diagonal 'c', and the Alison's circle simulate the diagonal of G2. This relationship is an order of magnitude more accurate than the one Alison noted. The simulation is 0.66 (0.6584) inches northeast off the original.
diag.17
Channeling the Solution of a Diagonal of Khafre's Pyramid
G2 - diagonal simulation # 2
Two capital lines, the long axis of the interim Rising Column (a), and the bottom line of the Horizontal Column (b), meet 0.66 inch southwest of the diagonal in Petrie's plan. This creates a point of insertion for a diagonal simulation (c). Remember it as G2 - diagonal simulation # 2.
Line 'd' in the diagram is 'diagonal simulation #1.
#1 0.6584 inch north-east of the diagonal
#2 0.6629 inch south-west of the diagonal
diag. 18
Petrie's diagonal itself does a good imitation of the axis of symmetry of the channel between the two simulations (diag. above). Considering the scale of operations, the axis is absolutely identical to Petrie's plan at 0.00226 inch, 0.0001 cubit, or 0.057 (1/17) millimeter to the southwest of the original. It is only fitting to let the axis serve as the Channeled Diagonal. Its distance from the channel sides is 0.66 inch.
The Vertical Axis of G2
Take the width of the Rising Column's duplicate, and mark it on the Horizontal Column from the top. The result is a line 0.935 inch south of the second pyramid's horizontal axis (cyan in the diag. below).
This line then meets the Diagonal Simulation #2 0.0028 inch, 0.07 millimeter from the original vertical axis of Petrie. That creates a point on the reconstructed vertical axis.
Diagram 19
The vertical axis with the Channeled Diagonal locate the G2 center a pinpoint away from Petrie's plan, at 0.0028 inch (1/358), 0.07 millimeter (1/14), or 0.00015 cubit. The reconstructed center is less than 0.0005 inch from Petrie's horizontal axis. That's better than 1 / 2000 inch., .
Diagram 20
The diagram above magnifies the top left of the preceding diagram. Although it will be about six times larger than life on most computer screens nowadays, it is still hard to see it as Petrie's plan and the reconstruction, side by side. Now, there is a tiny red circle visible around the pyramid centers. Make its radius nearly as long as the next diagram, at about 280 times lifesize, and the situation in the center becomes clear. Petrie's plan is in green color, the reconstruction is yellow.
Diagram 21
To sum up the reconstruction of G2, the center gets done with utmost accuracy, while the sides are within four millimeters of Petrie's plan, at 411.008 cubits long. On the scale of Giza, that is still bull's eye. For example, John Legon also happens to locate the same south side of G2 in his reconstruction, describing the result as highly accurate. Upon checking, the fault is 1.5 inches, almost ten times the 4 millimeters in this study (an order of magnitude). Of course, it is easy to understand Legon's, and everybody else's, perspective - what is an inch and a half in comparison to half a nautical mile, the north-to-south span of the pyramids? This is the place to humbly remember that even in this reconstruction there is one similar discrepancy - the south-west corner of G3 is 1.18 inches further west than the Petrie version, and consequently the western and northern sides of G3 are also that much out, even as the southern and eastern sides are in exact agreement with Petrie's plan. Yet, this point (the south-western corner) is pivotal to the reconstruction on several occasions. Its crucial purposes make it difficult to believe that it is not what the designers had in mind.
Conclusion:
Petrie's layout of the great pyramids of Giza can easily, yet accurately, be recreated from tabula rasa, beginning with the '13-step' construction of the regular 5-pointed star from a line segment. This method simply cannot be beat on simplicity, accuracy, and intellectual depth. Therefore, it must be essentially identical to the original Egyptian procedure of planning the Giza layout.
Jiri Mruzek
April 15, 2007
Vancouver, BC
©Jiri Mruzek
Appendix
Length of the Royal CubitThis reconstruction owes its units of measure to John Legon. He makes a 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, or 1.732. Later Legon also says that Egyptians may had targeted the more accurate root value obtainable by exact construction, if my memory serves me right.
Accordingly, I have tested the Pyramid Square side set to 1,732 cubits, as well as the accurate value of 1,732.0508.. from exact construction. Yet, it is at the value of 1,732.05 cubits for a side of the Pyramid Square, the first six digits of the square root of 3, where sensational value readings start popping up. Many measurements acquire values, which appear non-random, especially as a group. By this virtue, a good case is made for the exact length of the cubit used in planning Giza's layout.Scale as a factor
The bigger the implementation of a plan, the easier it is to achieve overall accuracy percentages to a preset unit. The Giza planners had exploited this principle 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 error had a proportionally greater role.
south-north distance = 35,713.1 inches = 1,732.05 cubits = one side of the Pyramid Square
1 cubit = 20.61897.. inches = 523.7218 millimeters
My comment is brief - it makes good sense for the builders to stop at the exact value of 1732.05 cubits (decimals would be part of the secret lore since at least La Marche). The digit 5 for centi-cubits is followed by a zero, which means no milli-cubits to deal with. As a milli-cubit is worth just over half a millimeter, the next digit involves correspondingly shorter lengths, some right at the boundaries of unaided human vision.
Note: Phi as 2.618 times 1.2 = 3.1416 Both the Phi and Pi values are impressively accurate, 3.1416 divided by 6 = 0.5236 (given by some as the digits of the royal cubit in millimeters)
Also: 1 - 0.6180339887 (Phi - 1) = 0.3819660113
0.3819660113 x 7 x 28 x 7 = 524.057 (given by some as the digits of the royal cubit in millimeters)
Drawing Inspiration From Ideas of Others
The Pyramid Square was my geometric response to having Petrie's ground plan in a CAD drawing (Computer Aided Design). This crucial step was something I did as a matter of course, because all my previous case studies in ancient science-art involved an all important square. The Giza pyramids dictate their containing rectangle, and it is often seen in various studies. They also dictate the square extrapolated from the containing rectangle.
The Pyramid Square & Khafre's Pyramid
Next, give this Pyramid Square a basic Golden Section grid (diag. below). Lines of the grid create a Golden-cross within the square. The square of the G2 (Khafre's pyramid) is then made the center square of its own Golden-cross, and the two Golden-crosses are superimposed over each other for comparison.
diagram a
In the diagram below, golden proportions added to the G2 in its real location seem to find some correlation to the south side of G1. Here as well, we encounter facsimiles of golden rectangles.
diagram b
These results called for more to be done. Not wanting to rediscover the wheel, I checked for sources on the subject of Giza layout. There is an informative article over at Jim Alison's site: http://home.hiwaay.net/~jalison/gpsp.html
It deals with work by John A.R. Legon, Chris Tedder, Robin Cook, and Jim Alison himself on various notions of a ground plan of Giza's major pyramids.
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
Jim Alison's rendition of certain ideas by Chris Tedder was holy water on my mill:
a) Perpendicular distances between the pyramid centers produce two golden rectangle facsimiles (ABCD, and DEFP).
diag.c
b) Alison authored the following key observation:
Alison's circle - The segment F-H is very close (0.8º) to holding the 45º angle from the horizontal. A circle, whose radius is the east-west distance between centers of Khufu and Menkaure pyramids, is drawn from the center of Menkaure's pyramid. It then divides F-H at G by the golden proportion:
22,616 inches / 13,954.114 inches = 1.621
The Breakthrough
Following the above directions, I added the design to my Giza plan, into the context of the Pyramid square. The result was spectacular!
Both golden rectangles create new golden rectangles with the Pyramid Square:
Tedder's Secondary Rectangle #1 An extension of Tedder's horizontal golden rectangle A-B-C-D to the west side of the Pyramid Square is itself a facsimile of a golden rectangle, the vertical rectangle C-D-O-K
C-D divided by C-K = 1.627
Tedder's Secondary Rectangle #2 - The I-point is where the Alison's ciircle intersects the extended diagonal of the second pyramid rising north due west. The distances I-J and I - L form the golden ratio.
I-J / I-L = 1.6199 less than 2/1000 off the true Φ value
The horizontal rectangle I-J-K-L is therefore an excellent facsimile of a golden rectangle. One more external idea integrates into my system, adding to its elaboration.
diag. d
So far, results indicate that the designer may have indeed targeted golden rectangles. There is a steep rise in the accuracy of the new and more complex position over the old one. The diagram below shows how the combination of the old rectangle with the newer one creates a horizontal column, which is more accurate as such (a combination of two true golden rectangles) than either of its components. Tedder's derivation is done in black lines. It is side-by-side with color lines of my reconstruction of the Horizontal Column out of two true golden rectangles, from the center of the Great Pyramid to the west side of the Pyramid Square. Visually, the two Horizontal Columns are as as one, but not the rectangles.
The vertical line from the third pyramid's center is visibly not the line that cuts the Horizontal Column into two golden rectangles. To showcase these facts, the diagram is page-wide.
diag.e
Notes
Petrie versus Cole
Re: Hidden circles
Posted by: Jiri Mruzek (IP Logged)
Date: April 24, 2009 01:13PM
MJ Thomas 2 Wrote:
--------------------------------------------------------
>
> That you have managed to create out of certain
> measurements selected from Petrie’s survey of the
> Giza Three a complex geometric pattern that makes
> some kind of sense to you is not disputed.
What bothers me? Your use of "a complex geometric pattern that makes
some kind of sense to you".
You are clearly saying that it (the pattern) does not "some kind of sense to you" make. Why should it make sense to you, as well? For one, "this some kind of something" does what it is supposed to do. It works for the given purpose, a purpose that precedes my own existence considerably, hence, this debate has none of my instigation. Second, my solution consists more of pure geometric theorems, and less of the follow-up (exact) transition into an accurate layout.
So, where along this simple path are you getting lost, MJ?
> 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.
a) The original plan would have been made with regular squares for the mantled pyramids, just like Petrie's plan.
b) If the objective were to build square pyramids, based upon carefully monitoring the progress, corrections may have been made to one of the sides to preserve the total perimeter.
>
> I don’t find this unreasonable, but then I have no experience, direct or otherwise, of laying out squares on such a vast scale and on uneven terrain.
>
> > > I am merely attempting to explain to you (by reference to the actual – as far as can be determined - dimensions and their means) that you are wrong in this.
> >
> > No, you are blindfolding yourself by denying me the acknowledgement that with respect to the average, provided by a reliable data source, the reconstruction works with great accuracy.
>
> But your ‘reconstruction’ does not take into
> consideration Petrie’s actual maximum and minimum
> measurements, and this – along with your inability
> to provide any evidence of intent - is what causes
> me to dismiss your theory as nothing more than a
> figment of your imagination.
>
>
> > The measuring for both surveyors was not done with cut and dry procedures, predetermined before the
> > actual field work. Both had to make choices, and Cole chose to rely on different data. Petrie did the survey his way, and Cole did likewise.
> > The fact that my accurate reconstruction fits Petrie, and not Cole
> > is a strong indicator that Petrie did more than a great job, he did an ideal job.
> >
> > I've read some, and in no way noticed
> > anything that would make Cole better than Petrie. To
> > the contrary, I found Cole's method more prone
> > to mismanagement.
>
> I have here a copy of Petrie’s Pyramids and
> Temples of Giza (1883) and a copy of Cole’s
> Determination of the Exact Size and
> Orientation of the Great Pyramid of Gîza
> (1925)
> IMO, what you are claiming is nonsensical and
> grossly misleading.
> It shows, again IMO, that you either have not read
> or have read but have misunderstood Petrie’s
> detailed account of how he surveyed the base of
> Khufu’s pyramid, and that you have not read Cole’s
> survey report in full.
I have stated some reasons why I think Petrie's results are more accurate than Cole's, but you have stated no reasons why it should be the other way. Can you do that?
> > >
> > > there is a difference of about 8 feet;
> > > it may well prove to be a case of this pyramid being built
> > > in a hurry and concern with accuracy to the degree seen
> > > in Khufu’s pyramid no longer existing.
> > >
> >
> > Eight feet? Ninety-six inches? What about these measures by Petrie?
> > G3
> > inches = cubits
> > West 4153.9 = 201.462
> > South 4157.8 = 201.649
> > East 4149.2 = 201.232
> > North 4153.6 = 201.445 !
> > Average 4153.6 = 201.447 !
> >
> > Is it not strange that north represents the average to a millimeter?
> > It could be a controlled event by the builders.
> > Anyhow, the average is what it is in my CAD
> > drawing. I do not depart from the data.
> >
> > The difference between east and west is 5 inches
> > the difference between south & north is 4 inches
> > There isn't any room here for eight feet.
>
> Once again you show your ignorance of the subject
> matter in general and your propensity for
> cherry-picking data.
> 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. 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.
Jiri
Re: Hidden circles
Posted by: Jiri Mruzek (IP Logged)
Date: April 24, 2009 02:17PM
Jiri Mruzek Wrote:
--------------------------------------------------------
> MJ Thomas 2 Wrote:
> --------------------------------------------------
> > But your ‘reconstruction’ does not take into
> > consideration Petrie’s actual maximum and minimum
> > measurements, and this – along with your inability
> > to provide any evidence of intent - is what causes
> > me to dismiss your theory as nothing more than a
> > figment of your imagination.
> >
Re G3, perhaps, we have some evidence of intent in that the north side is very accurate in representing the average, allowing the extrapolation that upholding the average, and therefore preserving the planned perimeter, had top priority. I see nothing strange in this approach, which allows for the correctness of my theory.
The divergences may also be intentional, to add another layer of meaning over the averaged out version. That is over my head though, definitely for the foreseeable future.
Jiri
Re: Hidden circles
Posted by: MJ Thomas 2 (IP Logged)
Date: April 25, 2009 12:12AM
Jiri Mruzek Wrote:
--------------------------------------------------------
> > 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. It pits him alone against two widely acknowledged
> professionals.
Oh dear, yet again you show your general ignorance of the subject-matter (e.g. Cole's report is about Khufu's pyramid not Menkaure's), and continue to stubbornly refuse to take into proper consideration any evidence that questions your theory (e.g. you continue to ignore the possible change to the initial planned size of the base of Khafre's pyramid).
You wrote, 'I have stated some reasons why I think Petrie's results are more accurate than Cole's, but you have stated no reasons why it should be the other way. Can you do that?'
I have referred you to both Petrie's and Cole's detailed accounts of their respective surveys of the base of Khufu's pyramid (both reports are freely available in full on the Internet*), read them, Jiri, and you will have your answer - and a lot more.
MJ
*
Cole: [www.kheraha.co.uk]
Petrie: [www.ronaldbirdsall.com]
Re: Hidden circles
Posted by: Jiri Mruzek (IP Logged)
Date: April 25, 2009 05:46AM
MJ Thomas 2 Wrote:
--------------------------------------------------------
> Oh dear, yet again you show your general ignorance
> of the subject-matter (e.g. Cole's report is about
> Khufu's pyramid not Menkaure's), and continue to
> stubbornly refuse to take into proper
> consideration any evidence that questions your
> theory (e.g. you continue to ignore the possible
> change to the initial planned size of the base of
> Khafre's pyramid).
>
a)
I know Cole didn't bother doing all three pyramids, but the fact that he came close to Petrie lends support to Petrie's credibility, not Lehner.
b) Menkaure is what it is, the final product counts the most.
> You wrote, 'I have stated some reasons why I think
> Petrie's results are more accurate than Cole's,
> but you have stated no reasons why it should be
> the other way. Can you do that?'
>
> I have referred you to both Petrie's and Cole's
> detailed accounts of their respective surveys of
> the base of Khufu's pyramid (both reports are
> freely available in full on the Internet*), read
> them, Jiri, and you will have your answer - and a
> lot more.
Do you really think I haven't read these reports? Well, I have, I must have, otherwise I would not give you my opinion on them, right? Keep presuming, buddy!
Stop evasive arguments, and state your reasons why Cole should be more accurate than Petrie, and I will point out why it is likely that Petrie did the better job.
And stop unfounded accusations of ignorance against me! After a while, it does comes across as rude, because you had your opportunities to establish my familiarity with the issues.
Jiri
Re: Hidden circles
Posted by: MJ Thomas 2 (IP Logged)
Date: April 25, 2009 03:17PM
Jiri Mruzek Wrote:
--------------------------------------------------------
> MJ Thomas 2 Wrote:
> --------------------------------------------------
> -----
>
> > Oh dear, yet again you show your general ignorance
> > of the subject-matter (e.g. Cole's report is about
> > Khufu's pyramid not Menkaure's), and continue to
> > stubbornly refuse to take into proper
> > consideration any evidence that questions your
> > theory (e.g. you continue to ignore the possible
> > change to the initial planned size of the base of
> > Khafre's pyramid).
> >
> a) I know Cole didn't bother doing all three
> pyramids, but the fact that he came close to
> Petrie lends support to Petrie's credibility, not
> Lehner.
> b) Menkaure is what it is, the final product
> counts the most.
>
> > You wrote, 'I have stated some reasons why I think
> > Petrie's results are more accurate than Cole's,
> > but you have stated no reasons why it should be
> > the other way. Can you do that?'
> >
> > I have referred you to both Petrie's and Cole's
> > detailed accounts of their respective surveys of
> > the base of Khufu's pyramid (both reports are
> > freely available in full on the Internet*), read
> > them, Jiri, and you will have your answer - and a
> > lot more.
>
> Do you really think I haven't read these reports?
> Well, I have, I must have, otherwise I would not
> give you my opinion on them, right? Keep
> presuming, buddy!
To be perfectly honest, the impression I get from you through your posts to this thread is that you have only read a summary of Petrie's measurements, and not read Cole's report.
> Stop evasive arguments, and state your reasons why
> Cole should be more accurate than Petrie, and I
> will point out why it is likely that Petrie did
> the better job.
As not doing so would get us nowhere I bend the rules to suit you, and post the following extract 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, whereas those of Prof. Petrie are of 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.”
Perhaps you now care to explain how and why, in your opinion, Cole's survey is inaccurate - that Cole's findings do not fit your theory is not a valid explanation.
> And stop unfounded accusations of ignorance against me!
> After a while, it does comes across as rude, because
> you had your opportunities to establish my familiarity with the issues.
The only knowledge of the Giza necropolis you have so far displayed is a summary of Petrie's 1880-2 survey of the site, which suggests to me that you are ignorant of a great deal about this place and, more to the point, the culture that created it.
MJ
Re: Hidden circles
Posted by: Jiri Mruzek (IP Logged)
Date: April 26, 2009 08:23AM
MJ Thomas 2 Wrote:
--------------------------------------------------------
> Jiri Mruzek Wrote:
> --------------------------------------------------
> -----
> > MJ Thomas 2 Wrote:
> > 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 about 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 for an incredibly accurate regeneration from a clean slate using the noble Section, can by no means be discounted. With this reconstruction of the Giza-layout everything clicks into place. A click in the reconstruction - a booming cannon shot across the bow of Egyptology! The units used, the Pi, and the intelligence behind it all, everything works to perfection. This success is anomalous, unprecedented, and cannot be discounted. Anyone working on the same issues should have knowledge of the fact. Then there is that La Marche - Nazca - Giza connection! Should the fantastic fact be discounted, and not included in information available to other researchers?
A conclusion I make here is that your reaction is more concerned with damage control and fully discounting this successful reconstruction than with dispelling my 'ignorance'. Funny, how the very deed, the very endeavor, and the ensuing success of it (the reconstruction) meant to you that I am ignorant of Egypt's ancient culture. Quite a shortening of the circuits, wouldn't you say?
Jiri Mruzek
p.s. - I have to do some traveling, so a couple more posts, and I will have to take a few days off, maybe a bit longer.
Sorry, you do not have permission to read this forum. (alternative geometry & Numerology)
Little did I know that the above was my last post. The message in red ink above showed just minutes later. I could not even view the post, and I could not view anything to do with this forum. Someone had thrown the switch, without a note from the moderator. I can read and probably post in other forums, however. Hence the objective must be to shut me up on the above subject, I don't see other explanations. No rude behaviour on my part had caused the penalty, after all. But, wait, I used the word "BULL" That teaches me to talk like that :)
Re: Hidden circles
Posted by: Warwick L Nixon (IP Logged)
Date: April 26, 2009 08:54AM
"Funny, how the very deed, the very endeavor, and the ensuing success of it (the reconstruction) meant to you that I am ignorant of Egypt's ancient culture."
I honestly do not think anyone here is accusing you of not being knowledgable of AE culture.
All I and others have been asking is how you interface your findings with that knowledge
a technical note re the casing..a simple planing technique when adzing the freshly quarried, thus very workable ,tufa casing stones, would yield an amazingly flat surface as a matter of course(pun not intended).
Warwick
If I'm eating cheese do I need to see the cow?
Re: Hidden circles
Posted by: MJ Thomas 2 (IP Logged)
Date: April 26, 2009 12:32PM
Warwick L Nixon Wrote:
--------------------------------------------------------
> I honestly do not think anyone here is accusing
> you of not being knowledgable of AE culture.
I am, and very much so.
MJ
Edited 1 times. Last edit at 04/26/09 12:33PM by MJ Thomas 2.
Re: Hidden circles
Posted by: MJ Thomas 2 (IP Logged)
Date: April 26, 2009 12:30PM
Hello Jiri,
Thank you for explaining why you believe Petrie's survey of the base of Khufu's pyramid is more accurate (and therefore more acceptable to you) than Cole's.
I don't have a theory about the layout of the Giza Three, so I happily settle for the fact that AFAIK most Egyptologists and many Alternative researchers go with Cole's survey.
Interestingly, there are a number of Alternative theories about the layout of these pyramids that are based on Cole's results - yet more cases of cherry-picking data that fits the theory?.
With advances in surveying techniques (satellites and lasers, and other high-tech paraphernalia that passeth my understanding) will perhaps one day come the definitive survey of Giza, but until then I'll stick with Cole for the actual dimensions and orientation of the base of Khufu's pyramid (but with Petrie always in mind).
-------------------------End of discussion ------------------------- -------------------------------------------------------------------------------------------------------------------
Integration of ideas by Legon, Alison, Cook and Tedder into the Pyramid Square published 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 in order to check out various theories. The subject seemed heavily explored by numerous scholars, hence I had no expectations of finding anything new. But squaring (extending) the rectangle often used by experimenters to enclose the three pyramids into the Pyramid Square, gave me a so far untested approach to Giza. 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..
Another special effect is shown below. Line-g runs between the intersection of the two circles to the south-east corner of the Great Pyramid duplicates the true golden diagonal to 0.0015º, a very fine value. The true golden diagonal 'g' drawn down from G1's SE corner comes to within 0.66'' of the intersection of the two circles.
diagram f
Petrie's value of 9,068.8 inches, or 230.348 meters, or 439.82782340 cubits of the actual average side by the standard of this study, is about 2.5 mm off the desired perfect value for Pi. An error of less than 3mm was allowed for by Petrie. Hence he could be off by that much, and the pyramid may have been built perfect. If we let the facts prevail, the conclusion must be that the average pyramid side was designed with the true value of Pi in mind.
John Legon writes:
"In terms of the Giza royal cubit of 0.52375 metres, the actual mean side of 230.364 metres corresponds to 439.8 cubits, with an average variation in the sides of only 6 cm or 0.1 cubit. Petrie suggested that an adjustment may have been effected in order that the perimeter of the base should express the so-called 'pi-proportion' in relation to the height of 280 cubits, with greater accuracy than the value for pi of 22/7. In this case, the theoretically exact mean side-length would be 439.822... cubits. It seems that the builders achieved this result while retaining the round number of 440 cubits in the south side."
I agree with Legon's assessment. The same measurement has both its exact, and rounded out values, if that is what he meant. The need for a side of 440 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...) Legon abstracts a cohesive system from the Giza position dealing with square roots. But, while Giza only approximates Legon's system, it is still deliberately so, in my opinion. It is just another layer of information.
Another long-time researcher, Robin Cook adopts the right approach in observing all strong relations as possible coincidences first, and then asking, which of the mutually exclusive relations might be the intended ones. Cook is right, because without the illuminating background of the 'Pyramid Square', such ideas are a bit like Plato's shadows dancing on a cave wall. In such situation 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.
Petrie is said to have had some difficulties with 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 Egyptians had started out with regular squares. Adjustments were then made to create a new medium to encode more data. Another possibility is that the corner was deliberately misplaced at the final step to make the plan less obvious, and to accomodate something else as well, perhaps to do with equinoctial precession.
Acceptance of my solution as essentially the original Egyptian 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 fine differentiation between drawing objects, these key differences are tiny enough to be invisible on any drawing board. Since the plan cannot be worked out by drafting methods, it had to be worked out by calculation. Thus, the knowledge of mathematics guarded by the temples had to be on a level unreachable to a neolithic society less than two millenia removed from the hunter-gatherer stage.
In this case, speculation about advanced prehistoric science that had somehow survived to an unknown degree until dynastic Egypt under secret guardianship of the temples, simply cannot be avoided. Consistent coincidences (an oxymoron) are typical for the Great Pyramid, and Giza. This reconstruction satisfies all the criteria, which differentiate it from random. 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 reproduced by an exact, and sophisticated method.
A skeptic's stand - from a discussion on Randi's (an internet-fortress of skepticism).
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.
Someone nicked Wollery voids the value of circumstantial evidence. According to Wollery, the truth would never prevail no matter how much circumstantial evidence supported it, if his unreasonable demands for proof were not met. I would call this approach a premeditated murder of truth - an intellectual crime. Is it not true that considerable amount of circumstantial evidence can cast heavy doubts over any conclusions, which would disregard it? Is it not true that people are sent to jail, even to the electric chair, on such evidence? In that vein, statements to the effect that the Giza pyramids were not part of a unified plan look totally false next to this analysis.
In fact, if Petrie's measurements are correct, considerable Golden Section clock-work is inherent in the Giza architectural arrangement. By Ockham's razor, then it is also how it was originally designed. In combination with other data on design characteristics we have on Egyptian architecture, and Giza in particular, one must take for granted that certain Egyptians had extensive knowledge of the Golden Section, and had utilized it in sacred architecture.
There was an interesting change in my sentiments, as the study progressed. I marvel more at how close Petrie's plan comes to the reconstruction than the other way around, as I used to do. Why? Well, it seems most reasonable to presume that anyone able to design a plan as sophisticated as the Giza layout, would also be capable of implementing it with great accuracy.
Some low hanging fruit in this position: Draw a circle from the center of the interim G3. Pick the distance to the south-west corner of the Horizontal Column as the radius. Then this circle overlaps the G2's diagonal due SE (diagram below) by less than an inch. This gives another simulation of the given diagonal. By the way, the same circle idea implemented on the original position misses the same diagonal by more than half a cubit, quite a contrast.
Precise Values
Distances given in cubits unless otherwise specified
One side of the Great Pyramid - interim version:
439.50009258.. cubits less than 1/10,000 cubit from an exact half-cubit
311.005 half the diagonal of the reconstructed Great Pyramid
411.008 side of G2
314.502574
- 201.502644
= 112.99993
516.005
1.0001
1787.50088661 = Distance between centers of reconstructed G1 and G3
25,920 / 1787.50088661 = 14.500...
1 inch = 25.4 mms
If Giza is to involve equinoctial precession designs, the line of centers between G1 and G3 should have a role, perhaps, as a radius or diameter of the celestial mill. The exact distance between centers of reconstructed G1 and G3 = 1787.50088661 cubits]
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. Please, accept my apologies. Use CAD to verify my results, not the gifs..
