-
Exact Reconstruction of the Layout of the Great Giza Pyramids
| Introduction
First, let me caution the reader about developing prejudice against this work on account of my 'literary' English. Upon landing in Canada as a young adult, mustering a few sentences in broken English was all I could do. So, bear in mind that if I violate English, it is all in the cause of Science :)
Since 1883, the year Sir William Flinders Petrie published his geodetic survey of the three great Giza Pyramids, it has been analyzed for geometric, and numeric significance. To show that the three pyramids are parts of one overall plan, there have been attempts to generate the layout of the three great pyramids from pure ideas. Giza abounds with inspiring relationships. But, to safely pin ideas on the designers, those have to be tested in redesigning of the plan, and found to be compellingly accurate, if indeed the builders had been capable of such accuracy. Since there is a number of examples showcasing the knack of Giza builders for accuracy, such as that found in the Grand Gallery, or in the total north-to-south distance between the pyramids, our standard should be as demanding. Here, it is time to mention John Legon .
Legon's reconstruction works with such accuracy along the east-west axis for lines passing through the east and west sides of the Second Pyramid, and the west side of the Third Pyramid. However, the north-to-south division produces a much bigger fault of 1.25 inches. Although Legon deems it completely accurate, why did the Egyptians not achieve comparable accuracy along this axis? This should be the warning sign that Legon's theory is too simple, because it uncovers a rather small, even if highly important, part of the overall solution.
Be it as it may, it seems clear to me that Legon is correct, when stating that axial spacing between the pyramids approximates the square roots of the first five prime numbers, 1, 2, 3,5, 7, given in 250 x, and 1000 x multiples of cubits. In fact, his reconstruction comes so close to Petrie's plan that had I been familiar with it earlier, I might have accepted it, albeit reluctantly. That would be a disservice to the ancient builders, because Legon's interpretation puts them on a level everybody expects, i.e., they had a simple plan, and executed it somewhat unevenly. For this reason, I find it hard to understand, why Legon's theory does not get more attention and, or approval from the academy.
Yet, I discovered a method of producing the plan of Giza by exact geometric construction, which complies perfectly to Petrie's measurements. This geometry bases on the secrets of Golden Section (Sacred Geometry), so unlike the mundane square roots, it is of the most appropriate type for sacred grounds like Giza. How glad will the skeptically leaning academy be to hear of it? I suspect, not very. In reaction, it will probably resort to embracing Legon's theory, as the simpler explanation.
Legon's theory - Competitor or Contributor?
Since my own theory draws some important benefits from Legon's work, he has my gratitude, and admiration, but it is necessary to subject his work to a critical assessment, a brief peer review of sorts. First of all, our respective approaches differ greatly. While my reconstruction is done completely from scratch, starting from just a line segment, Legon copies the ground dimensions of the Great Pyramid (G1) directly from Petrie. Then he derives the rest. He starts out with basic geometry, but from there it is all a march of numbers.
My reconstruction is essentially visual, it is a progression of pictures, each producing special effects, which imply further experiments. It is an action-packed movie.
It has long accepted Legon's idea that the axial north-to-south distance between the pyramids expresses the square root of 3. But, it takes this distance as two decimal points of √3 more accurate. To Legon, it was intended to be 1732 cubits exactly. I had to test the notion that it was intended to be 1732.05 cubits. Setting units this way had immediately produced spectacular results.
Reconstructing Legon's reconstruction - Legon does not give the actual procedure. Rather he states several possibilities, which all are supposed to lead to more or less the same accurate result. Consequently, I select those that really yield the best result. Because there are two possible scales of reconstruction (N-S = 1732 , and 1732.05 cubits), I had to carry out a reconstruction for each.
Also, Legon gives the impression that he measures from the center of the Great Pyramid's theoretical version with an even number of cubits per side (440), while everything actually works from the center of the version, which is as given by Petrie.
.
Differences in Legon's plan from Petrie's plan - When N-S (35,713.1 inches) = 1732 cubits
Second pyramid Cubits Inches Millimeters
South side 0.0606 1.25 31.76
East side 0.0012 0.025 0.63
North side 0.0482 0.995 25.26
West side 0.0136 0.280 7.13
Center 0.0549 1.132 28.77
Third pyramid Cubits Inches Millimeters
South side 0 0 0
North side 0.0603 1.24 31.6
West side 0.0082 0.169 4.3
East side 0.0521 1.075 27.29
Center 0.0373 0.682 19.54
Procedure: Start by drawing the square base of the Great Pyramid, as given by Petrie. Next, extend the east side to 1732 cubits (35,713.1 inches). This will be the axial north-south distance from the north side of the Great Pyramid to the south side of the Third Pyramid.
Mark 1101 cubits from the north southwards on the extended east side of G1 (the Great Pyramid). This will be the axial distance to the south side of the Second Pyramid.
Mark 433 cubits ( 250Ö3 = 433.0127..) from the center of the Great Pyramid westwards. This is the axial distance to the east side of the Second Pyramid. Since the positions of two sides, south and east are already known, and because Legon postulates exactly 411 cubits as the intended length per side, the square base of the Second Pyramid can be now be drawn in full.
Next, if we make the axial distance between the western sides of the Second and Third Pyramids 250√2 given to two decimals (353.55 cubits), it is just 0.008 cubit short of the western side of G3. This is a very accurate result. Since the positions of two sides, south and west are already known, and because Legon postulates exactly 201.5 cubits, as the intended length per side, the square base of the Third Pyramid can be now be drawn.
Differences in Legon's plan from Petrie's plan - When N-S (35,713.1 inches) = 1732.05 cubits
This scale goes to two more digits in the fractional part of Ö3 than the one Legon uses. The same procedures used previously on 'his' scale to gain the division along the east-west axis, achieve great accuracy here as well, but with one important difference - instead of rounded out values for the square roots, they work with the actual exact root values!
On the whole, the same procedures work even better in my reconstruction. The procedure for the eastern side of the Second Pyramid gives an ever so slightly better result to Legon's reconstruction (by 0.00025 cubit), but the one for the western side of the Third Pyramid works much better for my reconstruction. However, both procedures on this more accurate scale work so well that I adopted them as instruments for the final adjustment.
Legon's division along the east-west axis is accurate to half a millimeter for the east-side of G2, and half a centimeter for the west-side of G3. For the division along the north-south axis, Legon uses two procedures, which he deems both accurate. Each procedure works better on one scale and worse on the other, but neither works nearly as well as the procedures along the east-west axis. This unfurls cautionary flags that perhaps the builders didn't use these particular procedures at all..
Second pyramid Cubits Inches Millimeters
South side 0.0724 1.49 37.96
East side 0.0010 0.020 0.53
North side 0.0482 0.994 25.26
West side 0.0252 0.521 13.23
Center 0.0618 1.274 32.35
Third pyramid Cubits Inches Millimeters
South side 0 0 0
North side 0.0545 1.12 28.6
West side 0.0102 0.212 5.4
East side 0.0648 1.336 33.9
Center 0.0464 0.956 24.29
Differences in my plan from Petrie's plan - When N-S (35,713.1 inches) = 1732.05 cubits
The Great Pyramid
Cubits Inches Millimeters
South side 0.0005 0.010 0.26
East side 0 0 0 forced
North side 0 0 0 forced
West side 0.0005 0.010 0.26
Center 0.0003 0.007 0.19
Initial Second pyramid
Cubits Inches Millimeters
South side 0.0081 0.167 4.24
East side 0.0085 0.176 4.47
North side 0.0091 0.187 4.76
West side 0.0087 0.179 4.53
Center 0.0005 0.010 0.26
vertical axis 0.00007 0.001 0.04
The reconstructed vertical axis is extremely close to Petrie's original
Second Pyramid after final adjustment
Cubits Inches Millimeters
South side 0.00168 0.035 0.88
East side 0.00126 0.026 0.66
North side 0.00070 0.014 0.37
West side 0.00112 0.023 0.59
Center and axes remain, as before.
Third Pyramid before final adjustment
Cubits Inches Millimeters
South side 0 0 0
North side 0.0573 1.18 29.99
West side 0.0570 1.175 29.84
East side 0.0003 0.006 0.15
Center 0.0403 0.83 21.10
Third Pyramid after final adjustment
Cubits Inches Millimeters
South side 0 0 0
North side 0.0004 0.009 0.23
West side 0.0007 0.015 0.38
East side 0.0003 0.006 0.15
Center 0.0006 0.011 0.29
The tables given above show that aside from several accurate results, Legon's reconstruction does not reflect Petrie's data with consistent accuracy. Is this due to the builders' inability to stick to the plan, or was the plan different? In general, the academy scoffs at the very idea of an original plan. With so much smoke around, this is the burning question:
Were the three Giza pyramids laid out as parts of one grand plan?
A diehard skeptic will try to nip the idea in the bud, and say that even if Giza could be cleanly translated into a meaningful geometric design, which is what I have done, it would mean nothing without the discovery of original blueprints, or a statement from the architect. So much for extreme prejudice..
A true solution stands out among others because of its accuracy, and meaningfulness. There can be only one such solution, and that's why all other solutions must be imperfect.
If the layout is truly random, there will be no grand unifying idea, and no efficient solution .. There is no way to efficiently describe a random position of three squares on a scale such as that of Giza, as accurately as this superfine reconstruction of Giza's layout. In all cases, the Great Pyramid in my plan differs from Petrie by 1/100" (inch), or less. Faults on the Second Pyramid range from 0.001" to 0.035". On the Third Pyramid the range is from 0.009" to 0.015".
How miniscule are these faults? The maximum north-south distance between the pyramids is 35,713.1 inches. To show a fault of 1/100" in a plan of Giza on a computer screen we need to make it at least a pixel. The total resolution has to be 3,5 x 3.5 million pixels in order to show the fault. Imagine a 36" square as a computer screen containing all those pixels. Its lines would sport about 100,000 pixels per inch. The fault of 0.01" would show on this screen as 1/100,000". Backed by these comparisons, it can safely be said that the solution is microscopically precise. By this token, it is the original method, the scientific soul of Giza.
A tribute is due to Petrie's brilliant achievement. Although academic opinions tend to believe that measurements of the Great Pyramid by Cole supersede those by Petrie, I am confident in the belief that Petrie did a better job than Cole, by the virtue of choosing the better method under circumstances. That would explain why there is massive precise agreement between my reconstruction and Petrie's plan, and none with Cole's.
Framing - the Pyramids
The secret to successful , and quick reconstruction of Petrie's plan from scratch (tabula rasa) was in the synthesis of existing observations on the subject into the framework of an original concept. The solution was born the minute I had extended the enclosing rectangle (blue) of the three pyramids, so ubiquitous in other studies, into a square.
Simple, yes, but this background enhances the meaning of certain geometric elements.
Divide the square by golden section (green lines). The second pyramid's vertical axis is a lot closer to the axis of the square than its diagonal to the big square's diagonal (the Great Pyramid's diagonal). Just as obviously, the square base of the second pyramid obviously resembles the small square in the center of the green cross.
Do an experiment: Place the second pyramid's square (base) in the center of the cross, and extrapolate its own golden square, and a cross from it, as in the diagram below. Now, there are two sets of everything (the Great Pyramid is also scaled).
The result is an almost perfect ilusion at this magnification. The program translating the exact CAD drawing into a raster drawing got fooled, and shows most double lines as if they were single. Could this illusion be symbolic language, implying that the design of Giza has something to do with the Golden Section?The Pyramid Square concept proves productive. It is unusual that as far as I know, no researchers had worked with it. After all, extrapolating a square from a rectangle is about the simplest geometric operation that can be done with it. It was the first thing I had done once I had Petrie's ground plan in a CAD drawing (Computer Aided Design). To me this step was elementary, because all my previous case studies in ancient science-art involved an all important square. The Giza pyramids dictate their containing rectangle. The rectangle dictates this square. The square constitutes the proper context for analysis. To illustrate what I mean:
Chris Tedder sees two golden rectangles given by intersections of lines in cardinal directions emanating from the center of each pyramid. He proceeds to propose a possible plan, the Egyptians might have used. Unfortunately, these rectangles have poor accuracy, a fact that would reflect negatively on the level of Egyptian skills.
Tedder's site: http://www.kolumbus.fi/lea.tedder/OKAD/Gizaplan.htm
Undeservedly so! See what the top rectangle does in the context of our Pyramid Square. The effect of combining the previous image with the Pyramid Square is that we get another rectangle, ABOK. This rectangle is absolutely noteworthy, because it is indistinguishable (here) from another rectangle based on an exact idea! The distance from the center of the Great Pyramid to the western side of the Pyramid Square is now the length of a rectangle composed of two exact golden rectangles ( ABCD, and CDOK).
Moreover, the golden diagonal emanating from O is indistinguishable from a line made from O as a tangent to the circle incribed into the Great Pyramid's square base. Armed with these facts, we can experimentally reconstruct the Great Pyramid's square base in relation to the Pyramid Square. The pyramid's side will then be shorter than it is in reality (Petrie's measurements) by on this scale invisible seven inches. If we suppose that the Egyptians evolved the Giza plan in this way, their skills suddenly look vastly superior to the level, at which Chris Tedder perceives them. Anyhow, this procedure forms the backbone of the geometry evolving my replica of Petrie's plan.
Mirrored Illusions Become Reality
The layout of Giza is extremely rife in illusion creating coincidences. The fact makes decryption tricky (it even seems to lead to hallucinatory states of mind in some authors:), and would have been one of the major reasons for its selection. In a spiritual sense, false paths obscuring Giza's recreation (I blundered down a few of those) may well symbolize the soul's perilous journey through the labyrinth of life to fulfilment.
Testing illusions geometrically may at first be somewhat disappointing, but, even as old illusions are dispelled, new ones appear. This may sound cryptic, but reality then is the axis of symmetry between illusions.. The reader will see this for his, or her self.Length of the Royal Cubit
This reconstruction more or less accepts units of measure theorized by 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.
South to North between the pyramids = 35,713.1 inches = 1,732.05 cubits = a side of the Pyramid Square
Accordingly, I have tested the Pyramid Square side set to 1,732 cubits, as well as the 1,732.0508.. from exact construction, and thirdly, I have tested it at the value of 1,732.05 cubits, which is extremely accurate, as it goes to six digits of the square root of 3. Here, some sensational value readings pop up in the reconstruction. A whole group od measurements looks definitely non-random. By this virtue, a good case is made for the exact length of the cubit used in planning Giza's layout. It is 1/1000 " shorter than the cubit of 20.62 " , given by Petrie.
1 cubit = 20.61897.. inches = 523.7218 millimeters
It makes good sense for the designer to stop at the exact value of 1732.05 cubits (decimals exist since at least the time of La Marche, Stone-Age France). The digit 5 for hundredths is followed by a zero, which means no thousandths (milli-cubits) to deal with (1/1000 cubit is just over half-a-millimeter). The next digit already deals in tenths of millicubits, so, this is a natural cut-off point for the north-to-south dimensioning of Giza. As a representation of the square root of 3, this value differs from the true by eight ten-millionths of a unit - 1.73205080..
Petrie's Royal Cubit
Petrie's measurements at Giza, and inside the Great Pyramid had produced many differing cubits. In the end, he settled for an average of 20.62" . Our Giza cubit (20.61897) rounds to 20.619". Amazingly, this is a diference of only 0.001" .
Measuring Success of the Reconstruction
The first criterium is how close the reconstruction gets to Petrie's position. The reader saw that the faithfulness of the reconstruction to Petrie's plan is in a class of its own, and nothing else comes close.
Precise Values
Another aspect of the reconstruction, which puts it into a class of its own is the undeniable extreme proximity of many of the resulting measurements to whole or half cubits.
.
Distances given in cubits
439.50009.. a side of the interim G1 - less than 1/10,000 cubit from an exact half-cubit
622.009.. a diagonal of the reconstructed G1
411.007 a side of the initial G2
201.50275 a side of the initial G3 - less than 3/1000 from being a perfect half cubit
516.0055 from the reconstructed SE corner of G3 to the SW corner of the Pyramid Square
1787.5005 distance between the centers of reconstructed G1 and G3 (before final adjustment)
1642.002222.. line 'd' ( diagram 5)
314.50275 from the reconstructed SW corner of G3 to the SW corner of the Pyramid Square
-201.50246 from the reconstructed SW corner of G3 to the SE corner of Petrie's G3
= 113.00029
39.00000319 from the adjusted SW corner of G3 to the SW corner of the Pyramid Square
2.001.. difference between the radii of the e-circle, and its lookalike (diagram 7)
1150.626180 distance between adjacent tips of the 13-step-star, or one side of the smaller star (diag. 6), five consecutive digits of Phi squared
142.44316181 half the diagonal of the final version of G3, first four digits of Phi, in the fraction.
439.82732 side of the reconstructed G1, which yields a very good value for Pi ( 3.1416..) with half-height of the pyramid (140 cubits) 439.8273 / 140 = 3.1416..
How to Recreate Petrie's Giza Scheme from a Line Segment
The Giza plan evolves from a solid theoretical foundation of the Golden Section. It showcases knowledge of the simplest (fastest) construction of the regular 5-pointed star. The strange thing is that this construction is not to be found in any books on geometry. Believe it or not, it is my favorite prehistoric construction, for I had learned it from the geometry of Nazca Monkey.
Start with the below classic construction. It begins with a horizontal line, and takes ten steps. Two of the steps involve help circles (to draw the axial cross), which are not shown. The eighth step gets the key Golden-circle ('c' in the diag.), which is centered in the bottom tip of the axial cross. On steps nine and ten, lines are drawn from the top of the axial cross, as tangents to this circle. These lines create an angle of 36 degrees exactly (like on a 5-pointed star).
Diagram
1
Diagram below:
Three more steps finish the construction of a regular 5-pointed star (pentagram): On step eleven, a circle from the point Q (the Q-circle) is drawn through the top and bottom of the axial cross. This creates two points (marked by small circles), which are then connected by lines to the bottom tip of the axial cross (the bottom corner of the diamond square). This is the fastest such construction in geometry, I believe. It takes only thirteen moves from start to finish (its simplicity is 13).
The unique element of this construction is the Q-circle (or its mirror image from the other arm of the original 36º angle) since other constructions generally produce differing stars. After the Q-circle, there is a choice of things to do. By the way, take a good note of 'P' in the diagram, it is strategically positioned for a key role in this reconstruction later on.
Diagram 2
The position below is based on the above diagrams, but is reoriented to Giza. There is a lot of wondrous goings on in it. .
Diagram 3
a)
The points A-B-F-G-H mark four segments in a row, where each segment forms the Φ-ratio with the neighboring segment.
b)
The south-pole of circle-3 (circles 2 and 3 are golden circles) gives the south side of the Third 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.
The larger golden rectangle of the Horizontal Column was identified 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 is a vertical Golden Rectangle. This was a crucial piece of intelligence.
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.
f)
Not shown in the diagram, a line through the intersections between circles 2 and 3 has the angle of a diagonal in a vertical golden rectangle.
. The "13-step" construction projects the initial Great Pyramid several times. The two Q-circles together set a square, which is also the square basis of the Great Pyramid on the opposite side.(diag. below). The line of centers between the two Q-circles (points 4, and 5 - which is also a pentagon's side in diag.2) is equal to one side of the initial Great Pyramid.
Diagram 4
The Four Corners of the Great Pyramid - stage 1
Diagram 5
The diagram above shows at once four ways to project the Great Pyramid onto the "13-step" construction.
a) Two of these are lines 'g'and 'h', which are given by points of intersection of the two Q-circles with the sides of the diamond-square. When extended these lines intersect the diagonal lines of the pyramid at its corners (the lines are known, because the NE corner of the pyramid is at the NE corner of the golden rectangle).
b) The lines 'd', 'e', and 'f' are golden diagonals. Line 'e' is tangential to the inscribed circle of the pyramid, and that also allows the reconstruction.
Line 'd' will form the radius of the key 'e-circle'.
Line 'e' reads 1642.00222202 cubits, a typical measurement in cubits for this reconstruction, approaching even cubits.
Each of the several above-mentioned procedures leads to the same initial pyramid. The design selection hints at the designers' familiarity with the entire spectrum of possibilities.
In fact, if the north-east corner of the interim Great Pyramid is exact in this blueprint, then the other corners are over 6 inches short of Petrie's locations, but that distance shrinks to virtually nothing on any drawing board. The initial pyramid sets the stage for exact reconstruction of G1.
The Pyramid Square
After the initial pyramid (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 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. Repeated Internet searches for instructions on how to construct a pentagram, yield none as fast. The fastest I came across were 15-step operations, but then in February of 2010, I got a letter from Mike Ivsin about his original construction of the regular pentagon. Applying Ivsin's ingenious original idea to pentagram construction, I found that it is a 14-step operation. In the below diagram, the first five steps are in black color, the following five are blue, and the rest are red. Two of the circles appear as Vesica Pisces in order to clean-up the image.
Magic happens, when Ivsin goes on to add just one circle ('c') to the basic construction, whose radius equals the length 'a' of the star's arm (diagram below). Now, extend line 'b' so it meets 'c', and draw the line 'd'. What we have here is the cross-section of the Great Pyramid!
The angle between lines 'e' and 'd' is 51.82729237 degrees, which rounds out to 51º50'.
A quote from Petrie: On the whole, we probably cannot do better than take 51º 52' ± 2' as the nearest approximation to the mean angle of the Pyramid..
* Hence, 51º50' complies to the lower ± limit set by Petrie.
* The ratio of 'd' to 'e/2' is the exact value of φ (Phi). This is better than the usual Phi formula for the Great Pyramid: 356 / 220 = 1.6181818
Not only the Giza layout, but also the Great Pyramid's cross-section seem derived from Golden Section designs, namely, the construction of the regular pentagram needing the least number of steps, 13, and possibly, the only other such construction needing less than 15 steps. In any case, the nature of each construction is consistent with the other, and thus augments its credibility.
Giza broadens the Nazca - La Marche connection (between the Nazca-monkey, and one Stone-Age engraving from the rock shelter of La Marche, near Lussac-les-Châteaux, France, with 1,500 masterfully engraved stone tablets, now kept in museum vaults, and largely unknown to the public). Why do these three ancient works from different ages and continents encode the same construction? Is there some connection to the apparent imagery of the so called Abydos Helicopter, which encodes yet another rigorous construction of Golden Ratio?
The North-South division - Locating the south side of G2
Diag.6
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-square meet 4.2 millimeters above the south side of G2. This fact is another indicator that the "13-steps' method is the one that is involved, and not another. Now, if we had the center, we could already reconstruct this pyramid's layout with unsurpassed accuracy..
A funny coincidence!
A side of the pentagon drawn between the tips of the 13-step-star, or one side of the smaller star = 1150.626180 cubits. We see five consecutive digits of Φ squared (26180). This is another indication that the units used are correct:
Initial Reconstruction of Menkaure's Pyramid (G3)
Two Look-alike Circle Pairs
Much like the ubiquitous instruction guides for "dummies", the Giza designers give an easy to follow guidance in the form of pictorial clues. All one has to do is just get on the right track. Having finished the initial Great Pyramid, and on the look-out for clues to further continuation, the following catches the eye: The Giza containing rectangle, which is widely used, and the Pyramid Square we use here, share the same south-eastern corner, and so I wonder, how many people had drawn an experimental circle from that corner to touch either the Great Pyramid's circumcircle, or the south-eastern corner of G3 to find out that it then seems to touch the other object, as well. Moreover the circle appears to be the same size as the e-circle.
diag.7
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.
In the diagram however, the pyramid's circumcircle has been replaced by a circle, which complements the e-circle. While it is concentric with the circumcircle, it is a tangent to the e-circle. So, we have here two pairs of circles, which look the same from high up above Giza.
Another special effect is shown below. Line-g drawn between the intersection (on the left) of the two e-circles , and the south-east corner of the initial Great Pyramid then duplicates the angle of the other golden diagonals to 0.0015º. This is a very fine value. And conversely, the true golden diagonal drawn from the same corner then comes to within 0.66'' of the same intersection.
diagram 7*
Very Special Effects
This latter e-circle-pair wheels the entire reconstruction fast forward, as there are more special effects. First, an amazing effect, involving the big circles from each pair (described above), rivets our attention.
Shattering the Giza Record
diagram 8
The diagram above is a close-up from diagram 7 of the situation in the south-east corner of G3, the Menkaure pyramid. Lines 1 and 2 are sides of the pyramid. Lines a and b are the e-circle, and its look-alike, the tangential circle to the Great Pyramid's circumcircle in Petrie's version.
Both are almost exactly equidistant to the pyramid's corner!
The centerpoint of the distance between the circles is 3.4 millimeters to the east of the pyramid corner, as given by Petrie. The distance between them is 1.0005 cubits, a fascinatingly round value.
Since we are going to precisely reconstruct the Great Pyramid's position, we shall also be able to pinpoint the southeast corner of Menkaure's pyramid with this great accuracy.
The same trick as above, but using the circumcircle of the interim G1 instead, produces the SE corner of the # 2 version of the interim G3. It is 3.68 inches to the west of the original.
Operation Rising Column
Channeling
Given two tentative versions of the original, which create a channel of parallel, or concentric lines, often, the solution is simplicity itself - a single step:
"Go down the Middle of the Channel!"
This is also the method we'll see work for a fantastically accuraterepositioning of the Great Pyramid (all we need is another corner in addition to the already known NE corner). But there is more! In one fell-swoop, the same procedure puts Menkaure's SE corner half-an-inch (0.506"), west of its location in Petrie's plan.
A method has to be elegant, and this is elegance!
diagram 9
The e-circle pair hugs the Menkaure pyramid from both east and west in diagram 7. This leads straight to the first interim reconstruction of G3, and exact positioning of the Great Pyramid.
To digress a little, the symbolic language of coincidences around the e-circle lets us follow the trail, but at one time it left a question in my mind over the selection. Why the e-circle? Then I saw it in the context of the '13-step' construction, as in the diagram below.
A 135 degree line through the NE corner of the Horizontal Column (the pyramid's diagonal) meets the line through points 1, 2, 3, 4 at the NW corner of G1. This is yet another method for simple reconstruction of the interim G1. So, this line through points 1,2,3,4, the radius of the e-circle, serves construction just like its circle. Since the 'e-circle' dominates this stage of the reconstruction, it matters a lot that it is originated by such classy fundamental geometry.
diagram 10
Menkaure's initial vertical axis ('go down the middle')
Next, some elements of the position are manipulated in several simple operations. The midpoint of the gap between the circles sandwiching G3 in diagram 8, is 4.2 inches to the west of Petrie's vertical axis for the pyramid. Draw a vertical axis from there for our initial G3.
The Initial G3 (#1)
a)
* Center a circle at the point, where this vertical axis crosses the Horizontal Column.
* Have it touch the far side of the little circle in the column's SW corner.
The little circle is already a part of the position, since its diameter is a side of the inner pentagon of the '13-step star', and it is also the inscribed circle of the initial G1.
b)
The section of the vertical axis below the new circle is taken as equal to one side of G3. On the basis of this assumption, the pyramid is completed around the axis.
diag.11
The NW-corner of the interim G1, and the SE corner of G3 in the above #1 version, are crucial ingredients for a simple operation, which makes the reconstruction of the Great Pyramid identical to Petrie's plan.
Robin Cook says that if we enclose the pyramids between two 45º lines (perpendicular to their N.W. by S.E. diagonals), as in the diagram below, the long axis of the resulting column (the Rising Column) is almost exactly the same as one of the second pyramid's diagonals.
diag. 12
But in Petrie's plan, the axis actually runs 13.82 inches east of the pyramid's diagonal. This relationship does look accurate on computer screens, or paper, however.
Meanwhile, I noted another special effect in this position, one three times more accurate: the width W-Z of the Rising Column is just 4.32 inches more than the width A-B of the Horizontal Column. Comparing, or substituting the reconstructed columns for each other could therefore be of interest, and is next on the agenda.
The idea that the bottom side of the reconstructed Horizontal Column should also be the horizontal axis of Khafre's pyramid (G2), is a 10.12 inches miss. Marking the actual (Petrie's) thickness of the Rising Column straight down from the top side of the reconstructed Horizontal Column gets to 0.94 inch south of the second pyramid's horizontal axis.
Exact Repositioning of the Great Pyramid
Substitute the width of the reconstructed Horizontal Column (a combination of two golden rectangles) for the width of the Rising Column (axis over axis). Then the situation in the Great Pyramid's NW corner looks like the following diagram.
diagram 13
'go down the middle'
The long axis of the channel between the western sides of the two rising columns is identical to the western side of the same in Petrie's version. Or, a line drawn vertically down from the northwest corner of the wider Rising Column becomes one with the western side of the Great Pyramid in Petrie's version.
On the line 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 0.0101.. inch (1/97), or 0.26 millimeter, or 0.0005 cubit. The pyramid centers are 0.007 inch, or 0.19 millimeter apart. In other words, the two versions of the Great Pyramid are identical - the Petrie's plan, and this reconstruction, whose final value is: 439.827 cubits (439.82732...) or 9,068.8 inches ( 9,068.79) per side.
439.82732 / Pi = 140.001.. The pyramid needs to be about 1 millimeter higher than its theoretical height of 280 cubits, to be perfect with respect to Pi, and this reconstruction..
Petrie's value: 439.828 (439.8278).. cubits or, 9,068.8 inches
Petrie's plan is under .005 cubit (about 2.5 mm) longer than the optimal value of 439.823 for Pi encoding, while this reconstruction is half a thousandth closer:
Pi times half the pyramid's height = 439.82297150.. or, 439.823 rounded
Operation Rising Column also yields the same benefit in the diametrically opposite corner, as it produces so far the best reconstructed value for the length of G3 : This is # 3 version of initial G3. (the SW corner remains constant in all three versions of G3).
diagram 13b
In the diagram, a and b are sides of the #1 initial version of G3, and 'c' is the reconstruction of the eastern side of G3.
Note: Neither the Great Pyramid nor Menkaure's pyramid requires Khafre's pyramid for reconstruction.
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 8
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 3.1 millimeters to the east of the pyramid corner as given by Petrie.
Of interest is the distance between lines 'a' and 'b'. It is expressed by the cyan circle, whose radius is 1.0005.., a rather exact cubit.
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 striking accuracy. The reconstructed SW corner of G3, and using the cubit, as given in this study, is all we need. 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.
diag. 15
The location of the SW corner of G3, the third pyramid, yields some notable readings in cubits.
a)
However, first, we note that 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 0246 or 201.5 cubits (the small change 0.0026 is worth over one millimeter)
c)
THe distance between the SW corner of reconstructed G3, and the SW corner of the Pyramid Square:
314.5 0275The fractional parts in the above 201.50246 and 314.50275 are very similar:
314.50275 from the reconstructed SW corner of G3 to the SW corner of the Pyramid Square
-201.50246 from the reconstructed SW corner of G3 to the SE corner of Petrie's G3
= 113.00029
Put into words, if we flip over westwards the distance between the reconstructed SW corner of G3 and Petrie's SE corner, it is short of the SW corner of the Pyramid Square by:
113.0003 cubits
1/6 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 0.0003 cubit, and an absolute zero on the Giza scale.
Reconstruction of the Third Pyramid's SE corner
Mark exactly 113 cubits from the SW corner of the Pyramid Square towards the SW corner of G3. The remaining gap to the corner becomes the radius of a circle centered in this SW corner of G3. This circle then locates the SE corner, as given by Petrie, to within the above mentioned (1/6) millimeter, or 0.0003 cubit. In plain English, the two locations are perfectly identical. The accuracy is superlative, and never seen before in any analysis of Giza.
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.
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. Without it, there would be no exact reconstruction of the SE corner of G3, however. We need that one for the exact reconstruction of the Rising Column. The latter then takes part in the finding of G2's center.
http://www.ronaldbirdsall.com/gizeh/petrie/c10.html
The Layout of Khafre's Pyramid (G2)
Lumber in the Yard
Earlier the reader saw a way to position the south side to within four millimeters of the original Second Pyramid. Given the center, the reconstruction would come out very nicely. Some usable data is mentioned in the note under diag. 19: 'marking the actual thickness of the Rising Column (W-Z), downwards from the top side of the reconstructed Horizontal Column, gets to within 0.94 inch south of the Second Pyramid's horizontal axis'.
We can duplicate the Rising Column, and in so doing, get to within 0.94 inch of G's horizontal axis. Yet, although this is a nice approximation for the horizontal axis, its true function is to take part in locating the vertical axis.
Channeling the Solution of the NW by SE Diagonal of Khafre's Pyramid (G2)
G2 - diagonal simulation # 1
Alison's circle
Jim Alison had read the position in the diagram below, as saying that a circle (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 Great Pyramid's inscribed circle, and one of its diagonals.
This line runs between from the center of the #2 interim version of G3 to the point of intersection between the inscribed circle of the initial Great Pyramid, and its diagonal. Plainly visible in the diagram below, Alison's circle finds the intersection between the golden diagonal 'c', and the second pyramid's extended diagonal (point I). So, let the golden diagonal 'c', and the Alison's circle set a simulation of the diagonal of G2. This relationship is an order of magnitude more accurate than the one Alison noted. The simulation is 0.66 (0.6592..) inches northeast off the original.
diagram 16
G2 - diagonal simulation # 2
I was surprised at finding several more simulations of the same diagonal of G2, which are all well within an inch of Petrie's plan. For instance:
Two capital lines: (a) the long axis of the initial Rising Column (was key to Great Pyramid's duplication)
(b) the bottom line of the Horizontal Column
meet 0.66 (0.664..) inch southwest of the diagonal in Petrie's plan. This creates a point of insertion for a diagonal simulation (c).
diag. 17
The channel axis between the diagonal simulations #1, and #2 , falls mere 0.002 inch below (southwest of) the diagonal as given by Petrie.. It is therefore identical to Petrie's diagonal.
There is also another way of replicating the same diagonal, which comes to within 0.006 inch of Petrie's version. It is really simple .
diag.18
The big red circle is actually double, with centers in both the #2 and #3 interim versions of G3: From the pyramid-center of each version, draw a circle through the SW corner of the (golden) Horizontal Column. Each seems tangential to one axis of G2. In fact, both come close.
G3 #2 version - the circle is 0.8286 inch short of the axis, while in
G3 #3 version - the circle is 0.8163 inch past the axis.
The axis line of this channel is 0.006 inch, 0.0003 cubit, or 0.16 millimeter to the southwest of the original G2 axis given by Petrie.
The Vertical Axis of G2 & the Center
Once more, column substitution does the job. Previously, the Horizontal Column substituted for the Rising Column let us establish the Great Pyramid's exact position. In a reversal of that process, take the width of the reconstructed Rising Column, and mark it on the Horizontal Column from the top. Its bottom line then falls 0.94 inch south of the second pyramid's horizontal axis (cyan in the diagram below).
This line then meets the Diagonal Simulation #2 0.0014 inch, 0.035 millimeter, or 0.00007 cubit east of Petrie's vertical axis. Thus, the reconstruction utterly succeeds in getting a point on the vertical axis.
Diagram 19
The vertical axis and the channeled diagonal together locate G2's center a pinpoint away from Petrie's plan, at 0.01 inch (1/100"), 0.26 millimeter, or 0.0005 cubit. We know the position of the southern side since diagram 5, hence G2, the Second Pyramid can be recreated with great accuracy (4 millimeters) in the plan.
Final adjustments - the Third Pyramid
Having the initial Second Pyramid permits testing Legon's ideas.
a: Legon says that the east-west axial distance between the west sides of the Second and Third Pyramids equals 250√2 cubits, or 353.55339.. . This works really nicely for this reconstruction, locating the west side of the Third Pyramid 0.006 cubit (3.3 millimeters) to the west of Petrie's version. It doesn't work too well for Legon's reconstruction.
b: Legon also implies that counting westwards from the east side of the Second Pyramid, 250√2 + 411 cubits locates the west side of the Third Pyramid. The same formula works wonders for this reconstruction, locating the west side, or its southwest corner, 0.0007 cubit (0.38 millimeter) east of Petrie's version. The other corner of the south side is given with even greater precision, and this way, the Third Pyramid can be recreated exactly as given by Petrie.
The formula also works nicely for Legon's reconstruction - as it gets to 0.01 cubit (5.4 millimeters) east of Petrie's version.
A Sensational Dilemma
It is natural to choose the better result, but which one is it? The closer one? Its special effect is its property of being exactly on the dot , the SW corner in Petrie's version. It competes with the version 'a' which 'only' locates the same corner to 3.3 mms. Its special effect, however is spectacular. A line 250√2 cubits long (353.55339059..) drawn westwards from the west side of the Second Pyramid is 314.553387 cubits short of the west side of the Pyramid Square. Now, subtract the shorter distance from the longer:
353.55339059
- 314.55338740 = 39.00000319
The result is a fabulously precise 39 cubits! the fault being just over three microns, 3×10−6 m, 3⁄1000000 m.
Not only are the two fractional parts same to five zeros, but a significant square root value is involved, as well, and for the umpteenth time.
This fact offers another algorithm for the reconstruction in Legon's style. Since both fractions are so alike, simply draw the distance of 250√2 - 39 cubits, eastwards from the SW corner of the Pyramid-Square. It will end 3.3 millimeters west of the Petrie's version of the SW corner of G3.
In (a special) effect, this is the second instance of a high-precision number expressing distance between an element of the Third Pyramid and the western side of the Pyramid Square. Remember 113.0003 ? That special numerical effect leads directly to the exact location of the SE corner. Now, we have the same thing in spades leading to the reconstruction of the SW corner. Is this a sign of consistency in the solution?
The fact is that there are at once three accurate solutions for the west side of the Third Pyramid, or its SW corner. All three are equally accurate, inasmuch Petrie's points are in reality small circles, or dots, as he qualifies them with a +- value. The beauty of this reconstruction is that it hits most of Petrie's dots in the very middle. Multiple solutions are an attribute of the position. This attribute had played a major role in the selection of this particular design for Giza, and not another.
It is irrefutable that the Giza position is abloom with deeper meaning, and special effects, especially, when the North to South distance between the pyramids is taken as 1732.05 cubits (1000 √3 given to six digits). This scale is what makes the solution work its magic.
Final adjustments - the Second Pyramid
Draw a line westwards from the vertical axis of the reconstructed Great Pyramid, whose length is 250√3 , or 433.0127.. cubits). Make this the axial distance to the east side of the Second Pyramid. The fault from Petrie's version then is 0.00126 cubit, or 0.66 millimeter. Since the center of this pyramid is already located with utmost precision, the adjusted Second Pyramid is identical to Petrie's version. This method works even better in the Petrie version, although only by a small fraction of a millimeter.
Conclusion:
Petrie's layout of the great pyramids of Giza can be accurately recreated from a clean slate, beginning with the '13-step' construction of the regular 5-pointed star from a line segment, with some involvement of the basic prime number square root values. This method has simplicity, accuracy, beauty, and intellectual depth. Therefore, it must be essentially identical to the actual Egyptian procedure of planning the Giza layout.
Its acceptance would raise high the bar of Egyptian knowledge of mathematics since the method does not work with the classical tools of geometry, at least not without a microscope, given the scale of Giza, and the hairline differences between drawing objects. The plan had to be worked out mathematically. Thus, the knowledge of mathematics guarded by the temples was clearly on a level unattainable in a neolithic society less than two millenia removed from the hunter-gatherer stage.
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.
Jiri Mruzek
Vancouver, BC
©Jiri Mruzek April 15, 2007
Notes
Drawing Inspiration From Ideas of Others
Intial Observations - The Pyramid Square & Khafre's Pyramid
In an experiment, the Pyramid Square gets a basic Golden Section grid (diag. below). Lines of the grid create a Golden-cross within the square.
Next, another Golden-cross is extrapolated from the square of the G2 (Khafre's pyramid). The two Golden-crosses are then 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. None of these sites link to my study, however. It looks like these veterans detest a Johnny-come-lately 'eating their porridge'.
http://www.legon.demon.co.uk/gizaplan.htm - Legon's site
http://sevenislands.tk/ - Cook's site
http://www.kolumbus.fi/lea.tedder/OKAD/Gizaplan.htm - Tedder's site
Jim Alison's rendition of certain ideas by Chris Tedder was holy water on my mill:
Perpendicular distances between the pyramid centers produce two golden rectangle facsimiles (ABCD, and DEFP).
diag.c
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 two new golden rectangles with the Pyramid Square:
Tedder's Secondary Rectangle #1 - An extension of Tedder's horizontal golden rectangle A-B-C-D to the west side of the Pyramid Square is itself a facsimile of a golden rectangle, the vertical rectangle C-D-O-K
C-D divided by C-K = 1.627
Tedder's Secondary Rectangle #2 - Alison's circle intersects the extended diagonal of the second pyramid rising north due west at the I-point. The distances I-J and I - L form the golden ratio.
I-J / I-L = 1.6199 less than 2/1000 off the true Φ value
The horizontal rectangle I-J-K-L is therefore an excellent facsimile of a golden rectangle.
diag. d
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.
The diagram compares the Horizontal Column based on Tedder's rectangle, and 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). Note that visually, the two Horizontal Columns are as as one. Their component rectangles are not. Tedder's rectangle (black) is obviously inaccurate, as 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
diag.6
Starting from the Horizontal Column
This was my original method of reconstruction of the initial Great Pyramid:
Draw two golden rectangles, a vertical one on the left, and a horizontal one on the right, diagonals radiating from their corners,. The rectangles form a single column - the Horizontal Column.
C divides A-K so that if C-K equals Φ - 1, then A-C equals Φ, and A-B equals 1.
The length of the combined rectangle (the Horizontal Column) then is 2Φ - 1.)
1) The center of the Great Pyramid is at the top right corner of the Horizontal Column.
2) The left side of the Horizontal Column is the western side of the Pyramid Square.
3) The diagonal 'a' is tangential to the inscribed circle of the Great Pyramid (diag.6). This circle is then enclosed in a square, i.e., the pyramid sides.
4) The lines through the north and east pyramid-sides are corresponding sides of the Pyramid Square. With three sides of the square known, so is the fourth - the bottom of Square. The south side of the third pyramid lies on the square's bottom (diag.7).
diag.7
Naturally, the golden rectangles above could be preceded by a number of different starting positions. But, only the '13-step' stars produce a close location for one side of G2.
Petrie versus Cole-------------------------------------------------------
> What irks me, Jiri, is your persistent refusal to address the question of intent; you have yet to
> provide any separate evidence that this pattern was created intentionally by the designers of the
> Giza pyramids and temples.
Whenever you reconcile the obligatory appreciation for the eternal beauty of the geometry involved, and the fact that this is Giza, the issue of intent may become clearer. . Such a design certainly places the layout into the highest category of sophistication.
> According to Lehner (The Complete Pyramids 1997)
> the base of Menkaure’s pyramid is 335 x 343 feet.
> Yes, this is contrary to Petrie’s measurements
> (mean 346.13 x 346.13) but does this mean that
> Lehner is wrong?
It most certainly does, considering that his figures differ wildly from not only Petrie, but also Cole, who after all did come close to Petrie in measuring the Great Pyramid. It pits him alone against two widely acknowledged professionals. plus, Lehner was caught cheating in a drama of his "This Old Man Pyramid", if I have the title right. A mechanical shovel was used to move some blocks, but no mention of the fact was made in the flick.
I found that typical of the PyramiPhobia, which so torments some academicians they will sell their soul to the devil.
> > from Cole's survey report:
>> "These differences in azimuth are due to the fact that the new azimuths are found from the actual directions of the sides determined from the excavated pavement,
Cole took series of measurements of the available sections of the excavated pavement. The various hypothetical lines were then averaged out, and extended until they met near the corners. Considering how close Cole comes to Petrie, he did a great job!
> .. a hypothetical base obtained by computing “a square that shall pass
> through the points of the casing found on each side, and having also
> its corners lying on the diagonals of the sockets.”
a)
Yes, the acclaimed and anomalously accurate casing! The quality of work, said Petrie, equaled modern opticians, but on the scale of acres. Determination of lines from the plane of the casing blocks, and their projection down to the pavement to produce an averaged out line in the pavement should naturally be superior to relying on measurements from a single line, as Cole had done.
The descending gallery shows similar accuracy in that it deviates from its axis over the course of 350 feet by a quarter inch (6 millimeters) side to side, and only one tenth inch (2.5 millimeters) up and down. In using the anomalously accurate plane of the casing blocks to obtain a hypothetical line along the pavement, Petrie seems to have achieved even greater accuracy.
b)
Corners lying on the diagonals of the sockets:
This was another logical decision by Petrie, which gave his method further advantage over Cole's. Clearly, the sockets were the target the builders were aiming the sides at.
Petrie was a consummate professional. In my eyes, his drive to provide the most accurate data on Giza surpassed Cole's. After all, Cole could only be bothered to survey one pyramid. Thus he forfeited further experience with measuring at Giza, which might have given him a little more insight.
The fact that Petrie's measurements set the position up for an incredibly accurate regeneration from a clean slate using the noble Section, can by no means be discounted. With this reconstruction of the Giza-layout everything clicks into place. A click in the reconstruction - a booming cannon shot across the bow of Egyptology!
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.
Petrie's value of 9,068.8 inches, or 230.348 meters, or 439.82782340 cubits of the actual average side by the standard of this study, is about 2.5 mm off the desired perfect value for Pi. An error of less than 3mm was allowed for by Petrie. Hence he could be off by that much, and the pyramid may have been built perfect. If we let the facts prevail, the conclusion must be that the average pyramid side was designed with the true value of Pi in mind.
John Legon writes:
"In terms of the Giza royal cubit of 0.52375 metres, the actual mean side of 230.364 metres corresponds to 439.8 cubits, with an average variation in the sides of only 6 cm or 0.1 cubit. Petrie suggested that an adjustment may have been effected in order that the perimeter of the base should express the so-called 'pi-proportion' in relation to the height of 280 cubits, with greater accuracy than the value for pi of 22/7. In this case, the theoretically exact mean side-length would be 439.822... cubits. It seems that the builders achieved this result while retaining the round number of 440 cubits in the south side."
Naturally, a measurement dictates both its exact, and rounded out values. The need for a side of 440 rounded out cubits arises from the perspective of Phi. The apothem divided by half the side, or 356/220 equals Phi to the first three decimals (1.618 181818...)
Legon abstracts a cohesive system from the Giza position dealing with square roots. Although some of these readings are somewhat approximate, Legon suggests logical reasons, why there were adjustments. Legon is right. He correctly identifies an additional layer of abstraction in the position. Compared to the "13 steps" reconstruction, it is considerably less accurate, yet it makes perfect sense in a rounded-out way. The more meaning, the more reason for selection of this plan.
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 situations it is easy for a theorist to become convinced that his recreations mirror the Egyptian planners, before the builders strayed from the plan somewhat, just as expected, or before the plan got changed for reasons unknown.
It makes sense that the designers had started out with regular squares. Adjustments were then made to create a new layer of meaning, and to encode more data. Equinoctial precession would be very fitting as a subject.
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..