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       Giza Pyramid Temples & the Golden Section 

Egyptology on the whole denies that ancient Egyptians had knowledge of the Golden Section (Φ). Documenting presence of Φ in the Giza temples' floor-plans would bear strongly upon the issue of Φ elsewhere at Giza, and as well, in the controversial Abydos Helicopter. The Summation Series (Fibonacci series) in Khafre's pyramid temple is the only study of one of these temples from the viewpoint of Golden Section. I could find, so far. Unfortunately Mr. Gadalla's solution does not quite pass testing in CAD, as we shall see. Of course, should my analyses of the temples coincide with any works I am presently unaware of, then they serve as independent confirmation. In any case, I've enjoyed discovering these things for myself. The temple diagrams in this study are downloaded from the web. They are originally from "The Pyramids of Egypt" by J.E.S. Edwards, who compiled them from research done on the ruins by J.P. Lauer, U. Holscher and Col. H. Vyse.

You can read a bit more on Giza temples at this site

•  Khufu's Pyramid Temple

The outside walls

Tthe floor-plan of Khufu's temple is a rectangle based on its long side. In order, to check it for golden-section, we use a technique, which completes the rectangle into a square
, which is then given the basic golden-section grid. The diagram below is a partial example of that, where verticals are omitted to eliminate clutter. Present in this grid are three golden-rectangles, marked by diagonals.

Diag. 1

When we superimpose this diagram over the base of the temple, the second line from the top coincides closely with the inside edge of the main western wall .

Diag. 2
                       the side of the temple adjacent to the Pyramid is proportioned by the Golden-ratio

• The temple, when simplified to a rectangle, is defined by the golden-section.

• The short wall at the top gives Φ-proportion with the
temple's width. 

This relationship looks very accurate on the right side, and slightly off on the left.

We also see golden-circles in the diagram, which determine lines of the (golden) section. This allows us to see some further Φ relationships in the temple's architecture. The entire top is divided into a series of such relationships, in addition to the one already pointed out. However, these are approximate only.
Notably, the golden-section line resting atop of the big circle in the center of the square also closely coincides with wall-edges on both sides. Such parts of the original plan we can recreate as defined ideas.

Walls of the inner enclosure

There is a surprise in the form of Φ-rectangle based on two main courtyard walls (base, right side) and, the short western walls.
The Φ-rectangle based on these then defaults to the near side of a vertical row of seven columns. 

Diag. 3

The small white circles mark the corners of this rectangle. We see the wall on the left of the inner enclosure base is set slightly higher than the one on the right.

•The base of the rectangle is set by the outside edge of a wall on the left.
•The height is set by the edge of a wall on the top right.
•The right side of the rectangle is also set by
the outside edge of a wall.

Big part of the plan revealed

See the
the three pairs of concentric circles spanning the rectangle's length? These are "golden circles", because they divide the rectangle in Φ-proportions. There is a vertical line on each side of each circle. The middle pair is hi-lited in color for easy orientation.
Every single one of these eight vertical lines coincides with the temple's plan! They coincide with edges of walls, and in a pretty effect, with the trio of halls at the top, whose drawing looks like the Stanley Cup with wings, as it might look a thousand years from now. Additionally, a pair of major diagonals (of the Φ-rectangles in the position) passes through the lines of the winged formation's base.  This deserves a closer look below.

Diag. 4  



With the full golden grid of the rectangle superposed over the diagram of the temple, one soon realizes that it is possible to reconstruct the winged formation from this grid, as well as reconstruct most of the temple's plan. 

Diag. 5


• Menkaure's Pyramid Temple

Diag. 6

                    The Menkaure pyramid's temple floorplan is dominated by the (Φ) Golden Rectangles

The main building's layout, the outer walls basically form a square.  The inside wall enclosure looks on paper like a big rectangle  with a 3-step pyramid attached on top. 

he first Φ-rectangle comprises the entire colored area. It is given by the outside lines of three of the outer walls, and by the top of the first-step of the 3-step pyramid. 
•The second Φ-rectangle is white. It bases on the outside lines of three walls of the inner enclosure, and on the inside line of the same wall, whose outside gave us the first rectangle.
•The third rectangle shares its right bottom corner with the big rectangle, and is extruded to the left and the top outside lines of the big rectangle of the inner wal enclosure.  It would be all green, if not partially screened by the white rectangle. There are other smaller Φ-rectangles here, but for now we just focus on the main big ones, in order to show the influence of Φ on the main overall design. 
In that respect, starting from the eastern side of the building, which is shown as the horizontal bottom in the diagram, we have used four successive horizontal walls. The fourth wall level up was used twice, once on the inside, once outside.
Next up is the fifth horizontal wall, marked in the diagram by the red horizontal line.  To get to this level we have to first inscribe a circle within the square, and then inscribe a 5-pointed star within the circle. Then the horizontal arm of the star gives us the red line. Note that even the tip of the pentagram fits the position nicely.

The Khefren's Pyramid Temple

Part 1

A look at the claim of Summation (Fibonacci) Series expressed by the temple

While searching web for information on the Giza temples, I came across an extremely interesting claim by Moustafa Gadalla regarding the Fibonacci Series, or as Gadalla says "Summation Series, given by Khephren's temple.  According to Gadalla, these series are encoded into the architecture of numerous temples in Egypt. Somewhat unexpectedly to me, he provides a single low resolution example on his site, and the better resolution images are reserved for paying customers. Since he provides an example of the pyramid temple, which I had already done some work on, I was extremely curious.

Gadalla says: "There is evidence about the knowledge of the Summation Series, ever since the Pyramid (erroneously known as mortuary) Temple of Khafra (Chephren), at Giza, built in 2500 BCE, i.e. about 3700 years before Fibonacci.

The essential points of the temple [shown herein] 
comply with the Summation Series, which reaches the figure of 233 cubits in its total length, as measured from the pyramid, with TEN consecutive numbers (Bold-sic) of the series."

As I ponder Gadalla's diagram under magnification, it suffers all the faults that my work has ever been charged with unjustly by opponents at Randi's, and elsewhere. I just cannot see any coincidence between the temple's forms, and the overlay by Gadalla. Is it not what my opponents ascribe to me?
 The difference between my methods and Gadalla's escapes them.
Yet, my results frequently show ridiculously perfect agreement between exactly drawn geometry, and the underlying ancient art, or architecture. Critics still singing the same note after convincing demonstrations to the contrary must be hardwired. I rest their case.
Regarding the Fibonacci sequence, I was eager to check it in CAD. Gadalla's diagram shows the temple's width as 89 cubits. Accordingly, I scaled my image of the temple to the same width. Next, I measured out the given distances along the longitudinal axis, and only two out of nine were close enough to temple features,  34, and 55.
So, Gadalla is wrong, right? Well, he is definitely misleading. His show in its present state is all wrong, but if we take the temple's width as an uneven number of cubits (88.66 cubits), suddenly, some of the Fibonacci numbers start coinciding with temple features at the appropriate levels. We can see it in the diagram below: 13, 34, 55, and 89, but not 5, 8, 21, 144, and 233. Although my diagram lacks the pyramid edge, the difference between Gadalla's indication of the 144 cubit level and mine is about 4 cubits, so by the time we get to the pyramid's edge the difference will be even greater, and my reading of the distance is going to be about 228 cubits. As is, Gadalla has five of nine values wrong in principle, but seven of nine wrong in his actual execution. Remember, his placement of the 34 and 55 levels is different from mine. How he can speak of ten consecutive values is a mystery to me since the diagram shows nine. Internal inconsistency makes me wonder. Perhaps, Gadalla is not the author of the study, but he neglects to make a mention of it. Who is the author then, if not Mr. Gadalla? I'd like to see the work of the original author of the idea of universality of golden section proportioning in ancient Egyptian temples. Is it Lubicz? My results so far indicate that the idea becomes correct, after we eliminate the word 'universal' and instead simply specify that it was of major importance in the design of many temples. To determine how many, we would have to do a comprehensive study. As for my own research, I had looked at :

• Abydos temple of Seti I, and Ramses II (but only in the Abydos Helicopter section)
• three pyramid temples 
• a valley temple connected to one of the three abovementioned pyramid temples

Out of these five samples only one does not indicate the Golden Section as a major planning principle - the valley temple of Khafre's pyramid. This is rather unexpected,  because Khafre's pyramid temple abounds with the Golden Section.
We also get three
consecutive Fibonacci values, and four of five in the part of the temple up to the courtyard. The 200 cubit level is also clearly indicated.

Diag. 7

The theory, which maintains that Egyptians did not know the Golden Section geometric procedure also predicts that the same cannot be found consistently playing a dominant role in the design of a number of temples.

Finding the Cubits used in Khafre's pyramid temple

With the units set to my own model, many features of the temple indicate whole numbers of cubits in their elevation above the base. In the diagram below we get the following numbers: 7, 13, 17, 19, 25, 34, 41, 45, 47, 55, 64, 84, and 89 (adding up to 540). That's a total of thirteen levels of the temple in its first section, which correspond to  whole numbers.
The conclusion, I trust, is obvious - since so many levels of the temple work out to whole numbers of cubits on the scale used in my model - that scale must be essentially the same as the original scale used
by its Egyptian architect.

Diag. 8

       cubit units indicated in the second Giza pyramid's temple

Part 2

Plethora of golden rectangles in Khafre's pyramid temple

The temple is a long rectangular structure, the ratio between its length and breadth is 2.35. In the diagram, an exact square is measured off from the eastern edge of the temple. When this is done from the left edge, which is little higher to the west, then this square extends almost to the very end of the entrance hall, whose two parts, look like an inverted T. 

Next, drawing in a horizontal Φ-rectangle from the top of the square involves a horizontal line, which divides the square into the golden Φ-ratio. This line then coincides with the temple's plan accurately.  It is marked by the lower pair of small purple circles.
•  Doing the same from the bottom of the square, results in a horizontal line marked by the higher pair of purple circles. This  line then coincides with edges of two columns.
•  From the existing grid of golden-section lines in the square it is also evident that drawing two vertical golden rectangles side by side across the top of the square downwards results in a line marked by the three little green circles. This line too coincides with edges of the antechamber's walls, and thus the temple plan. The height of these rectangles equals the combined height (or depth) of both parts of the entrance hall, including the access doorway.
• The four main vertical golden-section lines coincide with the temple's plan. This alignment is also evident in the rest of the structure, as can be seen from following diagrams.

Diag. 9


Golden Diagonals

Diagram below:
Lines along walls, and rows of columns in the entrance hall form six more Φ-rectangles. I leave you the pleasure of finding them yourself. This is an easy task, just follow the diagonals, as all the rectangles have corners on one of the two diagonals. This skill should come especially handy for diag.13a. So many rectangles in the diagram have their corners on these golden-diagonals that one might wonder if the temple's architects had such a tool for drawing golden rectangles. The basic tool is easy to visualize: A small rectangle with a protruding, or extendable diagonal. The two diagonally opposed corners slide along the rectangle's diagonal. It moves on the drawing board back and forth like an inchworm.

Diag. 10


The Twin and Coaxial Φ-rectangles

On going deeper west into the temple, we find a stack of two temple-wide Φ- rectangles. The first one, whose 
horizontal sides are accurately given by major wall lines, envelops the entire courtyard. In it, I found (only) one coaxial Φ-rectangle, which looks accurate even at considerable magnification  (the white lines). The two green Φ-rectangles within seem to fit the column lines, but are a little off from up close.

Diag. 11


The top horizontal line of the second twin rectangle again accurately coincides with the temple. This rectangle is even better than the first one, because it contains a coaxial rectangle, whose main features are ideally suited to the temple's blueprint:

• Height of the five chambers just west of the courtyard is equal to half the height of this rectangle
•  Height of the row of five smaller chambers above divides this smaller rectangle into Φ proportions

Once again the two biggest Φ-rectangles originate from the same point, but this time it is in the lower left corner. Both rectangles seem accurately matched to the temple walls. The smaller major rectangle is further divided in the middle both ways, and some of its vertical golden-section lines are also marked in by wall-edges. I leave it up to the reader to determine which line is which simply by inspecting the diagonals marking the numerous golden rectangles here.



There is yet another Φ- rectangle also encompassing the courtyard.
It is temple-wide, and bases along the eastern wall of the courtyard. It is extraordinary, because it has a number of coaxial rectangles inside.  
Its top horizontal side is set by:
                                                • a turn in the second wall from left
                                                • tiny niches and turns in the next seven walls
Its base is set by the eastern edge of the eastern courtyard wall:
Within the rectangle are its diagonals, and a horizontal (yellow) line dividing it into halves. This line coincides with, or is given by eastern edges of two columns.
The diagram below of this rectangle comes with an example of coaxial golden rectangles
set by column, and wall edges within the above mentioned rectangle: Two lines are shown, one vertical, one horizontal, which both coincide with column edges. These lines happen to meet in a (circled) point on one of the diagonals. This mutual coincidence means that the aforementioned two lines section off two golden rectangles of unequal size in conjunction with the sides of the main rectangle. 



The Golden Promenade - a field of coaxial Golden Rectangles

The same temple-wide rectangle is shown with all of its coaxial Φ-rectangles, all anchored by corners to the two diagonals. The anchoring points are marked by small circles.
• Starting from the top left, six rectangles are set by lines along column edges until we get to the outside of the wall on the north side of the courtyard. That line sets a very accurately looking Φ-rectangle with the line drawn as the eastern limit of the courtyard columns. Altogether nine rectangles reside on this diagonal. Seven of these seem very accurate, two (#2, #3) a little less, see for yourself. 
• Following the diagonal
from the bottom left upwards, there are ten golden rectangles to see, all set by columns. Three of these are slightly off, the rest I find quite accurate, and even the less acccurate rectangles are still reasonable facsimiles of the golden rectangle. Overall, I find this plethora of twenty Φ-rectangles along the two diagonals nothing short of overwhelmingly convincing! Actually, the count is twenty-four, when we include the four rectangles quartering the big main rectangle, as two columns divide it in half horizontally with admirable accuracy, while the the long vertical axis is also set elsewhere in the temple.


Below: Still more golden Φ-rectangles given by lines along major walls in the same area.



The Columns - Twelve Little Golden Rectangles

Diagram 16 presents another major golden-rectangle. It is almost temple-wide, between the outside of the second wall from the south, and the inside of the outer north wall.  It is further subdivided by the golden-section, whose lines coincide with lines given by the courtyard columns.  All of this is very accurate.
• Also in the diagram are small golden-rectangles superposed over the twelve imbedded columns.  The fit certainly is good enough to allow identification of these columnal bases as golden-rectangles.


the ten columns in the courtyard of Khephren's pyramid temple are all golden Φ-rectangles

Observation: For any major horizontal, or vertical line set by the temple there is a corresponding trio of temple lines
, with which it produces a true to form golden-rectangle.

Conclusion: It is evident that architectural planning of the three pyramid temples makes extensive use of the Φ-proportion. Since the IV. dynasty's architects had used Φ in planning of the Giza temples, we ought not to be surprized at finding it i
ncorporated in the Great Pyramid, part of the same complex. By the same token, finding Φ in Egyptian temples more than a thousand years afterwards (specifically the Abydos temple) should also pose no surprise.

Jiri Mruzek

June 21, 2007

Vancouver, BC

Note: The rasterization module in my vector driven program  has a stubborn kink, which elongates the rasterized images vertically by about three and a half percent (1.034 to 1.035). This is a recent problem, dating from  "The Giza Pyramid Temples & the Golden Section. I am not willing to readjust raster images, it just does not seem right. But, such adjustment will produce the optically correct ratio.

p.s. This analysis is not yet by any means complete, but I decided to publish what there is now, to mark the passing of summer solstice, and keep adding more material later. This could take years! :)

August 4, 2007

How Khephren's pyramid sets the width of the adjacent  pyramid temple

       Golden Section of a side of the Khephren's pyramid sets the width of the adjacent pyramid temple

In the diagram above, the width of the yellow section is the same as the width of the adjacent pyramid temple.

If a side of the Khephren's pyramid is taken as Φ squared (2.618..), then the width of the temple is 1/Φ (0.618..)

Caveat Emptor - better resolution map of the temple in relation to Khafre's pyramid is sorely needed to confirm the preliminary, and tentative observation above. 

 The Monkey from Nasca & the Golden Section   Abydos helicopter & the Golden Section
Stone-Age Horsemen
 The Hex-Machine
The Seal of Atlantis
next:  the Ground-plan of the Great Giza Pyramids - explanation & exact recreation