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'
floorplans 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
floorplan of Khufu's temple is a rectangle based on its long
side. In order, to check it for goldensection, we use a technique,
which completes the rectangle into a square,
which is then given the basic goldensection
grid. The diagram below is a partial
example of that, where verticals are omitted to eliminate clutter. Present
in this grid
are three goldenrectangles, 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 temple, when simplified to a rectangle, is
defined by the goldensection. • The short wall at the
top gives Φproportion with the temple's width.
This relationship looks very accurate on the right side, and slightly
off on the left.
We also see goldencircles 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 goldensection line resting atop of the
big circle in the center of the square also closely coincides with
walledges 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 Φrectangle based
on two
main courtyard walls (base, right side) and, the short
western walls. The Φrectangle
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 Φproportions.
There
is a vertical line on each side of each circle. The
middle pair is hilited 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 Φrectangles in the position) passes through the lines of
the
winged formation's base. This deserves a closer look below.
Diag.
4
Reconstruction!
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 main building's
layout, the outer walls basically
form a square. The inside wall enclosure looks on paper like
a
big rectangle with a 3step pyramid attached on top.
•The first
Φrectangle
comprises the
entire colored area. It is given by the outside lines of three of the
outer walls, and by the top of the firststep of
the 3step
pyramid.
•The second Φrectangle 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 Φrectangles 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 5pointed 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] http://www.egypttehuti.org/articles/sacredgeometry.html
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 (Boldsic) 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
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 Φrectangle from the top of the square
involves a horizontal line, which divides the
square into the
golden Φratio. 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 goldensection 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 goldensection
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 Φrectangles. 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 goldendiagonals
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 Φrectangles
On going deeper west into the temple, we find a stack of two
templewide 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 Φrectangle, which looks accurate even at
considerable magnification (the white lines). The two green
Φrectangles within seem to fit the column lines, but are a
little
off from up close.
Diag. 11
Below: 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 Φrectangles
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
goldensection lines are also marked in by walledges. I leave it up to
the reader to determine which line is which simply by inspecting the
diagonals marking the numerous golden rectangles here.
diag.12
There is yet another Φ rectangle also encompassing
the courtyard. It
is templewide, 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.
Diag.13
The
Golden Promenade  a field of coaxial Golden Rectangles
The same templewide rectangle is
shown with all of its coaxial
Φrectangles, 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
Φrectangle 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 Φrectangles along
the two diagonals nothing
short of overwhelmingly convincing! Actually, the count is twentyfour,
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.
Diag.13a
Below:
Still more golden Φrectangles given by lines along
major walls in the same area. Diag.15
The Columns  Twelve Little Golden Rectangles
• Diagram 16 presents another major
goldenrectangle.
It is almost templewide, between the outside of the
second wall
from the south, and the inside of the outer north wall. It is
further subdivided by the goldensection, whose lines coincide with
lines given by the courtyard columns. All of this is very
accurate.
• Also in the diagram are small goldenrectangles superposed
over
the twelve imbedded columns. The fit certainly is good enough
to allow identification of these columnal bases
as goldenrectangles. Diag.16
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
goldenrectangle. Conclusion: It is evident
that architectural planning of the three pyramid temples makes
extensive use of the Φproportion. Since the IV.
dynasty's architects had used
Φ in planning of the Giza temples, we ought
not to be surprized at finding it incorporated 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
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.
