Great Pyramid
Footprint
James D. Branson
© 2005-2009 Copyrighted by J D Branson. All Rights Reserved
This paper presents analytical models to strongly
conclude the following:
- The Great Pyramid was
not designed to be square with all 90 degree corners and the construction
simply missed the design by a small amount. It appears the NW corner was in fact
designed to be 90 degrees and it was constructed so close that surveys are
not really sure whether it is exact or the surveys are slightly in error.
- The Great Pyramid base
had a very specific design and that design was executed with great
precision in the construction stage of the process. There is abundant evidence provided
herein that the entire base was constructed with the same amount of
precision as the NW corner.
- Modern design and
construction capabilities have never equaled the Great Pyramid as
described in this analytic model to my knowledge.
- This Year Model seems to indicate
that the Great Pyramid design criteria may use the changes in earth
orbital characteristics from precession to establish dates in the past and
perhaps into the future for either significant events or time periods as
the solar system revolves around the Galactic Core. This Year Model opens up all sorts
of venues for future research and new theories.
Does
anybody think that the Great Pyramid
was just haphazardly thrown together by camel jockeys or done without a
plan? I am sure that anyone who has
visited the area is astounded with the still visible accomplishments
today. Archeologists seem to steadfastly
hold onto the idea that these pyramids were tombs designed and built by
Pharaohs with a primary desire to entomb them when life was through, even
though no remains were ever found inside the pyramids. But even if this was the case, it is still
clear from analysis herein that a great deal of thought went into the design
and construction of these structures. And while it is obvious to an engineer
that the function of the Great Pyramid is far beyond our understanding, the
message built into the design layout may have been particularly designed for
our consumption.
The
readers need to ask of themselves, “How did the lengths of sides visualized by
the designer get transferred to stones on the ground?” Did they have long ropes or vines? Seven
hundred fifty feet is a fairly long rope.
Did they use short lengths and simply put them end to end? Can you achieve precision doing it that
way? Even in surveys in the 20th
century, steel chains are not of that length so they must be added to at least
some degree. The coefficient of
expansion of steel in the hot desert or contraction in the cold evenings can
change the length dozens of millimeters when trying to achieve precise
measurements and certain precautions must be taken by engineers to minimize
these influences.
So
why would somebody bother to even survey the structures at all? It has been done many times by different
people so there must be something about the structures that attracts detailed
investigations by technical people. Maybe people just have a gut feeling that
there is something more than meets the eye there. Or…perhaps a very simple
explanation.
One
of the most widely respected surveys of the exterior of the Great Pyramid was
performed by a British surveyor named H. Cole in the 1925 era. He provides the dimensions of the sides of
the base to the nearest millimeter and even suggests how much error is possible
in each of those numbers. He goes on to
detail the precise difference in the angles from true 90 degree corners. The world has long marveled at how precisely
close to 90 degrees this construction was and how marvelous these ancients
could echo our modern technologies, as if our modern methods are really
anything to write home about.
This
paper will show that the base of the structure is more precise than most
investigators can even imagine and far more precise than we typically perform
in modern construction. The typical
investigator simply does not have sufficient construction experience to
appreciate the accomplishments. It seems
curious to me that nobody, including myself before now, has put the numbers
into an accurate drawing and analyzed them as is done below. One could not
expect Cole to have done much in 1925 because the equipment of the times was
very limited. The simple mathematics of huge precision just would not have been
done. The manipulation of precise powers
of logarithmic bases would have been a major chore and the chance for error so
high that few would have confidence to even get started.
Just
using a few of the Cole numbers, the north edge is laid into this first stage
precise drawing at 230253 mms and at a difference from due east to west of
.04166666 degrees. The west edge is
likewise laid in at 230357 mms and at a difference from due north to south of
.4111111 degrees, just as Cole suggests.
The east and south sides, however in this first drawing, are simply the
intersection of arcs at the lengths given by Cole of 230391 mms and 230454 mms respectively. Now we can check the lineal type of
information against the remaining survey data and see if they all mesh with any
precision. This is something engineers
do typically to see if the data fit together in the manner suggested by the
survey. If the lengths do not match up with the angles, then there is obviously
an error in one or both of the data streams.
In
the graphic below (1.0) one can compare the Cole measured angles shown in
colored text along with the dimensioned angles that result when the
lengths are used to determine the layout.
He explains the relative errors that he thought could occur and his
suggested error analysis is a much greater percent than this analysis indicates
actually happened. Since he measured
both the lengths and the angles, whatever surveying error did occur is largely
attributable to the fact that he was measuring “suggested casing stones” that
were generally no longer in place (only five are remaining). One must conclude
from the analysis herein that he did a very good job making the survey and
calculating the angles.
The
largest deviation is at the SE corner where he indicates the angle measured was
89.94083333 and his
lengths make it be 89.9421377, a difference of just under 5
seconds. If we divide 89.9421377 by
89.94083333, it suggests a surveying error of .00145 percent. I believe this
represents a top notch surveying effort for the times and equipment and I
suspect we could not do better today because there is likely more error in
“finding the measuring point” than these small amounts. This, however, does
provide solid indication that the Cole Survey could easily be off by up to five
seconds and that is more than the 2 seconds he thought the NW corner might be
out of square.
When
I found that the product of the diagonals in this drawing was nearly exactly
the double of the area (expected in exact squares but not in skewed rectangles
nor quadrangular shapes) it seemed possible that the layout may have been very
accurately planned and executed with extreme precision. If this is thought to be remotely true, the
question then arose as to how to see if there was some way to find the design
criteria no matter how complex or intricate.
Nobody credibly thinks that the layout was haphazard or random. So how did the builders decide what lengths
and angles to use? If they couldn’t measure precisely and couldn’t place the
blocks precisely, how might they end up precise? But we know they placed 12-ton
blocks together precise enough that the seams were almost impenetrable with
thin steel feeler gauges. This is a strong hint that precision did in fact
exist at the time of construction. Even in modern times it is far more
difficult to get two large flat surfaces together than most people are able to appreciate.
In
building log cabins, the next log was set atop the previous and then a
cross-cut saw with the teeth turned out a little was run between them leaving
the two surfaces pretty near parallel and when taken off the blocks, fit quite
precisely. We can envision in modern times holding the new block with a crane
and sawing between them similarly and then they fit together. There are a whole
bunch of reasons why this technique won’t work on large rocks nor easy methods
to let loose of the new rock.
If
one wants to know how several numbers such as the sides of the pyramid may have
come into existence by human design, then one must mimic human typical
activities. There are really a limited
number of ways the numbers might easily come into existence as design criterion,
and not the result of errors. In this case, dozens of mathematical discovery
methods were employed and only one was successful. It was discovered that the ratios of one side
to another are somewhat easily converted to powers of each other. The approach was to put the lengths in a
spread sheet and check ratios. It was discovered that three of the sides were
very nearly equal to the powers of the north side using 2, 3/2 and 1 as
exponents.
The Year Model
In
the following mathematical model, the reader should look at the first three
equations first as a group. These are
simply dimensionless ratios so
that any measurements of the sides using any units would result in the same
exponent. One doesn’t need Pyramid Inches or units of bat wings or hog’s
hooves. And any inaccuracies in measurement of a single side can be made more
accurate by combining it with inaccuracies in the other number which may be off
in the same direction. For example,
errors due to expansion of the measuring device would be reduced by placing the
results in a dimensionless ratio such as these.
When you think about it, someone wanting the world to discover their
efforts would have to use dimensionless ratios.
Please
note that the only change from one to the other is the exponents of 2, 3/2 and
1. If the numbers were random or
accidental, this exponential relationship should not exist. The first number
3.15570 hits the standardization number 3.15569 to six digits, a dream come
true to any mathematician trying to find a relationship. All three hit to the fifth digit. There is a
good reason to be explained later why they don’t hit exactly.
Using
exactly the same basic formula, the ratios produce the same result 3.1557xxx
which is totally remarkable by any standard.
In processing analysis, if we hit this precisely we would be in hog
heaven. The fact that it is the initial digits of a hugely important number
like 31,556,925.97 seconds in the earth’s orbit in the year 1900 AD, set as
standard for the definition of the second in the 1950 era.
The
fourth equation hits the annual orbital time to five digits at 3.1556 and sets
up the trial assumption for finding a mathematical model. If we use this
equation to define the north boundary length, it has 5 digits of precision
already built into it. Since it divides the north boundary in millimeters by
25.4 (conversion to inches) then we might be alerted to solutions that might
pop up in inches. So we will make the model input trial such that north uses the
precise earth obital time is 3.1556926 and the north side becomes 230252.0625
millimeters.
In
order to convert the base number to something more easily recognized by the
general public, we will divide by the decimal shift of 86,400 seconds in a day
and result in 365.24xxxxx type numbers. We can use a resource such as the Supplement
to the Astronomical Almanac as reflected above and find yearly time periods
such as the tropical and anomalistic years which I have simply selected because
they produce results below that one can see by observation. Later other combinations of yearly periods
may provide additional results.
Similarly
we calculate the north side labeled b8
in the detail calculations for The Year
Model below. One must note that
the terminology is log
for the logarithm
to base 10 versus ln for natural logarithms used above in the discovery equations. The use of logarithms will drive a lot of
academics nuts who don’t feel the Egyptians even knew logarithms. But they did do “summations of fractions”
which have a very similar look and feel to logarithms. It is a little like
trigonometry. People argue that there
was no sign of trigonometry yet the ratios of right triangle legs are used
repeatedly in Sumerian tablets like the Plimpton322 and all over the earth. It
doesn’t take knowledge of trigonometry to divide one leg of a triangle by the
other. I think they just skipped the
step we call trigonometry and went from “ratios of fractions” to triangles and
there was no need for “thinking in angles”. And not everybody needed to
understand the plan for it was far too complex for the average person of the
times and even now for the average layman in modern times. Consider what
percent of the population actually know the difference between logarithms to
the base 10 or to the base e=2.71828?
The
Year Model will use two annual
periods from the Astronomical Almanac.
In the graphic 3.0 below, the first two relationships and the fourth one
listed therein, use the tropical year for the results. The third equation develops the south side
alone and uses the anomalistic year (perihelion to perihelion, or maximum to
maximum). It seems appropriate that the longest length of the pyramid on the
south side might use the longest measure of the earth year.
This
is just a trial and the reader will need to understand that we only can accept
these particular assumptions if other checksum results provide confidence these
assumptions are justified.
One
can see in The Year Model
Detail Calculations 3.0 below that the north side (230252.06XX versus Cole =
230253) and the west side in the center column are less than one millimeter
different from the Cole Survey numbers on the far right and each use the
tropical year which is again the definition of the second of time. Only the south side (d8) is calculated using
the anomalistic year. If we were attempting to describe some complex chemical
process, it would not matter whether this has any scientific basis….it simply
produces very good results. Note in the yellow highlight that the 4th
equation above is simple turned around to solve for the length of the north
side and it yields 230252.0625.
The
maximum deviation of The Year Model
with the Cole lengths is less than 2 millimeters. In comparison the other three
are almost exact. The circumference of The
Year Model is just 0.07 millimeter greater than the sum of the Cole
numbers. Just imagine how incredibly difficult it would be to set 12 ton rocks
to that sort of precision, if The Year
Model is sustained. One must
keep in mind the total perimeter is nearly a million millimeters around the
base of the pyramid. That is not something we would attempt today no matter how
many cranes and computers one may be able to employ. As an experienced project
engineer and manager of multi-million dollar projects, I simply would not agree
to even attempt such a project without a direct pipeline to Fort Knox
Let’s
back up and summarize what has been given up to here in this analysis. We developed the north length based on
nothing but the defining tropical year of 1900 AD and it “happens to use
logarithms to obtain the Cole Number” but nothing says it had to be inches in
the beginning or even use logarithms.
Two of the three remaining sides also use the tropical year as a ratio
of the north side. The south side uses
the anomalistic year as a ratio of the north side too. Nothing in these assumptions says the design
used inches in the beginning. We use apparent
inches only to hit the Cole numbers which were done in millimeters and
make it easier for the reader to remain mostly in our modern mental
process. This is just a way to choose a
very precise trial estimate.
What
are we to think about how close The
Year Model numbers are to the Cole numbers? If I had also done the Cole Survey and
generated the survey numbers everyone would be calling foul that it was likely
fudged or something. Even if the
builders could have designed and executed the construction with precision, it
seems hard to believe that Cole could have even surveyed it and come out this
close considering most of the outside casing blocks were removed centuries ago
and he was relying on his interpretation of the corner socket marks. By this analysis, his assumptions may have
been more accurate than even he believed in the final report. Cole would not
have reported the numbers to the nearest millimeter if he had not believed that
it could just be possible that the pyramid was built with that type of
accuracy. Many people seem to have this type of feeling when doing
research of the Giza Complex. Certainly the models in other areas of the
pyramid indicate exceptional precision.
If
this trial Year Model is going
to gain respect in the academic community, there will need to be a revelation
of major brilliance that is overwhelmingly convincing. What could even be contemplated in that
regard? If we assume that the design
basis may have been developed by a mind similar to a savant, then we should
expect a serious type of puzzle to unfold.
And it seems like it would have to be based upon dimensionless constants
at least initially.
At
least in the modern era, it is relatively easy to try various things with great
precision and therefore the discovery may not require too many trials. It would be best if we could find things that
seem to continue the Year Model’s
basic concept. And if we can offer proof
of that, we need to find relationships that indicate why intelligence would use
this type of system for their design criterion.
The
following graphic The Year Model
Secondary Calculations 4.0 is provided for only the most inquisitive reader and
one who is mathematically proficient. If
you don’t fall into those categories, just skim over it and rely on others to
find any errors or omissions. It is just like the graphics above but uses the
Astronomical Almanac data to further calculate Year Model Secondary values for diagonals and areas.
It expands the simple model with the law of sines and cosines from ninth grade
trigonometry and algebra. The original builder
design did not need these tools and they are used here simply to communicate
with the reader in modern day experience.
The
following graphic Dimensionless Ratios 5.0 gets at the central issue of the
confirmation of the design criterion.
These dimensionless ratios
using widespread secondary results capture the essence of a major
communications effort by the design and building of the Great Pyramid. All these calculations result from the
initial single assumption that the north side is measured in units that are a simple
function of the tropical year. The units
don’t really play a role in these ratios. If one removes the square root of
ten, 86400 and 10^7 then it simply says all kinds of ratios are
equal to each other. The fact that they all can be converted to a year-like
number is a secondary benefit towards convincing the reader of “design intent”.
One
can see that the dimensionless ratios come up with very similar results
comparatively using five different methods but in a sense very similar in
format. Note that the diagonal ratio seems to point
to the other two area ratios with the same result of 365.248. The term areanorth and areaeast are above one diagonal and areawest
and areasouth are below the same
diagonal. The area ratios are squared
terms and the diagonal ratio
is a fourth power. Just as if one was
trying to keep it all dimensionless. This demonstrates the versatility of the
potential design and something that one would not expect in numbers resulting
from errors and accidents.
There
is no chance to fudge the numbers since the original input into the Year Model come from the Supplemental
to the Astronomical Almanac. There is
nothing to manipulate with a computer program or fancy graphics. Essentially, we start with a year-like number
and perform geometry upon it and end up with a year-like number to demonstrate
it was intentional. What could be more simple and eloquent?
It
seems possible that the pyramid not only does something physical but may use
the systematic precision to do something spiritually. It seems the precision is
higher than required if the acoustical and electromagnetic wave functions
actually perform some complex process. Then I realized that the designers would
probably not know just when our modern civilization would awaken enough to be
able to analyze this construction. If
they guessed to the nearest thousand years, it would still provide the results
above. But why pick 365.247XX instead of
say a projection of our modern number of 365.242XXX?
Then
I remembered that the earth orbital period is constantly changing a little
amount every thousand years or so. What
if this number 365.247XX was the tropical year many millennia in the past? If I went back and substituted astronomical
data from years far in the past into the same basic Year Model and these ratios then became the same number
reflected in the dimensionless ratios above, then that might be the message the
designers intended to send. It could be as simple as reflecting when the
pyramid was designed or the pyramid design basis or even some significant time
event they felt worthy of calling attention to.
The
reader with astronomical training can use their own means to find the tropical
and anomalistic year for periods in the past.
I used the JPL program available online but somewhat complicated and
easy to make an error. There is also a
website that provides a spreadsheet calculator to do much of the same and I
used that to check my original data. It
really doesn’t matter if the date in the past is -71,000 or tens of thousands
of years different. It is the order of
magnitude that is of primary interest.
This is, however, about the time it is widely thought that the last ice
age began and it is also when a major earthquake happened in the Pacific. But there is still a lot of room to search
for why the Great Pyramid designers would put this type of communication
potential together. I suspect it is only the beginning of a very long academic
journey for us.
It
simply could be a central message that is developed further with the details
inside the Great Pyramid. There are
thousands of numbers inside that likely have connecting meaning. This Year
Model simply uses the four sides of the base of the pyramid as surveyed
by Cole.
In the chart below one can see that the year -71000 was used to develop
the tropical and anomalistic year. These
two numbers were then substituted into the identical equations from above and
revised Year Model numbers
developed to compare with Cole and also for new ratios further below.
The
Dimensionless Ratios 7.0 above is the same diagram as the prior 5.0 but simply
substitutes the year values into The Year
Model from -71000. The idea is
to see if when one uses “input years from -71000” do these “output dimensionless ratios” become more closely equal to
the input and they in fact do.
In mathematics this is called convergence. This step indicates that the design is much
more likely to have been deliberate and follows the “puzzle mentality” of the
designing intelligence.
In
case the reader is still not convinced that these dimensionless ratios cannot be
just accidents, one can check below using a parallel in perimeters (perinorth =
perimeter of north triangle) instead of areas for the same four triangles. Note the top left is the very similar
365.246XXX. The others are shown primarily to demonstrate that they exist and
that they don’t change very much with time like the areas did above.
One should definitely not
expect these ratios to occur accidentally.
The bottom three are again dimensionless ratios but the first one is
not. It is becoming quite apparent that
the design was purposefully set to have a huge amount of “discovery curiosity
opportunities”. For someone
investigating the Great Pyramid Footprint dimensions, there is a high
probability that curiosity would be piqued or tweaked.
In almost an attempt to “make
this system fail” one might attempt to combine the dimensionless ratios from
the areas with those of the perimeters as is done below.(area/perimeter squared
is dimensionless) Here I have left out
the conversion back to a year type number and just concentrated on showing how
they compare with each other. One can
see that the base area numbers seldom agree to more than three digits yet the
combined ratios are identical to six and seven digits.
If
the Great Pyramid had been intended to be square and at 90 degree corners
throughout, but was made slightly inaccurately as many contend, then there ought
to be only rare cases where these dimensionless ratios are so consistently
equal and related back to other earth years.
Who would even think before reading this paper that such a simple in
appearance base footprint for the Great Pyramid could possibly relate to earth
years via dimensions, areas and perimeters?
Here
is what I believe is shown by the number systems above. The designer has a mental attitude most
commonly associated in modern times with a savant. There is simply beauty in the putting
together a number system whereby all sorts of curious dimensionless
relationships are developed. And these
relationships are quite easily related back to the earth year which would be
knowledge that many investigators would have at their fingertips. In other words, the design’s primary purpose
is to catch the attention of people who are studying the Great Pyramid with
precision measurements. Once the captivating numerical fullness of such a
simple geometric figure is realized, then the investigator is locked into a
frenzy to find more and more similar puzzle solutions. Of course, this process
is just communications, not functionality.
It
is the shape of the area that is the ultimate conveyor of information. The shapes of the triangles are controlled in
entirety by the 90 degree northwest corner and the exponential relationships of
the west, south and east sides relative to the north side. That is the whole essence of The Year Model. At least a dozen dimensionless ratios result
from these simple assumptions and have been posted herein. There are probably at least another dozen yet
to be fully explored.
What
about the issue of precision? How could
it be possible to be this precise without the use of modern day laser measuring
equipment? How can one even make a
straight-line for 750 feet? How can one
make a perfect 90 degree angle? In fact, it is much easier to make an angle
perfectly 90 degrees than it is to make one slightly off 90 degrees. Anybody
with a rope longer than half the length of a given line can strike arcs from
each end and the point of intersection is perfectly perpendicular to the
initial line. The north and west boundaries were the easiest to design.
How
could one measure the distances with precision?
Do you really need to measure anything to have precision? All sides are a simple relationship with the
north side. If we have something like a
wire that is the length of the north side, the other three sides are easily
calculated increments of much more manageable lengths. But just like we face in modern times, if we
want a calculated length, we need to know the length of something else so we
can make the distance a multiple of the known length. This is the same loop of uncertainty we experience
in the field of measuring today. As in
the Heisenberg Law of Uncertainty, nothing can really be measured without
knowing the measurement of something else, such as our standard units. In modern times we are gradually moving away
from carefully stored iridium standard bars to definitions based upon the
wavelengths of certain isotopes. This
will make everything dependent upon the speed of light and our definition of
the second. This makes The Year Model’s use of the
definition of the second seem much more important.
Hipparchus
is credited with being the first person “known to have dealt with earth’s precession”
even though all of his actual writings are lost to us now. We know about his work only through the
writing of people like Ptolemy who referenced the early work from the second
century BC. Hipparchus’ work seems to reflect
upon the importance of a tropical year being 365 + ¼ - 1/300 = 365.24667. I am guessing most researchers figure he
meant for that tropical year to be of his own era around 135 BC but maybe he
actually was referencing something written about by others much earlier. Note this value for the tropical year is
amazingly close to the -71,000 tropical year (365.24667484) used to demonstrate
The Year Model influence via time.
We
must keep in mind that we are only talking about the basic dimensions of the base
of the Great Pyramid. There are so many
more dimensional characteristics involved inside and from pyramid to
pyramid. Just think if four little
dimensions can be intertwined so completely, what about everything else. And the total capabilities of these four
dimensions have much to be explored simply by changing the type of year and the
period of time represented by the years. Other models inside the Great Pyramid
confirm that the entire structure is built with near infinite precision.
The Year Model sets the background for essentially completing the
entire outside of the Great Pyramid. One can just reduce the north side length
as you move up the pyramid and have solutions for each elevation. The model at different levels could be
altered to account for the concave action but of course this has no impact on
the socket marks on the bottom which were used in the Cole Survey.
We
are now no longer married to the Great Pyramid and requiring ever increasing
surveys with greater precision. This
model is now purely academic and can be floated with variables in a computer
program and perhaps something fundamental about the earth orbital mechanics
will be discovered. Perhaps we will be
able to more fully decipher what the pyramid designers had in mind when they
left us this splendid monument to all humankind.
For
those who suggest there is not one shred of evidence that the pyramids were
built with advanced knowhow, I suggest they dig into the analysis I have
presented here and on other blogs and then raise all the technical issues they
can. This paper in no way suggests that
the builders of the Great Pyramid of Giza were necessarily alien or of alien
influence. I do strongly suggest that
the Giza Pyramids were designed by someone with outstanding intelligence and that
intellectual thought was executed into the construction. Certainly The Year Model
solidly suggests a basis for the design and suggests the potential reason why
such a basis might be employed for the communication to future investigators.
Conclusion
- The Great Pyramid was
not designed to be square with all 90 degree corners and the construction
simply missed the design by a small amount. It appears the NW corner was in fact
designed to be 90 degrees and it was constructed so close that surveys are
not really sure whether it is exact or the surveys are slightly in error.
- The Great Pyramid base
had a very specific design and that design was executed with great
precision in the construction stage of the process. There is abundant evidence provided
herein that the entire base was constructed with the same amount of
precision as the NW corner.
- Modern design and
construction capabilities have never equaled the Great Pyramid as
described in this analytic model to my knowledge.
- This Year Model seems to indicate
that the Great Pyramid design criteria may use the changes in earth
orbital characteristics from precession to establish dates in the past and
perhaps into the future for either significant events or time periods as
the solar system revolves around the Galactic Core. This Year Model opens up all sorts
of vistas for future research and new theories.
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