Author Clive Ross examining the granite base stones of Menkaure’s pyramid 1988.
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Curious minds are filled with detailed stories and myths relating to the Giza pyramids…where, when, how and why did it all begin? Who were the instigators of these fables and what was their gain?
Egyptologists theorize that the three largest pyramids on site were dedicated to three successive kings who reigned during their construction. The pyramids vary in size and style; within each laid the body of the deceased. They claim that the three large pyramids at Giza were designed as final resting places…as majestic tombs built in memory of the people’s living god…their king!
Many “alternates” question the assumed sequence of events and challenge these well-established beliefs. They bring with them as many new theories as they do questions. Their quest is never-ending as they persistently battle for individual supremacy to be the first to uncover the true meaning and reason why the ancients constructed the Giza pyramid complex. Respectfully, these additional theories must also be weighed and investigated to the fullest until all aspects are reviewed and critiqued. And they are…extensively!
The wonders of the Internet have allowed the professional and amateur researcher to compile and interpret all evidence that is presently available…including most recent discoveries. It has opened doors that would otherwise be closed or inaccessible to the average individual.
Computer programs are becoming the tabletop “tools” of modern science; they are required to prepare graphic, artistic, scientific and literary papers. From the home or office to the World Wide Web, their finished product is presented instantly to an eagerly awaiting audience.
What are we to believe…who are we to believe…and…how can it be proven true or false?
Two beliefs…many stories…but few options.
A) If Egyptologists are correct with their assumptions then any and all alternate theories are irrelevant…the chapter is complete!
Their organized approach has successfully recreated the history of Egypt, unraveling the many mysteries from a long-forgotten civilization. Centuries of dedicated work, analyzing and deciphering papyrus scripts, excavating and examining ancient sites and their remains…all combined to create one unique picture of “Egypt the ancient”. Still, there are many gaps remaining and holes to be filled where only theorized events can be used in an attempt to complete their book of history.
Steadfast, this group of professionals defends its beliefs by introducing “hearsay” evidence usually in the form of assumed or irrelevant quotations from other associates or renowned Egyptologists. But the majority of documentation used to support the Giza “built-for-kings” hypothesis is circumstantial. Are these professionals attempting to support a theory that was developed from a mythical story contrived thousands of years after the pyramids were constructed?
Any theory void of unquestionable and irrefutable evidence is far from factual…it is a weak theory at best.
B) If the pyramids were not built to entomb kings then is it possible that the workers were no more than eager stonemasons demonstrating their ability to carve into a hillside, cutting and pulling stone blocks in an organized fashion, creating a staircase to the heavens above? Unfortunately a visit to the site quickly demolishes this concept.
The smallest of the three large pyramids has interlocking granite casing (see introduction photo), confirming that each layer of stone was fully completed before proceeding to the next level. If the goal of the builders were to construct high monuments then why would they build this pyramid half the base measure and height of its neighbors? The completed base dimensions of this pyramid were set in place before construction commenced! Its small size was intentional…a very important observation to be noted.
C) Assuming for a moment that the first two theories are incorrect, then the pyramids were not intended to be tombs, neither did the builders attempt to reach the stars for the simple entertainment of stonemasons or kings. The only remaining option is that the structures and their contents were built as an offering, but nothing was discovered within. Since they were empty when first entered, then what was being offered and to whom?
The three main structures are built to an amazing accuracy in linear measure and align with Earth’s four cardinal points, indicating mathematical skills and astronomical knowledge. Enormous stones were raised to great heights; a puzzle to all who attempt resolving what method was employed. These three basic facts/features of the site provoke the curious to investigate the evidence closer.
Built to last an eternity, they are set on the hillside overlooking the development of civilization from generation to generation. Partially stripped of their external beauty, and haphazardly invaded upon, each of the three pyramids contained a single sarcophagus. And each sarcophagus was void of bodily remains, left for future generations to question why…but…the designers assumed incorrectly…we failed the test!
Patiently the monuments have waited for the development of mind and spirit, for a time when humankind would realize that the only commodity these unselfish ancients had to offer us of the future was…gained knowledge!
But how was knowledge expressed and why?
These two simple questions are impossible to answer unless the designers included a means of communication within the complex and we gaining sufficient intelligence to realize its existence.
What follows is a logical route that is to be traveled cautiously, to the very end…if we ever intend resolving the Giza mystery.
The simple fact is…
…It must be successfully proven that the ancients who designed these monuments were verse in mathematics…!
It is a bold and daring statement, but one that holds true!
Mathematics is “the” universal language; it can express factual stories void of the written word…and the pyramids contain not a single word!
The first item on any agenda of communication is to test the observer’s “reading” skills.
If you cannot see and read the mathematics used then you cannot visualize the stories held within, it is that simple, and it is the downfall of those who attempt decoding the site. Unfortunately the non-mathematicians succumb to the identical pains and woes of the illiterate; their mental vision is obscured and disappointingly…they are unable to see the light!
Modern society lacks both the understanding and skill to visualize what transpires throughout the evening skies. They are aware of the Sun, Moon and possibly several stars, but planet motion remains a great mystery to the average observer. Not so with the ancients!
This author presented a previous topic indicating that the Giza designers were competent astronomers. The proposal was presented in the year 2000 and remains unchallenged (see Perfect Alignment).
An illustration and quick review from that presentation:
a) The three Giza pyramids are set in place forming a distinct 191.4 degrees angle that has no mathematical connotation, yet this angle is identical to that traveled by Mars in one Earth year.
b) The center distances from each pyramid also coincide with the distances ratios of the four rocky planets: Mercury, Venus, Earth and Mars.
c) This brazen display of planetary motion is basic astro-mathematics; it demonstrates planetary distance and orbital ratios and we are aware that mathematics, in any form, is the simple comparison of numbers…ratios.
E.g. Earth orbits the Sun in 365.26 days…Mars, our closest outer neighbor, orbits in 686.98 days. Therefore Mars completes (365.26/686.98) orbits in one Earth Year. A very simple task in division produces an orbital ratio of a 0.532:1.
Our circular standard of measure is the 360 degrees circle; therefore Mars will orbit (0.532x360) degrees or 191.4 degrees…as indicated at Giza!
d) There are no complicated scientific computations. The only requirements were logical deductions from years of recorded data and the ancients were certainly dedicated observers of the heavens; their religious beliefs were entrenched in astrology… fully agreed upon by all!
e) Although the distances and angles are exact, there is no literary evidence indicating these people having the ability to calculate precise planetary data.
f) Egyptologists refute the claim declaring that planetary motion was definitely unknown to the ancient, although they do agree that the name Horus, having several subtitles: “of the East”, “West”, ”Horizon”…etc., indicates definite reference to the heavenly “wanderers”…the planets.
The above measures and ratios of our cosmic order may be incorporated within the Giza design, a unique demonstration of knowledge, but they could not have been expressed successfully unless the primaries of mathematics were fully understood. If the ancients were aware of orbital ratios then they must have attained a mathematical level above the scope of our acceptance.
In the spring of 2005 an oddity in measure at the Giza pyramid site was realized. The King and Queen’s Chamber dimensions from Khufu’s pyramid were used to standardize the Royal cubit (Rc) length, producing a southern base measure of this pyramid equaling 440 Rc (an accepted value). The same was applied to the overall measures provided by W. F. Petrie; all dimensions were transferred into Rc lengths.
It was noted that the actual limiting distances in the east-west and north-south directions of the complex had a “numerical” values equaling 1000 times the square root values of 2 (1.414) and 3 (1.732) respectively, actual measures are 1418.2 Rc and 1732.5 Rc. From these two lengths it can be determined that the hypotenuse would equal 1000 times the square root of 5…or 2236 Rc…the site measure is 2238.9 Rc, producing a minuscule error of 3 Rc or 0.13%. (Ill. 2).
Illustration 2. The measured extents of the Giza site are numerically equal to 1000 times the square-root values of 2 and 3 producing a hypotenuse value equal to the square root of 5.
After sharing these preliminary findings on the Internet, it was brought to my attention that another researcher, Mr. John Legon, had previously discovered and announced the identical measures in the late 1990’s…fifteen years previous to mine! What must be noted is that two researchers, working independently on the same project, realized the same coincidences in measures without knowledge of each other’s work.
There is voiced concern over the theoretical hypotenuse length of 2,236 Rc (square root of 5) compared to the site’s actual value being 3 Rc longer, while the north-south direction of 1732 Rc (square root of 3) shows no error in measure; it is perfect to the closest cubit length!
The answer is clearly evident.
The error in measure is the result of the east-west direction being 1418 Rc; it is 4 Rc greater in length than the true square root value of 1414 Rc.
This introduces a very irritating question:
If the designers were sufficiently skilled in measuring the north-south length to represent the exact value for the square root of 3 then why is there a 4 Rc error for the shorter distance representing the square root of 2? Was the Giza site prepared to demonstrate these square root values or are they no more than phenomenal coincidences?
Petrie’s noted observation of Menkaure’s misaligned pyramid (P3) was considered; this pyramid deviates furthest from Earth’s four cardinal points, possibly resulting in the 4 Rc discrepancy. His reasoning for the irregularity in side measures and alignment was the rushed attempt to complete the structure.
Petrie concluded that when the builders reached the one-quarter-height level (16th course) they ceased to include the external angular granite casing stones. Having little time remaining or the king possibly dying before completion, the structure was finished without granite casing stones enshrouding the upper three-quarters of the structure. But, as previously mentioned, the casing stones interlock with the core stones. Therefore each layer was completed before progressing to the next level, doubting that the initial layer was misaligned due to hurried construction.
The alignment of Earth’s four cardinal points can be obtained within any sunny eight-hour period…any day of the year! Furthermore, by combining all that has been observed of Menkaure’s pyramid and realizing the accuracy of alignment for the other pyramids on site, indicates this structure being intentionally offset from the true north-south direction…left for us to determine why.
Note: Before venturing further…
It must be realized that the three measures, if by design, confirm that the Giza pyramid complex was pre-designed before construction commenced. It introduces the possibility that the designers understood the square root value of basic numbers…and most relevant…they used the base-ten numerical system…two levels of mathematics never believed possible. Regrettably, it is only physical evidence; there are no documentations from this era to substantiate these findings.
The reader must understand what is being implied and question how the designers could have accomplished this feat.
Although difficult for us to comprehend, it was very possible for them to have discovered these values without mathematical formulas. Simple square root values can be attained using circular and linear geometry (Ill. 2).
Illustration 2. Using geometry to determine square root values.
Most complicated to the novice but common knowledge to the mathematician; geometric circles and squares produce the three numerical square root values that form the extents of Giza.
For the mathematically curious:
a) Draw a circle R1 of radius 1 unit.
b) Create a square of sides 1 unit
c) Draw the diagonal of the square to form the square root of 2 = 1.414.
d) Draw a circle R2 of radius 2 units.
e) Extend the base of the square to intersect circle R2 to form the square root of three = 1.732.
f) Draw the diagonal from the square to the diameter/circumference intersect of R2 to form the square root of 5 = 2.236.
The values were chosen wisely.
While mathematicians would normally select the 3-4-5 ratios to demonstrate a right-angle triangle, the Giza designers elected to use the lowest set of irrational numbers that become integers when squared. The intention was to demonstrate they understanding the same formula that Pythagoras introduced to the world nearly 2,000 years after Giza was constructed!
From Pythagoras (a2 + b2) = c2
becomes (1.4142 + 1.7322) = 2.2362
equals (2 + 3) = 5
To calculate the numerical value of a root number can be achieved by “trial and error”.
Find a number when multiplied by itself will equal 3:
1x1 = 1
2x2 = 4
…the answer lies between 1 and 4
17x17 = 289
18x18 = 324
…the answer lies between 17 and 18
173x173 = 29929
174x174 = 30276
…the answer lies between 173 and 174.
1732 x 1732 = 2999824
1733 x 1733 = 3003289
…the answer lies between 1732 and 1733.
The square root value of 3, to four decimal places, is 1.7325!
The pyramids are built to extraordinary precision, and building to this accuracy would have been a difficult undertaking without full knowledge of the right-angle theory, a point clearly identified by Petrie. Is it possible that the designers understood square root values and the right-angle correlation of squared numbers?
Basic geometry and simple multiplication was illustrated above to calculate the square root values, but to demonstrate geometrically the square root of 3 required the introduction of the circle, and the circle has its own set of rules.
Possibly the greatest mathematical breakthrough in ancient times was the discovery of the circle’s circumference ratio to its diameter…p(pi). Although it was introduced to the Greeks through the teachings of Pythagoras (600 BC) there is evidence indicating it was used in India several hundred years prior.
This fixed invariable ratio is used for all circular measures of circumference, area and volume of a sphere.
Irrelevant of a circle’s physical size, the distance around the circle (circumference [C]) will always remain 3.14 times longer than the distance across the circle (diameter [D]). Therefore 3.14 diameter lengths equals the length of the circumference …or 3.14xD = C. Pythagoras designated the Greek letter p to represent the 3.14 ratio leaving us with pD = C, or 2pr = C (r equals the radius of the circle) (Ill.3).
Illustration 3. Irrelevant of size, the diameter (D) can be stretched around the circumference (C) 3.14 times (22/7 or 3+1/7th).
There have been many attempts to prove that the designers of the Giza site were aware of the p ratio, but each proposal has been systematically rejected. The reason is clearly explained by Egyptologists: …there is no factual/documented evidence indicating that the ancient pyramid builders understood the circumference/diameter/p relationship and any ratio discovered at Giza is purely conjectural and/or speculative.
Although several fundamental measures produce this ratio, there are hundreds of combinations that exist. Therefore it is true, Egyptologists are correct; some ratios may only be coincidentally similar to the p value.
The responsibility is now in the ”alternate’s” court to supply unquestionable evidence (Ill.4).
Illustration 4. Two examples demonstrating the pi ratio at Giza…coincidence?
Both sides are deadlocked; there are no means to resolve this simple mathematical dilemma…who is right…who is wrong!
It is time to think like true designers…a time to take another cautious step forward.
If the ancients understood the p ratio then they too would realize that a single or double example of its value could be construed as coincidental…simple logic.
Secondly, this ratio would only demonstrate knowledge of the linear relationship between the circumference and diameter of a circle…very basic indeed.
Placing ourselves in their position then mathematicians would not choose the p ratio to demonstrate circular measure; they would elect to use the volume formula of a circle; it fulfills two requirements in one.
It proves that p is a known factor (initial objective) and it would refer to the most complex formula for a circle…volume of a sphere. But how does one express the complicated formula for a sphere (4/3)pr3 as a simple ratio?
This brings us back to the very beginning…
Recalling the three pyramid’s 191.4 degrees separation to demonstrate planet orbit ratios…these ingenious individuals duplicated the feat to express the complicated spherical volume formula that includes the p ratio!
Discovering the formula for spherical volume was no easy task, but logic tells us that the ancients confirmed their measures/findings by comparing its volume to other shapes of identical size.
Rephrased: What is the volume ratio of a sphere compared to a cube having identical external measures?
The answer is…6/p (Ill.5).
Illustration 5. Volume ratio of a cube compared to a sphere having identical widths.
And so it was done…the designers set their three pyramids into position using the volume ratio of the cube compared to the sphere. Their choice was restricted…it was the only logical ratio they could use that removes any doubt of their understanding “all” characteristics of the sphere.
All mass in the cosmos is spherical by nature, including the stars and planets. The cube/square cannot exist in space naturally; it will automatically revert to the shape of the sphere/circle…the universal ratio is…6/p (Ill.5).
Illustration 6. Measuring the center points of the pyramids in the diagonal direction produces the 6/p ratio of volume.
Coincidental…not at all!
To confirm, they chose an overall length (blue line) equaling 1774.5 Rc…or…1000 times the square root of p (1.773)!
How was it accomplished?
a) The designers initially set the 6/p ratio between the three structures to demonstrate p and the cube/sphere volume ratio.
b) They confirmed they knew the p ratio by making the distance between P1 and P3 centers equal to its square root value of 1773.
c) The 191.4 degrees angle alignment was used to demonstrate Earth and our closest outer neighbor Mars…(planet motion).
d) Then the pyramid’s base widths in the north-south and east-west direction were adjusted to produce the square root values of 3 and 2 (1732Rc, 1414Rc), demonstrating knowledge of the right-angle formula.
A magnificent and neat mathematical package…wizardry at its finest…but a story that is not complete…yet!
To be continued….
1. Special mention to Michael Saunders for his contribution and confirmation re: Petrie’s measures. http://www.archaic.freeserve.co.uk/
2. John Legon http://www.legon.demon.co.uk/
3. Exploration of the Universe…fifth edition……..Abell, Morrison, Wolf
4. “1o6” The Dawn of Man…sb.…..……………..Clive Ross
Copyright 1997 CIPO 459689 ISBN0-9686006-0-3