(Image: compliments of NASA)
Part 1 Insight
Dedication to Mr. Minoru Kodera
Using three circles, a second geometric ratio of pi:1 is discovered within the Giza site
A short question/answer section explaining what information we may consider leaving for future generations
Khufre’s pyramid is positioned and built precisely, guiding use to the size ratios for Mercury, Venus, Earth, and Mars
Part 5 Saturn tells her story
Galileo discovered and named the four large moons orbiting Jupiter, but what did he see when viewing Saturn?
There are many among us who are physically and psychologically drawn toward the greatest structures ever created by humankind…the majestic pyramids upon the Giza Plateau, Giza, Egypt.
From this group of people, there are several of us who continuously search for evidence to support our personal theories, but what are we attempting to uncover, and how can we present our findings in a logical and convincing fashion?
We risk having the hand of humility point directly toward us while our ideas, theories, and concepts are systematically attacked and verbally critiqued by the opposing side…known to us as the “non-believers”…or should we say those who believe that all is coincidental.
Historically, what has always been presented to the public is the simple theory explaining that the Giza pyramids were built to entomb the deceased kings who ruled during the time the Giza pyramids were constructed. However, there is absolutely no supportive evidence to establish the concept being true. No bodily remains from this pyramid-building era (IV Dynasty) have been found within the Giza pyramids, and there is one simple drawback to the concept of the coffins being pillaged by grave robbers…it is common knowledge that all recent discoveries of undisturbed sarcophagi at Giza have had their lids removed only to find that bodies were never placed within!
If the sarcophagi were not designed to contain bodies of the deceased, then we must ask why the ancients placed these stone coffins in designated areas within the Giza site? Although these theories of old appear to be sensible explanations, we must look in a different direction and evaluate this site using the tools of science in a logical, educated, and modern fashion.
…”The coffins within the Giza pyramids did not contain the bodies of deceased kings, therefore the designer(s) must have ordered these monuments constructed for an alternate reason!”…
Is it possible that the Giza pyramids were constructed in a particular location, position, and size with reference to each other in an attempt to inform us that a message is encoded within this site?
On many occasions, the following hypothetical statement has been presented…
…If the designer(s) were intellectually advanced beings then they would have designed the Giza site to include some advanced knowledge that only a future generation could interpret, but the messages could only be understood after the future generation has reached an equal level of intelligence as those who designed the Giza site…!
This is the most relevant issue to be considered.
Have we attained the level of knowledge required to decode the Giza site? Are we capable of finding any message(s) hidden within these structures? If we are capable then will we accept the possibility that a 4500-year-old civilization gained knowledge in the sciences equal to our present understanding?
Alternately, the second scenario is that we may have gained knowledge equal to what is stored within the Giza site. Unfortunately, we have never applied this knowledge to the pyramids because we automatically assumed that the ancients could not have known what we have learned most recently.
The most adventurous thought is to discover if the Giza site actually holds a secret within the design; if proven true, then we are confronted with the greatest mystery of all…who were the ingenious people that designed the Giza pyramid complex?
It was during the year 2000 AD when the late Mr. Minoru Kodera of Japan contacted me via the Internet asking for assistance to confirm his theory relating to the Giza complex. He had presented his findings to the ARCE 2000 Convention (American Research Council in Egypt) and his strong conviction was his total belief that the Giza pyramid site was pre-designed to fit within a set format before the first chisel struck the limestone hillside of the plateau.
His measurements required a most unusual “twist” to the Giza complex…it included a “mirror-image” view of the three large pyramids indicating the alignment of the Sphinx and a circle-triangle combination formed from this unique view of the site (Ill. 1 also visit topic1).
Illustration 1. The mirror-image theory presented to the ARCE 2000 Convention by Mr. Minoru Kodera. A line is drawn from the center of C7 perpendicular to the mirror-image line of L1. It points directly to the head portion of the Sphinx (see inset enlarged view, also visit topic1 ).
Mr. M. Kodera died in March of 2001 and he will be remembered for his great insight and devoted studies in the subject of Egyptian pyramids and monuments. It is my honor to dedicate this topic in memory of this distinguished and respected gentleman. The following will further substantiate his theory and delve deeper into the mirror-image view of the Giza site, to a point where hidden knowledge surfaces in abundance…with certainty you will be asking yourself…
…How did the ancient pyramid builders know?
Were the Giza pyramids built with reference to east and west?
To begin…the three large pyramids are named after three Egyptian kings from the IV dynasty, form the largest structure to the smallest they are as follows: King Khufu (Great Pyramid to the north), King Khafre (center pyramid), and King Menkura (smallest to the south). To reduce the confusion of names and simplify the ease in reading, the pyramids are numbered in order of decreasing size as follows: P1, P2, P3…
Topic 1 of this web site was designed in the later part of the year 2000. It includes the actual measures of the Giza site by Sir F. Petrie and confirmed by H. Cole of the Egyptian Survey Department. The main goal of topic 1 was to incorporate computer aided drafting (CAD) to the site’s dimensions and prove if M. Kodera’s theory was true or false. It was during this exercise that a closer examination of P1 indicated it being constructed almost in the form of a trapezoid, having the north and south sides built nearly parallel to each other. This instigated the experimental turning of the complete site to align the south side of P1 with the true east-west direction. Although the difference in measure was minuscule, the results were amazing, showing two circles drawn from the lower base line of the mirror-image triangle, one perpendicular to the south side of P1 and the other perpendicular to the south side of P2. Both circles have a radius equal in measure to the corner of their respected pyramids and both geometrically intersect at the same location…on the west center side of P3; a remarkable display of simple geometry within the Giza design (Ill. 2)
Illustration 2. The two circles intersect on the west side of P3. This is only possible when the Giza site is turned having the south side of P1 aligned in the true east-west direction, indicating that Earth’s axis may have moved since the Giza site was construction.
The two geometric circles (C4 and C6) meet on the west base line of P3. At this location a most peculiar shaped granite stone is positioned against the pyramid side. Did the designer(s) leave this large stone in its rough format intentionally for us to investigate the reason why? Does its location confirm the geometric circles as proposed above? (Photo 1).
Photo 1. Located at the center and several layers above the west base of P3, this odd shaped granite stone may have been positioned to confirm the location where circles C4 and C6 meet.
This is the first evidence indicating that the site may have been built using Earth’s east to west coordinates for constructing the pyramids, but insufficient to convince the skeptics…more had to be uncovered.
By combining Mr. Kodera’s triangle/circle alignment with the Sphinx (Ill. 1) and drawing the maximum sized circle using the center points of the pyramids as its boundaries, a new ratio of circular measure was discovered…the ratio of C6/C7 is equal to 3.14:1, also known as pi:1 (Ill. 3).
Illustration 3. The size ratio for the circle C6 to C7 is 3.14:1 (pi:1)
Is it possible that the designer(s) built the Giza site and intentionally located the position of P1 and P3 to form these geometric circles to produce the pi:1 ratio?
To confirm my measures, I asked Dr. J. Saringer, of Stouffville, Ontario, Canada to evaluate the above 52.09 degrees angle without him knowing the application.
The question was presented as follows…
…“What angular value will an isosceles triangle have when two circles are drawn; one circle to intersect the outer points of the triangle and the second circle to be drawn tangential to the side, base, and the perpendicular line drawn from the base to the apex; the outer circle having a ratio of pi:1 with the inner circle?”…
The question is most complicated to understand, however, Dr. Saringer calculated the answer almost immediately. The value was 59.39 degrees for the two equal angles and 61.22 for the apex. It was to my surprise that he had not agreed with my angular measure, I had assumed that only one angle could produce this ratio…now there were two!
After closer investigation, it was confirmed that only two isosceles triangles can be drawn within a circle to form a pi:1 ratio as described and their true base angular measures are 52.11 and 59.39 degrees. Now the challenge was to include the second triangle calculated by Dr. Saringer within the same circle (C6) and search for other alignments within the pyramid site. This would indicate the possibility that the ancients were also aware of the two angular measures producing the pi:1 ratio (Ill. 4).
Illustration 4. The second isosceles triangle fitting within the outer circle (C6) and having an inner circle (C8) drawn tangential to its sides. The ratio of C6/C8 is 3.14:1 (pi:1), a duplication of the C6/C7 example. Note the line L2, it is drawn tangential to the circle inscribed within P2.
The second triangle forms the geometric boundaries for C8 having a radius basically equal to C7. The 59.33 degrees angle is the true measure when the line L2 is intentionally drawn tangential to the circle formed within P2 (Petrie’s measures). The result in angular measure is a mere 0.1% from the value of 59.39 degrees calculated by Dr. J. Saringer.
What we are witnessing is the possibility that the designer(s) intentionally positioned the three large pyramids in a pre-designed location and constructed P2 having a base measure to accommodate the geometric triangles forming the pi:1 ratio of the outer circle to the inner circles. The first example of the pi:1 ratio is most visible, involving P1 and P3, but required the direction from Mr. M. Kodera and many months of calculations and comparison of measures before discovery.
Dr. J. Saringer calculated the second example, requiring extensive mathematical knowledge in geometry to accomplish. But it required a “trial-and-error” approach to discover the triangle being tangential to the geometric circle inscribed within P2.
Were the three large pyramids positioned to express these known angles formed from the pi:1 ratio? This would certainly indicate that the designer(s) fully understood the formula/ratio of a circle’s circumference to its diameter equaling 3.14:1…the numerical value of pi…the most logical numerical ratio any mathematician we would choose to represent circular measure.
This is the most difficult question to answer, because there are many answers that fulfill each individual’s needs. But it is here where we must weigh what is relevant to the cause of passing on knowledge.
If we were to build a structure(s) without any written guidance, yet indicate our gained knowledge, it must…by default…be a message composed from mathematical/scientific relationships and measures. It would be most foolish for us to attempt including names, places, and historical events created by the hands of humankind…this type of information would be most irrelevant. We would want to tell the future what we have learned…future generations would not be interested in what we have stumbled through during our evolutionary infancy period. They will be seeking the truth; searching for clues to indicate when we reached certain levels in the sciences; we are constantly attempting to seek the same through archeological investigation.
We would include basic information in the sciences, mathematics, physics, and measures, but most important…our growing knowledge of the unknown as we presently perceive or understand it.
Below is a simple example of demonstrating a known ratio:
…We learned from Archimedes that all materials have a specific gravity measure, and we compare these measures to that of water.
Were the designers aware of specific gravity?
There are three large pyramids at Giza: two almost equal in height, and the third built approximate one half the height of its neighbors. Was it the designer’s intention to demonstrate these “two plus one-half monuments” of stone to represent the number 2 ½…? Because the specific gravity of stone is approximately 2 ½ when compared to water…! That could be a coincidence.
Part 1 of this topic suggests the designer(s) being fully aware of the circumference/diameter ratio for a circle being 3.14:1 (pi), a simple means to indicate they understanding circular measure. This information can also be used to strengthen the theory that the side angle ratio of P1, the largest pyramid at Giza, includes the pi value. This giant structure has a measured side angle ratio equaling 4/pi (1.273:1).
Archeologists and Egyptologists are slowly and methodically presenting new evidence indicating that the ancients understood the number pi almost two thousand years before the “Father of Mathematics” (Pythagoras) was born…now the Giza site is beginning to confirm it! However, it is here where we must question the example illustrated above…
…Why would the designer(s) conceal the pi ratio within the Giza site, making it most difficult for us to discover?
The answer is most obvious to those understanding the sciences…
…If messages/codes are left within these structures then the single mathematical ratio of pi:1 would not be the main feature presented. There are many more important ratios of measure that we can choose to emphasize our gained knowledge, but there is one great drawback…coincidence!
If we know a ratio for a specific measure, then it is most probable that we could find a location within the Giza complex to demonstrate that ratio.
The first example above (Ill. 3) shows the pi:1 ratio (C6:C7)…this could be coincidental, but illustration 4 also uses C6 with a different inner circle (C8) reducing the possibility of coincidence in measure. The more examples of the pi:1 ratio that we uncover then the higher the possibility (less coincidental) that the pi ratio was known by the designer(s) and intentionally set within the boundaries of the site.
Knowing what is required when leaving a message(s) for the future, we can now proceed in creating our own design. From above we have learned that repeating a measure many times reduces the possibility of coincidence, but there is an alternate method that is more effective.
Rather than repeating a single ratio, we could elect to choose a set of objects having unchanging values that are easily compared to each other. This serves as a double function; it confirms our understanding of a particular situation while allowing us the opportunity to express our gained knowledge in this subject. But this presents the greatest obstacle of all…what information do we leave?
…The choice is yours !
Now that we understand how to express our gained knowledge using these techniques, we can begin to apply it at the Giza site and see if the pyramid designer(s) did likewise!
If the designer(s) were intelligent then we must begin to search the site for intelligence…using intelligence!
Topic 3 of this web site (The Perfect Alignment) illustrates the orbital ratio for the two planets: Earth and Mars; it also shows how the three large pyramids are set in a position to represent their respective distances from each other. The most odd feature was the physical size ratio for these three pyramids being similar to the measures of the planets: Mercury, Venus, Earth, and Mars…known as the inner planets. Unfortunately, the pyramid measures were not exact or identical to today’s measures. Therefore it is only fair that it be considered no more than coincidental (Ill. 5).
Illustration 5. From Topic 3…Using P1 to represent the diameter of Earth, then the physical size ratios for the remaining three inner planets are compared to the three large Giza pyramids.
Following the discovery of the second triangle forming the pi:1 ratio, instigated a renewed interest in the mirror-image view of the pyramid site as proposed by Mr. M. Kodera. After several weeks of investigation, it was discovered that the actual size ratios for our inner planets do exist within the Giza site. The ratios for the planets: Mercury, Venus, Earth, and Mars are drawn to scale as shown in illustration 6.
Illustration 6. The size ratio for the four planets: Earth, Venus, Mars, and Mercury.
Did the ancients know the actual size ratios for our inner planets? (Ill.7)
Illustration 7. The base lines for the two triangles used to demonstrate the pi:1 ratio are utilized to indicate the center locations of the two planets: Earth and Mars. The center of P1 is the center location for the planet Venus, while Mercury is discovered using basic geometry. All four planets have their circumferences cutting through the corner points of P2. Note: all numerical values are measured in Royal cubits (Rc) having P1 set at 440 Rc in wide.
It is without doubt that the designer(s) used an elaborate approach to demonstrate the size ratios for the first four planets from the Sun…they incorporated the four corner locations of P2. To be more precise, the designer(s) positioned and built P2 to a specific size for demonstrating the physical size of our closest planets.
Using the base line of the smaller triangle proposed by M. Kodera, the circle representing Mars is the first drawn and intersects the southeast corner of P2. Earth is drawn using the base line of the larger triangle calculated by Dr. J. Saringer and intersecting the northeast corner of P2. Venus is located at the center of P1 and intersects the southwest corner or P2, while Mercury, the smallest of the inner planets, uses three significant location points: the intersects of the northeast and northwest corners of P2 and the tangent to the base line of the smaller triangle.
Comparing these dimensional ratios with modern–day measures they read as follows:
Mercury/Mars 493/684.8 = 0.720…actual 0.718…total error = 0.28%.
Venus/Mars 1220.7/684.8 = 1.783…actual 1.781…total error = 0.08%.
Earth/Mars 1284/684.8 = 1.875…actual 1.878…total error = 0.16%.
Note: All measures are within one half of one percent error. This is certainly an ingenious means to display mathematical, geometrical, and astronomical knowledge.
If the ancients were fully aware of planet sizes then we are confronted with an issue to be resolved. How did they know these measures, because we only learned how to measure them to such accuracy in the latter part of the twentieth century?
Most important is the following: Venus has a dense atmosphere, preventing us from measuring its actual diameter unless radar or laser-measuring equipment is used…yet this is the most accurate ratio measure demonstrated at Giza…truly…a remarkable feat!
It was most exciting to discover these planetary ratios, but there was one measure that should have been presented if the ancients were aware of planet size ratios…
We are missing our closest companion…the Moon (Ill. 8).
Illustration 8. The physical size ratios for the planets: Mars and Mercury compared to our closest neighbor…the Moon.
To draw a circle representing the size of the Moon compared to Mars is most simple, but attempting to discover if it were implemented within the Giza site required a tremendous amount of trial and error. There are several possible locations; unfortunately their measures were greater than 1% from perfect.
Since the first four planets were associated with the corner locations of P2, it would be logical to assume that the Moon would also be geometrically involved with the same pyramid…and so it was discovered (Ill. 9)
Illustration 9. The size ratio for the Moon compared to the smaller planets: Mars and Mercury. The circle representing the Moon has four geometric locations: the northwest and southeast corners of P3, the tangent to the circle inscribed within P2, and the center location of the circle representing Mercury…a most complex means to illustrate the Moon/Mars/Mercury ratio.
Again, we compare the actual measured ratios with present-day measures:
Mercury/Moon 493/352.3 = 1.40…actual 1.403…total error = 0.22%.
Mars/Moon 684.8/352.3 = 1.94…actual 1.95…total error = 0.56%.
The size ratios for all of our inner planets, including the moon of Earth, are geometrically designed within the Giza site. If these measures are to be considered as coincidental then the mathematical chance of it occurring, within the accuracy stated, is infinitely high.
However…our search continues for greater knowledge and understanding.
Almost two thousand years after Pythagoras, the brave and courageous Galileo was to suffer the identical fate of his counterpart…both were sentenced to house arrest for the remainder of their days. These two men had committed much of their life to mathematics, science, and astronomy; both had also learned the workings of the cosmos. They knew and were determined to remain steadfast in their belief that the Sun was the center of our solar system, and our home (Earth) was no more than another “wanderer” in the heavens.
Their sentence was less severe than anticipated…blasphemy against God normally meant certain death by decree!
Pythagoras made claim to his findings following his second visit to the Giza pyramid site. Galileo was more fortunate; he was the first to use the telescope and record his observations of four moons orbiting the planet Jupiter. This was the final evidence required to prove that the “wandering” stars orbited the Sun.
Galileo named the four large moons of Jupiter in order of size and in honor of himself: Ganymede, Callisto, Io, and Europa.
From Ganymede… Ga
From Calisto… li
From Io… io
From Europa… Europe.
Combined we have Ga..li..io Europa …Galileo of Europe !!
Searching deeper into the heavens, Galileo was able to record the mysterious multi-colored rings encircling the planet Saturn. Similar to the four moons of Jupiter, these rings had eluded humankind since the beginning of time. Saturn is the furthest planet from the Sun that the average human eyesight can detect without the aid of special equipment; however, the rings of this planet are definitely invisible to the naked eye.
Many years have passed since these rings were first discovered; they are identified using a systematic method incorporating the letters of the alphabet. Their measures are from distinct locations using the radius of Saturn as a “ruler” for measure (Ill. 10).
Illustration 10. The major rings of Saturn drawn to scale.
It should be emphasized that we …Homo sapiens…are unable to visually identify the many distinct rings surrounding Saturn without the aid of magnifying equipment. Yet the illustration below shows the most amazing comparison of Saturn’s ring locations when placed upon the Giza site. The scaled drawing has Saturn positioned at the center of the base line suggested by M. Kodera with the planet’s outer dimension tangential to the larger triangle suggested by Dr. J. Saringer’s calculations. The rings fit within the Giza site design when using the identical points of measure as defined by modern astronomers! (Ill.11).
Illustration 11. The two triangles forming the pi:1 ratio are used to set the center and outer dimension of Saturn. The main rings of the planet are clearly identified within the Giza design. Astronomers use these identical points of reference for measuring the ring distances from the planet.
The question is not “How did they know?”…it should be…“Is it possible that our ancient forefathers had a clearer view of the heavens above than we have today, allowing for accurate measure of our neighboring planets and the rings surrounding Saturn?”
Answer…It is most improbable, but it cannot be dismissed as impossible.
Four thousand five hundred years have elapsed and now another chapter is added to the many mysteries locked within the Giza pyramid complex.
There are at least two sets of ratios presented at Giza indicating advanced knowledge embedded within the pyramid designs. The first example demonstrates the two distinct triangles formed within a single circle and producing the mathematical pi:1 ratio. The second example illustrates the physical size ratios for the first four planets from the Sun, including the moon of Earth and all measures involve at least one geometric reference location of P2.
Although we can assume it most improbable for the ancients to have known these measures, the Giza site includes other dimensions that were left for us uncover…the size ratios for the “invisible” rings of Saturn; the furthest visible planet the average human being can observe from Earth.
Were the ancient Egyptians fully aware of planetary size ratios by means of visual observation, or had the designer(s) of these pyramids learned from others?
Most important, we should be asking ourselves… “Could we have imagined implementing these planetary examples within our own demonstration of gained knowledge?”…
Personally…I doubt it!
What have we learned from above? …It depends upon us!
If our minds are closed then we have learned nothing. If our minds are open to suggestions, then we must continue to search for further evidence until all doubt is removed.
Until that time we must give the designer(s) credit for their efforts and consider they knew much more than first anticipated, if not…then we err by default by assuming they knew very little!
1) Abstract of The 51st Annual Meeting of the American Research Center in Egypt, pg. 57……………Minoru Kodera
2) “1o6” The Dawn of Man, 1999, sbe……………Clive Ross
3) Larousse Astronomy, 1987………………………….Philippe de la Cotardiere
4) NASA Jet Propulsion Laboratory……………….www.jpl.nasa.gov/