Interlocking granite casing stones cover the lower base area of P3.
1 Step 3:
2 Step 4:
3 Step 5:
The discovery of several circular and spherical ratios in measure involving pi and 6/pi indicate advanced mathematical knowledge. Unfortunately these ratios are fully dependent upon Petrie’s measures.
Although his work and that of others agree in accuracy within inches of each other, they all use the same reference corner markings. If any of these assumed locations are incorrect then it follows that most measures/ratios are also incorrect and this presents another issue to be resolved: are Petrie’s measures reliable?
Realizing the destructive forces of humankind, the designers intentionally locked the corner stones of Khufu’s pyramid into position to prevent removal. Petrie cleared sand, stone and debris that had concealed the corner locations; there was no doubt, the builder’s reference points had remained unaltered since construction. The task was demanding; almost fifty reference points were established throughout the site requiring several seasons to measure, re-measure then confirm (Ill.7).
Illustration 7. Petrie set approximately fifty reference points to establish each pyramid’s corner locations.
The designers were equally ambitious. Having incorporated numerous scientific and mathematical ratios within the pyramid structures they realized that these values could still be construed as coincidental; they had to convince those with reservations or refusal to accept the obvious.
The simple truth is that the whole of the Giza complex was designed before the first stone was ever placed into position; its boundary limits confirm this fact. In one simple move, the designers elected to incorporate the pyramids calculated width ratios to coincide with other extents within the site (Ill. 8).
Illustration 8. The north-south extents, with reference to the center of P2, produce the identical width ratio of P1/P2.
The north-south direction is unevenly divided by the center of P2; it confirms the intended base ratio of the two large pyramids. Measuring from the northern extent of P1 to the center of P2 (896.1 Rc) and comparing it to the distance from the southern extent of P3 to the center of P2 (837.4 Rc) produces a ratio equal to 1.07:1. The two pyramids share the identical ratio of 440/411.3 = 1.07:1.
If these two equal ratios were intended then it gives the investigator a reason to believe that an additional confirming measure for the third and smaller pyramid may also exist.
It required little effort to realize that a basic numerical ratio certainly does exist between P1 and P3.
The north-south distance from the center of P1 to P3 is 1411.8 Rc; totaling seven full lengths of P3 (7 x 201.5 = 1410.2) (Ill. 9).
Illustration 9. The north-south distance from P1 to P3 equals seven lengths of P3.
There is a distinct error of 1.3 Rc between both lengths, but Petrie explained of his difficulty measuring the precise location of P3’s northwest corner. Had he measured the west wall of P3 to be 0.2Rc (4 inches) longer then the ratio would be exactly 7:1. Perhaps it is a measure that modern surveyors may elect to investigate and finalize the complete set of dimensions for Giza…it would also satisfy the concerns recorded by Petrie!
The center-to-center distance from P1 to P3 is now set at seven pyramid lengths south of P1. However, the total corner-to-corner distance of P1 to P3 along the east-west axis remains at 1418 Rc, seven Royal cubits longer than the north-south direction. The designers wanted to produce an additional mathematical ratio using this horizontal length, unfortunately the 1418 Rc distance was invariable; it was set for other reasons.
The north-south distance is seven (7) pyramid measures in length and it is noticed that the east-west distance is seven (7) Rc longer. If the width of P3 was increased by 1 Rc (201.5 + 1) = 202.5, then it would also divide equally along the east-west axis seven (7) times, (7 x 202.5) = 1418. But how can a pyramid increase in width when its dimensions are fixed?
Answer: Rotate the pyramid !
As previously mentioned, P3 is unique in nature. Although it’s four sides are right-angled to each other they are offset from the four cardinal points of Earth. Basically, the base of this pyramid was intentionally rotated 0.28 degrees; the designers manipulated the structure to fit their needs (Ill. 10).
Illustration 10. One of P3’s four corners (red) showing the 0.28 degrees clockwise rotation from true east-west (black).
By rotating the base 0.28 degrees it produced an effective linear measure along the true east-west and north-south axis exactly one cubit longer than the actual measures given by Petrie. This forms a misaligned square pyramid base fitting neatly into an aligned square area that is 1 Rc longer on all four sides (Ill. 11).
Illustration 11. The measures recorded by Petrie including the 0.28 degrees rotation from the site’s axis (left).
The aligned N-S, E-W extents that the pyramid fits into (right).
Note: All side measures are effectively increased by 1 Rc.
We are now presented with an actual 201.7 Rc pyramid base that produces an effective 202.7 Rc length along the east-west axis.
Repeating the same multiplication as above it is now discovered that 7 times the effective 202.7 Rc length equals 1418.9 Rc, equal to the 1418.2 Rc distance measured from east to west (Ill. 12).
Illustration 12. The east-west axis is seven times longer than the effective length of P3.
An additional confirming linear ratio is discovered by following the small 0.28 degrees deviation of P3’s axis (Ill.13).
Illustration 13. The line A-B intersects line C-D at a distance 1417.7 Rc from the center of P1.
Line C-D is drawn from point C parallel to the southern base line of P3 (0.28 degrees S of E) to the eastern extent of P1.
Many will reject these suggested “unorthodox” procedures, but if careful consideration is given then it is realized that the designers were left with no other alternative. They had to confirm the intended size relationship of the pyramids and the precise coordinates used when preparing the site.
The pyramids width ratios were incorporated in the overall dimensions. They confirm the measures by Petrie and verify that the southern base of P1 was the primary axis used to prepare the complete Giza complex. It does not guarantee that P1’s base line was true to Earth’s east-west axis during construction, only that it was the reference line used for all measures.
Rotating the complete site in either direction from P1’s southern axis increases or decreases the 1411.8 Rc and 1418.2 Rc lengths. P3’s location would not produce the 7:1 ratio in both directions, indicating that this structure was intentionally positioned and rotated to establish the site’s exact coordinates.
This is the first of several reasons why the southern extent of the site was set at 1418 Rc; not the 1414 Rc (1000 times the square root of 2) that many have speculated.
Before proceeding into the 1418 Rc issue, I believe this to be a perfect opportunity to introduce the smallest pyramid structure on site…P10.
It can be noted from all illustrations that seven (7) additional pyramid (satellite) structures were built next to the major three.
Three are located on the southeast corner of P1, three at the southwest corner of P3 and one at the center south base of P2.
Unlike modern Egyptologists who identify these monuments as GP1a, 1b, 1c…GP2a…GP3a, 3b, 3c I have followed the pattern used by others before me; each pyramid is allotting a number according to size.
The largest is P1 (Khufu’s) through to P10, the tenth and smallest (base pyramid of P2…Khafre).
Using the center point of P2 and scribing a circle that intersects all four corners of this pyramid, it is found that the circle passes through the center location of P10, a radius of 290.85 Rc, or the square root of 2 (1.414) times the base width of 205.6
This small pyramid holds the key to all that is locked within this site and its exact location is crucial to further measures.
After establishing P10 as an additional reference point, the north-south linear measures are recalculated (Ill. 13).
Illustration 13. The smallest pyramid (P10) dictates the exact directional setting for the complete Giza site.
It is evident that the small discrepancies of P3’s four side lengths and the adjustment of its axis were by design. Measures from its corner locations provide three identical ratios that dictate the exact location of all corner points on site
Comparing the base measure of P1 (440 Rc) to the effective base measure of P3 (202.7 Rc) produces a 2.17:1 ratio. All north-south lengths shown, with reference to P10, share the identical value. And it must be remembered that P10’s location was determined by the radius formed from the corners of P2…a remarkable display of geometric mathematics.
The stage is almost set but for one small issue; it must be repeated. The designers had to remove all doubt…they had to prove that the site is an entity unto itself…a complete and unique package of knowledge.
If their intention was to provide convincing ratio measures, as indicated when measuring from north to south, then it is logical that they would also include others in the east to west direction (Ill. 14).
Illustration 14. Comparing identical ratio measures of P1-P2 and P3-P2 produce an additional common ratio equal to 3:1.
Elusive when presented with numbers, but obvious when illustrated. There are three (3) large pyramids at Giza and two sets of three (3) satellite pyramids…the obvious number is three (3).
The east-west distance from the center of P1 to P2 (638.5 Rc) is three (3) times greater than the space between these two pyramids (213.1 Rc), likewise are the measures when taken from the same reference points of P2 and P3 (458.3/152.4) !
We have now ventured from the first step that indicated the square root values of 2, 3 and 5 to a level of mathematics that now includes geometric relationships between each structure on site. Who designed this complex, what methods were used to calculate these interrelated ratios? These questions have yet to be answered. However, before departing from the mathematical stage of Giza, I will now offer absolute proof that this site was set using mathematical ratios.
To confirm all 90o corners of a square or rectangle, the two diagonal distances (corner to corner) must be equal in length.
From the Giza site, a line is drawn from the southwest corner of P3 to the diagonally opposite (northeast) corner of P2 (1292.8 Rc). The measured length corresponds to the vertical distance from the same point of P3 to the base of P1 (1292.6 Rc) (Ill. 15).
Illustration 15. The diagonal measure (1292.8 Rc) from P3 to P2 equals the vertical distance from P3 to P1 (1292.61).
The above distances are derived from the computer data presented by Petrie; they are precise to the closest inch measure.
It was suggested in illustration 4, Part 1 that the vertical height A was 3.14 (pi) times longer than the base width of P2. Remarkably, the designers duplicated the identical ratio by design. Length A is equal to length B; therefore B is also 3.14 (pi) times longer than the base length of P2.
To prove that it was by design, we simply compare what we began with to the one diagonal measure remaining. (Ill. 16).
Illustration 16. The ratio of the two diagonals is equal to the square root of three.
We have traveled a complete circle, from the first step where accuracy and coincidence were questioned, around and back. But now we are armed with evidence to prove that the “numerical values” of Giza were set by design.
The site’s diagonal length of 2238.9 Rc was intentionally close in value to the square root of five (2.236), but adjusted slightly to produce a more significant hidden ratio between the two diagonal lines linking these three majestic structures together…the simple square root value of 3 !
No words can describe the stunning accuracy of measure. These numbers, lengths and ratios cannot be rejected…they are set in stone and confirmed by the accurate measures produced by a man more than 100 years past…Sir. F. W. Petrie.
From the notes of Petrie:
Summary of probable theories:
178. “…It is possible that some parts may have been made intentionally varying in size, in order to include two different relations to other parts; but such is scarcely provable; and in a general statement like the following, it is better to omit some things that may be true, than it is to include a number of dubious theories which are not supported by a system of coincidences in different parts of the structure. And if some judge that this summary includes too much, and others think that it states too little, it must be remembered that the whole of the materials for forming an opinion are impartially provided in the previous chapters of this work...”
Unfortunately, Petrie never lived to realize that his dedicated works would be the tools used for unlocking the many mysteries of the Giza site. Let his written words be forever remembered.
To be continued….
1. Special mention to Michael Saunders for his contribution and confirmation re: Petrie’s measures.
Please visit and enjoy his site 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. The Pyramids and Temples of Gizeh…….Sir W.M.Flinders Petrie 1883
5. “1o6” The Dawn of Man…sb.…..……………..Clive Ross
Copyright 1997 CIPO 459689 ISBN0-9686006-0-3