PROPOSED SITE PLAN FOR THE THREE MAIN PYRAMIDS AT GIZA
The baselength of the great pyramid is 440 cubits. The following proposed design criteria for the size and relative locations of the three main pyramids at Giza is based on the 433/250 fractional expression of the square root of three. This is the lowest fraction in whole numbers that expresses the square root of three ratio of 1.732/1 or 1732/1000:
1. A 250 cubit north-south distance between the great pyramid and the second pyramid.
2. A 433 cubit east-west distance between the middle of the great pyramid and the east side of the second pyramid. The distance from the middle of the great pyramid to the west side of the great pyramid is 220 cubits, leaving a distance of 213 cubits between the west side of the great pyramid and the east side of the second pyramid.
3. The 213 cubit east-west distance between the great pyramid and the second pyramid is equal to the mean of the half bases of the great pyramid and the second pyramid. The half base of the great pyramid is 220 cubits. A baselength of 412 cubits for the second pyramid gives a half base of 206 cubits:
220 + 206 = 426
426/2 = 213
4. A 433 cubit north-south distance between the second pyramid and the third pyramid.
5. A 250 cubit east-west distance between the west side of the second pyramid and the middle of the third pyramid.
6. The 152 cubit east-west distance between the second pyramid and the third pyramid is equal to the mean of the half bases of the second pyramid and the third pyramid. A baselength of 412 cubits for the second pyramid gives a half base of 206 cubits and a baselength of 196 cubits for the third pyramid gives a half base of 98 cubits:
206 + 98 = 304
304/2 = 152
Given the size of any one of the three pyramids, the size of the other two pyramids and the relative locations of all three pyramids are fixed by these design criteria alone.
Because the north-south spacing between the pyramids is 250 cubits and 433 cubits, and the east-west spacing of 250 cubits and 433 cubits includes one half of the baselength of the great pyramid and one half of the baselength of the third pyramid, the line from the middle of the south edge of the great pyramid to the middle of the north edge of the third pyramid is a 45° diagonal. The other diagonal of the square formed by the middle of the south edge of the great pyramid and the middle of the north edge of the third pyramid exactly marks the diagonal of the second pyramid.
The 45° diagonal rectangle above is formed by the north-west corner of the great pyramid and the south-east corner of the third pyramid. Based on the size and relative positions of the three pyramids given in the previous diagrams, the midline of the diagonal rectangle exactly marks the other diagonal of the second pyramid. The intersection point of the diagonal of the square and the midline of the diagonal rectangle, marking the apex of the second pyramid, is 859 cubits west and 896 cubits south of the north-east corner of the great pyramid:
440 + 213 + 206 = 859
440 + 250 + 206 = 896
The lines drawn west from the apex of the great pyramid and north from the apex of the second pyramid intersect 639 cubits west of the apex of the great pyramid. The distance from the apex of the great pyramid to the north-east corner of the second pyramid is also 639 cubits:
220 + 213 + 206 = 639
433² + 470² = 639²
Arcing the intersection point of the lines west from the apex of the great pyramid and north from the apex of the second pyramid exactly marks the north-east corner of the second pyramid along the mid-line of the diagonal rectangle. A circle with a radius from the apex of the second pyramid to the north-east corner of the second pyramid marks the other corners of the second pyramid along the diagonal of the square and the midline of the diagonal rectangle.
The sidelengths of the rectangle formed by the apex of the second pyramid and the apex of the third pyramid are 737 cubits north-south and 456 cubits east-west. The ratio of the sidelengths is 737/456 = 1.61623, slightly less than the φ ratio of 1.61803. The sidelengths of the rectangle formed by the apex of the great pyramid and the intersection point west of the apex of the second pyramid and north of the apex of the third pyramid are 1095 cubits east-west and 676 cubits north-south. The ratio of these sidelengths is 1095/676 = 1.61982, slightly greater than φ. The average of the ratios of these two rectangles equals φ:
1.61623 + 1.61982 = 3.23605
3.23605/2 = 1.618025
According to the survey conducted by Flinders Petrie, the north-south distance between the great pyramid and the second pyramid is 250 cubits and the east-west distance between the great pyramid and the second pyramid is 213 cubits. According to Petrie, the baselength of the second pyramid is 411 cubits.
The first course of the second pyramid is approximately 52.52 inches high. The upper 41.52 inches of the face of the casing stones on the first course are at the pyramid angle of 53° 12'. The lower 11 inches of the face of the casing stones on the first course is vertical. The casing stones on the first course of the pyramid are also on rock that was leveled approximately 2-3 inches higher than the rock surrounding the pyramid. It has been assumed that raised paving stones were intended to cover the lower vertical portion of the first course of the pyramid, and that the height of the pyramid and the side lengths of the pyramid were intended to be measured from the point where the vertical portion ends and the sloping portion begins.
If the side lengths of the pyramid were intended to be measured from the base of the first course, with the slope of the pyramid either actually extended or symbolically extended to the surrounding ground level, then the baselength of the second pyramid would be 412 cubits. According to I.E.S. Edwards in The Pyramids of Egypt (1985), the baselength of the second pyramid is 412 cubits.
According to Petrie's survey, the east-west distance between the second pyramid and the third pyramid is 152 cubits. His measurement of the north-south distance between the second pyramid and the third pyramid was 429.5 cubits, based in part on his calculation of 201.5 cubits for the baselength of the third pyramid. Petrie was unable to measure the north side of the third pyramid because of debris around the pyramid, and he acknowledged that the rough unfinished condition of the lower courses and the generally dilapidated condition of the third pyramid made it difficult to determine the intended length.
Petrie based his determination of the dimensions of the pyramid on various measurements of the height of the pyramid and the angle of fragments of upper casing stones that were recovered from around the pyramid. Unlike the lower courses, the casing of the upper courses had been finished, although the casing of the upper courses is no longer on the pyramid. Measurements of nine fragments of the upper casing indicated an angle of 51° 58' for the slope of the pyramid. However, based on all of his measurements, Petrie concluded that the intended angle was lower. Petrie also believed that the granite portion of the outside of the pyramid rose to one quarter of the total height of the pyramid and he believed that the measurement of this height might be the best evidence of the intended total height of the pyramid.
Combining the angle of the pyramid from the finished upper casing fragments with a total height based on the height of the granite portion of the pyramid results in a baselength of approximately 196 cubits. Other modern published surveys of the baselength of the third pyramid have also ranged from under 196 cubits to over 200 cubits. Therefore, the 196 cubit baselength for the third pyramid that is given here is within the range of modern published survey findings for the intended baselength of the pyramid.
References:
John Legon observed the ratio of 1.732/1 between the 433 cubit east-west distance between the middle of the great pyramid and the east side of the second pyramid, and the north-south distance of 250 cubits between the great pyramid and the second pyramid.
Clive Ross observed the 1/3 ratio between the east-west distances between the pyramids and the east-west distances between the middle of the pyramids.
Chris Tedder observed that the diagonal of a square formed by the mid points of the south side of the great pyramid and the north side of the third pyramid marks one of the diagonals of the second pyramid.
Robin Cook observed that the midline of a 45° diagonal rectangle formed by the north-west corner of the great pyramid and the south-east corner of the third pyramid marks the other diagonal of the second pyramid. Cook also observed that the east-west distance between the apex of the great pyramid and the apex of the second pyramid is equal to the distance from the apex of the great pyramid to the north-east corner of the second pyramid.