Prof. William Matthew Flinders Petrie's topographic survey of Giza published in 1883, "The Pyramids and Temples of Gizeh", is generally considered to be very accurate. In Chap. 5. Co-ordinates, Petrie lists the raw survey data; and in Chap. 13. Positions and Orientation of the Pyramids, he provides the schema of the three big pyramids with distances given in inches.
Others, including myself, then use this schema for further study. Few, if any, use the the raw survey data.
Some here may be familiar with my reconstruction of Petrie's ground plan of Giza from a clean slate (with averaged out pyramid sides). This reconstruction succeeds in duplicating Petrie's figures for the sides of all three pyramids by a mathematical method.
Caveat Emptor
I checked my results against the figures for the plan given by John Legon, thinking, like Legon, that these were an exact restatement of Petrie's figures; however, Legon had inadvertently shortened the spacing between the pyramids by one-tenth of an inch altogether. With the pyramid sides remaining same, the North to South span of the pyramids shrinks from 35,713.2" to 35,713.1" in this version.
The change doesn't seem to matter much, since my reconstruction treats the North to South span as being 1,723.05 cubits, yet it works with the version translating to 35,713.1 inches perfectly, and definitely better than with Petrie's version.
So whose version is correct, if any? By Occam's Razor, the simplest explanation is that the version which reconciles with the fully-fledged bona fide mathematical model perfectly is the correct one.
If so, then by extrapolation, the pyramids had been built in ideal agreement with the plan, and Petrie had erred by just 0.1" over the distance of almost 3,000 feet!
Unfortunately, my satisfaction with the above heart-warming conclusion was not lasting.
Ronald Birdsall finds inaccuracies in Petrie's interpretation of the raw survey data
(http://www.ronaldbirdsall.com/gizeh/petrie/index.htm)
Birdsall had asked Stephen Brabin to feed Petrie's raw survey data into a spreadsheet, and the subsequent recalculation showed that Petrie, not having the luxury of a spreadsheet, had apparently made several slight mistakes in his own recalculations.
Of course, choosing a spreadsheet over a CAD program is puzzling, as CAD's superiority over a spreadsheet for the purpose is unquestionable. Obviously, checking out the checkers had become a necessity. Therefore, I fed Petrie's raw data (Chapter 5) into my CAD program, omitting the data irrelevant to the ground plan.
Surprises!
a) casing sides of the Great Pyramid
_____________ Petrie __________ CAD
N. ________ 9069-4 __________ 9,068.6 (-0.8")
S. ________ 9069-5 __________ 9,068.7 (-0.8")
E. ________ 9067-7 __________ 9,067.0 (-0.7")
W. ________ 9068-6 __________ 9,067.9 (-0.7")
Mean ________ 9068-8 __ —3'43" _9,068.05
Petrie consistently adds 3/4" (+/- 1/20") on each side. This is understandable; my data is calculated from his data on casing sides, whereas he adds the step of involving the diagonals between corner sockets. Frankly, it's not clear to me how exactly he did it, but since he had no interest whatsoever in doctoring the results, I accept his figures.
b) casing sides of G2
_____Survey units ______ Petrie ____________________ CAD
N__ 1666852.15604534____ 8,471.9" ____ - 5'31" ____ 0.09192240º__= 5.51534399'__= - 5'31"
E__ 1667494.22938522____ 8,475.2" ____ - 6'13" ____ 0.10623567º__= 6.3741402' __= - 6'22"
S__ 1667830.96112556____ 8,476.9" ____ - 5'40" ____ 0.09436309º__= 5.66178' ____= - 5'40"
W__ 1667565.31844033____ 8,475.5" ____ - 4'21" ____ 0.07260441º__= 4.3562646'___= - 4'21"
____________________Mean 8,474.9"_____ – 5' 26"____ CAD Mean___________________= - 5'.17"
Here both Petrie and CAD results for mean sides are the same; the azimuths agree with the sole exception of the East side.
c) casing sides G3
The below figures are based on chapter 5
survey units____________ inches
N__18077.64694374 ______4,158.0"
S__818055.85448244______4,157.8"
E__816359.11194523______4,149.2"
W__817255.86320197______4,153.8"
Total inches = 16,618.7978" = 4,154.7' average per side( 4,154.69946 )
___________________________ = 201.499 cubits = 201.5 cubit
But in chapter 10 we read: "The N. end of the W. side could not be reached, after several attempts; and hence the lack of knowing the length of the N. or azimuth of the W. side."
In this chapter Petrie reduces the list to:
N.... ??????
S _____ 4,157.8"
E _____ 4,149.2"
W _____ 4,153.9"
4,153.6 is Petrie's mean value, as is the mean CAD value
In chapter 21 Petrie says, "Its actual base is 4153.6 ± 1.8"
This "± 1.8" shows how uncertain he is about the G3 true values; yet, the value he cites is, once again, ideal for the mathematical model.
Direct distances between centers
In his determination of pyramid centers, Petrie does not provide (is miscalculating?) the true position of the intersections of diagonals:
_____________________________________________________________ from N _______ from E
Center of casing of 1st pyramid___given by Petrie _____ 2 758 515 ____ 2 295 478
0.03" to South off true value
0.22" to West off true value
Center of casing of 2nd pyramid___given by Petrie ____ 5 448 171_____ 4 939 173
0.1" to West off true value
1.2" to South off true value
Center of casing of 3rd pyramid ___ given by Petrie ____ 8 395 877_____ 6 856 742
1/18"to North off true value
1.1" to West off true value
Direct distances between centers
1st to 3rd
36,859.4__(36,859.42396) _ CAD measurement from the intersection of diagonals in the diagram of G1 sockets to the intersection of diagonals in the diagram of G3 casing sides
36,856.2__(36,856.159)__between true casing centers of G1 and G3
36,857.779 __ (36,857.8) average of the two values from above
36,857.7 at 37º 51' 6" __ Petrie gives this value ( looks like he is using the same average)
36,856.7 at 37º 51' 6" Ronald Birdsall calls Petrie's value a miscalculation (while his choice is the (miscalculated?) distance between Petrie's casing centers)
1st to 2nd
19,168.5__ socket center of G1 to true casing center of G2
19,167.6__ true casing centers in CAD
19,168.4__ both Petrie and Bird (both using Petrie's casing centers)
2nd to 3rd
17,873.65__true casing centers
17,873.1 __CAD measurement between Petrie's casing centers
17,873.1 __Ronald Bird - copies Petrie accurately
17,873.2 __Petrie misreads his own data - he is off 0.1"
Finally, the position is reoriented to true North.
Petrie says in chap.5: "The co-ordinates of the station marks, &c., are reckoned from a line beyond the N. side of the whole area, and from a line beyond the E. side of the area: thus there are no minus quantities. The azimuth of true North on the system of co-ordinates is East of the approximate North of the system, or the azimuth of its Eastern boundary, by
+ 1º 12' 22" ± 6" _____ (1.2061111111110534º )"
After rotating the position by the above value, we measure the span of the pyramids from North to South. The result is surprisingly far from the ubiquitous postulate of 35,713.2".
a) maximum North to South span between the pyramids ____ = 35,755.8"
b) minimum North to South span between the pyramids ____ = 35,731.2"
c) midpoint of G1 north side to midpoint of G3 south side = 35,743.5"
In each case, this is a big miss.
average azimuth in CAD: ____ G1 0.061974765 deg = 3'43" __ same as Petrie
____________________________ G2 0.0912813925 deg = 5'29" __ 3" difference
________________________________ average G1 + G2 = 4.6'
Petrie's values:
Great Pyramid, casing sides _ – 3' 43" ± 6"
Second Pyramid, casing sides – 5' 26" ± 16"
Petrie says: "On the whole, considering the various values of the data, – 5' 40" ± 10" may be taken as a safe statement of the suggested place of the pole, at the epoch of the Pyramid builders."
Petrie shows great confidence in the builders' capability to properly orient the pyramids.
I hold the same confidence in their ability to make the North to South span the equivalent of 1,732.05 cubits, i.e. 35,713.1 inches, the value which works ideally with the intrinsic mathematics of the ground plan.
Consequently, I reoriented the position so that the N-S span between the midpoints of G1's north side and G3's south side would be exactly 35,713.1".
My hypothesis was that in doing so, the pyramids' orientation will shift much closer to true North, and therefore indicate that this span was truly the builders' intention.
7,026,552.425 Survey Units = 35,713.1"
Rotate CCW by the midsides so the N-S span = 35,713.1"
angle of rotation = 1.12908817º = 1º7'45"
East side G1 angle in XY Plane __ = 89.98882923º _____(0.01117077º)_ =40"
North side G1 angle in XY Plane__ = 359.97856857º_____(0.02143143º)_ =1'17"
West side G1 angle in XY Plane __ = 89.98807670 ______(0.0119233)___ =43"
South side G1 angle in XY Plane - = 179.98433279 _____(0.01566721)__ =56"
average 54"
East side G2 angle in XY plane __ = 270.02921273 _____(0.02921273)_ =1'45"
North side G2 angle in XY Plane__ = 180.01489946 _____(0.01489946)_ =54"
West side G2 angle in XY Plane __ = 269.99558146 _____(0.00441854)_ =16"
South side G2 angle in XY Plane__ = 0.01734014 _______ 0.01734014)_ =1'2"
average 59"
The rotation to make the N-S span 35,713.1 " did indeed shift the pyramids' orientation much closer to true North, confirming the above hypothesis.
In comparison, the rotation to 35,713.2" is practically as close but comes in second.
Others, including myself, then use this schema for further study. Few, if any, use the the raw survey data.
Some here may be familiar with my reconstruction of Petrie's ground plan of Giza from a clean slate (with averaged out pyramid sides). This reconstruction succeeds in duplicating Petrie's figures for the sides of all three pyramids by a mathematical method.
Caveat Emptor
I checked my results against the figures for the plan given by John Legon, thinking, like Legon, that these were an exact restatement of Petrie's figures; however, Legon had inadvertently shortened the spacing between the pyramids by one-tenth of an inch altogether. With the pyramid sides remaining same, the North to South span of the pyramids shrinks from 35,713.2" to 35,713.1" in this version.
The change doesn't seem to matter much, since my reconstruction treats the North to South span as being 1,723.05 cubits, yet it works with the version translating to 35,713.1 inches perfectly, and definitely better than with Petrie's version.
So whose version is correct, if any? By Occam's Razor, the simplest explanation is that the version which reconciles with the fully-fledged bona fide mathematical model perfectly is the correct one.
If so, then by extrapolation, the pyramids had been built in ideal agreement with the plan, and Petrie had erred by just 0.1" over the distance of almost 3,000 feet!
Unfortunately, my satisfaction with the above heart-warming conclusion was not lasting.
Ronald Birdsall finds inaccuracies in Petrie's interpretation of the raw survey data
(http://www.ronaldbirdsall.com/gizeh/petrie/index.htm)
Birdsall had asked Stephen Brabin to feed Petrie's raw survey data into a spreadsheet, and the subsequent recalculation showed that Petrie, not having the luxury of a spreadsheet, had apparently made several slight mistakes in his own recalculations.
Of course, choosing a spreadsheet over a CAD program is puzzling, as CAD's superiority over a spreadsheet for the purpose is unquestionable. Obviously, checking out the checkers had become a necessity. Therefore, I fed Petrie's raw data (Chapter 5) into my CAD program, omitting the data irrelevant to the ground plan.
Surprises!
a) casing sides of the Great Pyramid
_____________ Petrie __________ CAD
N. ________ 9069-4 __________ 9,068.6 (-0.8")
S. ________ 9069-5 __________ 9,068.7 (-0.8")
E. ________ 9067-7 __________ 9,067.0 (-0.7")
W. ________ 9068-6 __________ 9,067.9 (-0.7")
Mean ________ 9068-8 __ —3'43" _9,068.05
Petrie consistently adds 3/4" (+/- 1/20") on each side. This is understandable; my data is calculated from his data on casing sides, whereas he adds the step of involving the diagonals between corner sockets. Frankly, it's not clear to me how exactly he did it, but since he had no interest whatsoever in doctoring the results, I accept his figures.
b) casing sides of G2
_____Survey units ______ Petrie ____________________ CAD
N__ 1666852.15604534____ 8,471.9" ____ - 5'31" ____ 0.09192240º__= 5.51534399'__= - 5'31"
E__ 1667494.22938522____ 8,475.2" ____ - 6'13" ____ 0.10623567º__= 6.3741402' __= - 6'22"
S__ 1667830.96112556____ 8,476.9" ____ - 5'40" ____ 0.09436309º__= 5.66178' ____= - 5'40"
W__ 1667565.31844033____ 8,475.5" ____ - 4'21" ____ 0.07260441º__= 4.3562646'___= - 4'21"
____________________Mean 8,474.9"_____ – 5' 26"____ CAD Mean___________________= - 5'.17"
Here both Petrie and CAD results for mean sides are the same; the azimuths agree with the sole exception of the East side.
c) casing sides G3
The below figures are based on chapter 5
survey units____________ inches
N__18077.64694374 ______4,158.0"
S__818055.85448244______4,157.8"
E__816359.11194523______4,149.2"
W__817255.86320197______4,153.8"
Total inches = 16,618.7978" = 4,154.7' average per side( 4,154.69946 )
___________________________ = 201.499 cubits = 201.5 cubit
But in chapter 10 we read: "The N. end of the W. side could not be reached, after several attempts; and hence the lack of knowing the length of the N. or azimuth of the W. side."
In this chapter Petrie reduces the list to:
N.... ??????
S _____ 4,157.8"
E _____ 4,149.2"
W _____ 4,153.9"
4,153.6 is Petrie's mean value, as is the mean CAD value
In chapter 21 Petrie says, "Its actual base is 4153.6 ± 1.8"
This "± 1.8" shows how uncertain he is about the G3 true values; yet, the value he cites is, once again, ideal for the mathematical model.
Direct distances between centers
In his determination of pyramid centers, Petrie does not provide (is miscalculating?) the true position of the intersections of diagonals:
_____________________________________________________________ from N _______ from E
Center of casing of 1st pyramid___given by Petrie _____ 2 758 515 ____ 2 295 478
0.03" to South off true value
0.22" to West off true value
Center of casing of 2nd pyramid___given by Petrie ____ 5 448 171_____ 4 939 173
0.1" to West off true value
1.2" to South off true value
Center of casing of 3rd pyramid ___ given by Petrie ____ 8 395 877_____ 6 856 742
1/18"to North off true value
1.1" to West off true value
Direct distances between centers
1st to 3rd
36,859.4__(36,859.42396) _ CAD measurement from the intersection of diagonals in the diagram of G1 sockets to the intersection of diagonals in the diagram of G3 casing sides
36,856.2__(36,856.159)__between true casing centers of G1 and G3
36,857.779 __ (36,857.8) average of the two values from above
36,857.7 at 37º 51' 6" __ Petrie gives this value ( looks like he is using the same average)
36,856.7 at 37º 51' 6" Ronald Birdsall calls Petrie's value a miscalculation (while his choice is the (miscalculated?) distance between Petrie's casing centers)
1st to 2nd
19,168.5__ socket center of G1 to true casing center of G2
19,167.6__ true casing centers in CAD
19,168.4__ both Petrie and Bird (both using Petrie's casing centers)
2nd to 3rd
17,873.65__true casing centers
17,873.1 __CAD measurement between Petrie's casing centers
17,873.1 __Ronald Bird - copies Petrie accurately
17,873.2 __Petrie misreads his own data - he is off 0.1"
Finally, the position is reoriented to true North.
Petrie says in chap.5: "The co-ordinates of the station marks, &c., are reckoned from a line beyond the N. side of the whole area, and from a line beyond the E. side of the area: thus there are no minus quantities. The azimuth of true North on the system of co-ordinates is East of the approximate North of the system, or the azimuth of its Eastern boundary, by
+ 1º 12' 22" ± 6" _____ (1.2061111111110534º )"
After rotating the position by the above value, we measure the span of the pyramids from North to South. The result is surprisingly far from the ubiquitous postulate of 35,713.2".
a) maximum North to South span between the pyramids ____ = 35,755.8"
b) minimum North to South span between the pyramids ____ = 35,731.2"
c) midpoint of G1 north side to midpoint of G3 south side = 35,743.5"
In each case, this is a big miss.
average azimuth in CAD: ____ G1 0.061974765 deg = 3'43" __ same as Petrie
____________________________ G2 0.0912813925 deg = 5'29" __ 3" difference
________________________________ average G1 + G2 = 4.6'
Petrie's values:
Great Pyramid, casing sides _ – 3' 43" ± 6"
Second Pyramid, casing sides – 5' 26" ± 16"
Petrie says: "On the whole, considering the various values of the data, – 5' 40" ± 10" may be taken as a safe statement of the suggested place of the pole, at the epoch of the Pyramid builders."
Petrie shows great confidence in the builders' capability to properly orient the pyramids.
I hold the same confidence in their ability to make the North to South span the equivalent of 1,732.05 cubits, i.e. 35,713.1 inches, the value which works ideally with the intrinsic mathematics of the ground plan.
Consequently, I reoriented the position so that the N-S span between the midpoints of G1's north side and G3's south side would be exactly 35,713.1".
My hypothesis was that in doing so, the pyramids' orientation will shift much closer to true North, and therefore indicate that this span was truly the builders' intention.
7,026,552.425 Survey Units = 35,713.1"
Rotate CCW by the midsides so the N-S span = 35,713.1"
angle of rotation = 1.12908817º = 1º7'45"
East side G1 angle in XY Plane __ = 89.98882923º _____(0.01117077º)_ =40"
North side G1 angle in XY Plane__ = 359.97856857º_____(0.02143143º)_ =1'17"
West side G1 angle in XY Plane __ = 89.98807670 ______(0.0119233)___ =43"
South side G1 angle in XY Plane - = 179.98433279 _____(0.01566721)__ =56"
average 54"
East side G2 angle in XY plane __ = 270.02921273 _____(0.02921273)_ =1'45"
North side G2 angle in XY Plane__ = 180.01489946 _____(0.01489946)_ =54"
West side G2 angle in XY Plane __ = 269.99558146 _____(0.00441854)_ =16"
South side G2 angle in XY Plane__ = 0.01734014 _______ 0.01734014)_ =1'2"
average 59"
The rotation to make the N-S span 35,713.1 " did indeed shift the pyramids' orientation much closer to true North, confirming the above hypothesis.
In comparison, the rotation to 35,713.2" is practically as close but comes in second.