Based on his survey of the bent pyramid, J.S. Perring gave 54 deg 14' for the slope of the lower part and 42 deg 59' for the slope of the upper part of the bent. Petrie found the slopes of the bent to be variable. He reported measurments for the slope of the lower part of the bent ranging from 53 deg 44' to 55 deg 23' and measurements ranging from 42 deg 39' to 43 deg 24' for the upper part of the bent. I.E.S Edwards gave 54 deg 31' for the lower part and 43 deg 21' for the upper part. J. P. Lauer gave 54 deg 27' 44" for the angle of the lower part of the bent. Mark Lehner gave 54 deg 27' 44" for the lower part and 43 deg 22' for the upper part. Lehner also gives the slope of the red pyramid as 43 deg 22'. Baines and Malek gave the same angles as Lehner for the bent and the red pyramid. Edwards and others have also commented that the angle of the red and the angle of the upper part of the bent was the same or approximately the same. Robins and Shute found the angle of the lower part of the bent to be 54 deg 27' 44".

The angle of 54 deg 27' 44" produces a nice round number seked slope of seven palms vertical over five palms horizontal. All of the authors that subscribe to this exact slope affirm that this is their desired seked 5. Considering the variation of over one degree in the various surveys and the variation of over one degree on the various surfaces on the lower part of the bent found by Petrie in his survey alone, I believe it is speculative to define the slope of the pyramid to the second in order to conform with and reinforce the idea of manditory seked calculations by the pyramid builders.

Based on his survey of the bent, Dorner gives 55 deg 0' 30" as the angle of the lower part of the bent, producing a rise/run ratio of 10/7, and he gives measurements of just over 55 degrees on all four sides of the pyramid to support his proposed angle. Legon proposed that the intended slope of the lower portion was a geometric combination of the seked 5 slope of 7/5 and Dorner's proposed slope of 10/7, and argues that Dorner ignores the lower angles found on various surfaces of the lower part of the pyramid. Legon's proposed slope is essentially an exact expression of the square root of two. Neither Dorner's or Legon's proposed angles are seked slopes.

Perring gives 44.9 meters or 85.7 cubits for the height of the bend in the pyramid. Petrie gives 1857 inches or 90 cubits for the height of the bend. Dorner also gives 90 cubits for the height of the bend. Petrie calculated the height of the bend in the pyramid by measuring the horizontal distance from the edge of the pyramid to the bend above, and then applying an average angle of the slope of the lower part of the pyramid to triangulate the vertical height of the bend. If Petrie had used a higher angle for the slope, his calculated height would have been greater. Had he used a lower angle for the slope of the lower part, his calculated height would have been less. When Petrie summarized the baselengths and height of the bend and overall height of the pyramid, he specifically said that he only regarded the baselength data as really accurate. Since Dorner used an even higher proposed angle than Petrie, if a lower angle was applied to Dorner's calculations it would also necessarily produce a lower height for the bend given Dorner's horizongal distance from the edge of the pyramid to the bend.

H.R. Butler summarized much of this info and concluded not unreasonably that more survey information is needed and that all of the surveys and conclusions to date do not provide a consensus or a certainty regarding the angles of the faces of the height of the bend.

Based primarily on Perring's survey data, Lubicz and Varille proposed a phi based solution to the angles and the height of the bend. Perring found the dimensions of the red pyramid to be 420 cubits in baselength and 200 cubits in height. This same measure for the red pyramid has also been confirmed and/or adopted by others. The subject of the angle of the red was recently discussed in a thread started by Jon B below. The proposed 200/210 rise/run ratio produces a hypotenuse (slant height) of exactly 290 cubits, based on the pythagorean triplet 20/21/29. Lubicz and Varille contend, as have numerous others, that the angle of the upper portion of the bent was intended to be the same as the angle of the red. Then they observe that taking the slant height of the upper part as the height of the lower part (a rise/run ratio of 29/21) produces an angle very close to Perring's angle for the lower part of the bent. Then they observe that the ratio 29/21 is almost identical to the phi ratio of phi squared plus one over phi squared: 3.618/2.618.

Petrie gives the height of the top of the bent above the high water mark of the Nile at 450 feet, or 261.8 cubits (phi squared times 100). If the 200 cubit height of the bent is subtracted, this leaves 61.8 cubits for the elevation of ground level above the Nile high water mark. .618 is the phi reciprocal (1/phi). Google earth gives the elevation of ground level above sea level at the bent as 190 - 195 feet. 192 feet is equal to 111.8 cubits. This is another factor of phi times 100. The square root of five divided by two is 1.118. Phi minus .5 is also 1.118. Since 111.8 cubits minus 61.8 cubits equals fifty cubits, the difference between sea level and the high water mark of the Nile in the vicinity of the bent is 50 cubits.

Because the base of the bent was not cleared when Perring surveyed the pyramid, his given baselengths were shorter than currently accepted. Additionally, his height for the bend is now regarded as too low. As a result, his findings, as well as the construction proposed by Lubicz/Varille, have generally been discarded. I have produced an new construction based on the phi angles of the sides, beginning with a half base of 261.8 cubits (phi squared times 100) and a height of 361.8 cubits (phi squared plus one times 100). The smaller circles below have a radius of one and the larger circles have a radius of 1.118. This is the standard method of geometrically constructing a phi squared segment of 2.618. On the left side, an additional circle with a radius of one is added to give 3.618 for the segment on the left hand side.



The segment on the left side is arced to the vertical line, giving a triangle with a height of 3.618 and a half base of 2.618.



The baseline of the construction is at sea level. Ground level for the pyramid is 1.118 above the baseline. The baselength of the pyramid at ground level is 3.618, exactly phi squared plus one times one hundred and exactly the same measure given by Petrie for the baselength of the side of the pyramid at ground level (361.8 cubits). The hypothetical height of the pyramid as extended to apex with the angle of the lower part of the pyramid is exactly 250 cubits.



The segment equal to the diameter of the smaller circle (200 cubits) is arced to the midline of the pyramid. This marks the height of the bend, 88.2 cubits above ground level. The remaining height of the hypothetical pyramid (161.8 cubits) is arced down to the midline of the pyramid, giving a slant height for the upper part of the pyramid of 161.8 cubits, or phi times 100. The total height of the bent pyramid is 199.9 cubits.



This construction explains the relationship between the slope of the lower part and the slope of the upper part of the pyramid. This construction also produces precise measures for the baselength and the height of the pyramid, and explains the measures of the baselength and the height, without any arbitrary pre-assumptions about the baselength or the height. This construction also gives a direct expression of phi in the slant height of the upper part of the pyramid, in addition to the expressions of phi in baselength and in the slopes of the lower and upper parts of the pyramid.

Note to mods: I constructed all of the diagrams as GIF images and the file size of all of the images are less than 10 kb each.