The Architect's Plan
By Leon Cooper
The Khafre Pyramid: Part I
The Basic Derivation Algorithm
This presentation examines the method by which the enigmatic interior layout design of the Khafre Pyramid can be accurately derived and accounted for. The accompanying essays in this series detail how this same basic method was apparently implemented in order to also derive the interior layout designs of the Bent, the Red, and the Khufu Pyramids.
In each of these four pyramids, a two-fold 'squaring of the circle' - a squaring in terms of the circumference of the circle and in terms of the area of the circle - was apparently used to create a geometric diagram which then provides the basis for:
1) the height above base level for each entrance passage;
2) the descending angle of each entrance passage;
3) the length along the floor of each descending passage; and
4) the lateral location of each entrance passage - and of the interior chambers themselves.
The methods to be described below lead to results which consistently prove to be within inches of, and in a few cases exactly match, published survey findings. The diagrams presented not only explain why the passageways and chambers were built where they are in the pyramids discussed, but these diagrams also give very strong clues as to where as yet undiscovered features are likely to be located.
I will repeat as we go along, but let me say at the outset as clearly as I can, that all of the mathematics and geometry to follow are achievable through very basic and simple empirical means. Although much of the discussion here will be in terms of modern understandings (for example, decimal unit notation is used and angles are often spoken of in terms of degree measurement), it will be shown that all of these undertakings are empirically achievable within the context of the 'royal cubit' measuring rod, and within the context of the mathematical - and angle measurement - systems which current scholarship acknowledges to have been in use as early as the Old Kingdom period. For those wishing to see a "modern" computational analysis for the lengths derived within each diagram, these can be found by linking to the Computations page.
We begin by noting that the Egyptians had indeed developed an empirical method by which they could 'square the circle' in terms of the circle's area. This means that they could - for any given circle - determine the size of the specific square which would contain the same area as that circle. Their method was to measure the diameter of a circle, find 8/9ths of this diameter length, and then square the 8/9ths amount.1 This simple algorithm is at once easy to remember, easy to implement, and remarkably accurate. The diagram below shows such an 8/9ths square centered on the circle's diameter.
Since each side of the square is equal to 8/9ths of the circle's diameter, centering the square around the circle's midpoint clarifies the finding that half of the square's side (S/2 = OR) will be equal to 8/9ths of the circle's radius (OB). The relevance of this distinction will become apparent shortly.
Seeing that the Egyptians had determined a means to represent the area of a circle in terms of a square, it is then reasonable to wonder if they were curious to learn whether this same square (or some other square) might not have (for all intents and purposes) the same total perimeter length as the circumference of the circle. There is no known written proof that the Egyptians had devised a "formulaic" method for determining the circumference of a circle, but it is credible that they would have sought a means to do so. As it turns out, they did have directly at hand a measurement system which would have easily and accurately allowed them to find the exact square having the same perimeter length as any given circle's circumference.
The foremost standard unit of measurement used by the Egyptians was the "royal cubit". This length, equal to about .524 meters, was marked out into 28 subdivisions which were called "fingers". If the diameter of a circle is made equal to a royal cubit in length, then this diameter will as a result be 28 "fingers" long. If a length of cord or twine is marked off into these same "finger length" units, it can easily be found that the circumference of this circle will then measure to be almost exactly 88 of these finger length units. If one then draws a square having a total perimeter length equal to these 88 units, this square will by necessity have a side that is 22 units long (from 88 divided by 4).
We see this "circumference" square centered on its circle in this next diagram. Note that since the diameter of the circle equals 28 fingers, the circle's radius (i.e., OB in Figure 2) will be 14 fingers in length.
By centering the square in this way we can see that half the side (i.e., OT) of the "circumference" square will be 11 fingers long, and hence will be 11/14ths of the length of the circle's radius (OB). It is therefore clear that by "squaring the circle" in terms of circumference we have derived a square that is not the same size as the square seen above for the circle's area correlation (which was a square whose half-side was 8/9ths the length of the radius). It is also clear that it would have been child's play for the Egyptians to have arrived at the above circumference result. A number of Egyptologists have indeed acknowledged this possibility - if not probability.2 (The reader may wish to try for himself the method described above in order to confirm how easily and accurately it can be done.)
The pathway to the derivation of pyramid interior design now lies in the direction of merely combining the above two 'squaring the circle' findings into a single diagram.
The above diagram reflects this
combination, and so contains the square having the same area, and the
square having the same perimeter, as the circle whose radius is
length OB. Therefore, half the side of the outer square (i.e., S/2 =
OR) will be 8/9ths the length of radius OB, and half the side
of the inner square (S/2 = OT) will be 11/14ths the length of
radius OB. This diagram is the derivation beginning point.
The Khafre Pyramid - part 1
Implementing the Algorithm
With the height of the Khafre Pyramid being 2/3rds the length of its full base side, the pyramid is in cross-section two back to back 3-4-5 right triangles (i.e., where the height = 4, the 1/2 base = 3, and the side = 5).3 However, the diagrammatic basis for the design of this pyramid appears to have been a bit more complex. In using the same approach here that was employed with the Red and Bent Pyramids of Dahshur it is found that - as with the Bent Pyramid - there is a veiled design element.
The Khafre design contains, on its conceptual level at least, a pyramid which has the same 14/11 proportions as that determined for the Khufu Pyramid - its sister pyramid on the Giza Plateau. Once this factor is taken into account, application of the 'squaring the circle' rationale quickly leads to the basic derivation of the layout of the Khafre Pyramid's interior.
In Figure 4 below, a circle has been drawn: 1) using the pyramid's half-base (OB) as its radius (and so radius "R" of this circle =1); and 2) a second circle is added using the pyramid's height (OC) as its radius (and so here R = 1.333..); and, lastly, 3) a third circle is drawn using a radius of 14/11ths x OB (and so R = OP = OL = 1.2727..).
There could now be drawn two squares for each of these three circles - a square with the same area, and a square with the same perimeter. However, all that will be needed at this point is: 1) a square having an area that is equal to the area of the outer circle, meaning that half the side of this square has a length of 1.3333 x .88888 units (and so S/2 = 1.185185..); and 2) a square which has an area equal to the area of the R = 1.2727 circle, and therefore half of this square's side (= S/2) will be 1.2727 x .88888 = 1.1313 units long.
The 'circle and square' intersection at Point P in Figure 4 defines a right triangle which has a base of 1.1313 and a hypotenuse of 1.2727. The angle made by Line OP with the horizontal is therefore 27°16'. The right triangle associated with Point L has a base of 1.185185 and a hypotenuse also of 1.2727. The angle made by Line OL with the horizontal therefore computes to be 21°22.6'. These two angles closely match the reported findings for the inclinations of the Khafre Pyramid's two descending entrance passages. Maragioglio and Rinaldi list "about" 26°46' for the upper passage and "about" 22°40' for the lower.4 Legon found 26°28' for the upper passage and 21°40' for the lower.5
However, as with the Red, Bent and Khufu Pyramids, a shift - or translation - must be effected. One initial unified, and fairly large, shift will accurately account for both of the entrance elevations and lengths, while an additional unified, and much smaller, adjustment made solely to the entire lower passage will then accurately account for its horizontal placement and for its peculiar layout. As seen in Figure 5 below, the initial shift takes the entire design pyramid and 'slides' it upwards along its southern side until the pyramid's base intersects with Point P, which is the 'circle and square' intersection location point where the 1.2727 radius circle intersects with the side of the S/2 = 1.1313 square..
This has been done in Figure 5, and consequently line OP has been extended to the north side of the now-shifted pyramid, here labeled Point D. Line OL has likewise been extended to Point E, which lies on the northern extension of the now-shifted pyramid's base level. Also added in Figure 5 is the square which has the same perimeter as the circumference of the outer circle (R = 1.333), meaning that half its side (S/2) equals 1.3333 x 11/14 = 1.0476 units.
Given the angles already mentioned for the lines OP and OL, all relative positionings are computable. (Refer to the Computation section.) When this is done it is found that the height of D above the pyramid's now-shifted base will be .11374 units in relative terms. With OB made equivalent to the 107.6 m actual surveyed length of half the base of the Khafre Pyramid, the diagram's elevation of D (that is, the pyramid's upper entrance) works out to be 12.24 m. This is in very close agreement with published findings.6
In Figure 6 below, we have expanded the view of the post-shift situation of the pyramid's upper and lower passageways as seen in Figure 5. All is the same as before except for: 1) the addition of line OF - drawn from the center of the original circle to Point F, the intersection of the (R = 1.333) outer circle with the 1.1313 line; and 2) the addition of the square having the same area as the circle on the pyramid's half-base (where R = 1, and so S/2 = .8888), the side of which is seen in Figure 6 passing through line OF at Point S.
As the line of the upper passage (DP) descends through Point P, it intersects the vertical 1.0476 line at Point V. The diagram's relative length of line DV is .34244 units. Again, with one unit being the actual pyramid's half-base length of 107.6 m, DV works out to be 36.85 m. Legon measured a length of 36.95 m for this interval in the actual pyramid.7
Similarly, the diagram provides a lower descending passage length (EL) of 35.18 meters. Legon measured this corresponding length to be 34.94 m.8 The lower passage next enters a horizontal section, seen as length LT in Figure 6. Legon measured this portion to be 15.76 m. (1989, 30), and the diagram has it at 14.8 m.
The diagram next has the upward sloping section of TQ meeting the R = 1 circle directly across from, and at the same elevation as, Point S. Line section TQ therefore computes to be 25.14 m long. Legon measured this sloping section to be 24.34 m. (1989, 30). Interestingly, the diagram's combined measurement from Point L to Point Q gives almost exactly the same length as found by Legon for this combined distance.
Those who are familiar with the Khafre Pyramid may have noticed that Figure 6 shows Point V (that is, where the upper descending passage turns southward to the horizontal) to be directly above Point T (where the lower passage turns upwards from its horizontal section), and that this is not the alignment seen in the actual pyramid. Legon's findings (1989, 30-32) also place Point V higher than Point T, but show that it is 3.7 m to the south (that is, to our left) of the point at which the Lower Passage begins sloping upwards (this point labeled here as Point T). The reason for this departure from the diagram has to do, I believe, with the as-constructed location of Point E, the entrance to the lower passage.
The diagram places Point E (the entrance to the Lower Passage) at a distance of 5.63 m from the northern base edge of the pyramid. However, Legon (1989, 31) has measured this actual distance to be 9.33 m. The 3.7 m difference can possibly be explained by noting that the roof of the actual lower descending passage, and not its floor, was constructed to coincide with the 5.63 m point. (See Figure 7 below.) Maragioglio and Rinaldi found that the interior vertical height of the Lower Passage has an average length of 1.32 m, meaning that due to the passageway's angle of inclination the horizontal distance from the passageway's roof to its floor is very nearly the 3.7 m distance in question.9
It is not clear if the roof placement at the 5.63 m mark was due to an intentional alteration of the original plans (most likely), or to some form of error that was made at the time of construction (less likely). The net result, however, is that relative to the upper part of the pyramid, Points E, L, T and Q (of Figure 6) have all been pulled approximately a 3.7 meter distance to the north. (Note that 3.7 meters is almost exactly 7 royal cubits. It is possible that the rationale for the exact magnitude of this shift was based on achieving a 7 royal cubit correlation. See Part 2 of this essay for more on this.)
This portion of the analysis will conclude with a look at the placement of the main chamber of the Khafre Pyramid, which sits on the horizontal passage that is represented in Figure 6 by the line SG. The south wall of this chamber has been surveyed to lie 1.17 m north of the pyramid's central east-west axis.10 The chamber itself is 4.98 m wide north to south, and therefore its midpoint is located almost exactly 3.7 m north of the pyramid's center.11 It is reasonable to suppose, then, that the center of the main chamber was placed directly beneath the pyramid's east west axis in the original plans, but that it - along with the immediate passageway leading to it - was pulled the same 3.7m distance north as were the Points E, L, T and Q, and that this was all done in response to the repositioning of the entrance to the pyramid's lower passageway so as to align it relative to the original design along its roof rather than along its floor.
Legon's survey results imply a distance of 64.86 m from the bottom end of the upper descending passage to the center of the pyramid.12 The diagram predicts this distance (line VG) to be 65.66 m.
Maragioglio & Rinaldi delve into a general discussion of possible changes made at the time of the pyramid's construction to the original plans for both the upper portion of the pyramid and its lower passage.13 The 'squaring the circle' theory offers opportunities for further detailed analysis of the upper and lower passages, and for insight into the rationales for these changes. These issues will come into consideration in Part 2 of this essay.
Table 1 gives a comparative summary of the findings that have been given in the text for the Khafre Pyramid's upper and lower passageways.
The Khafre Pyramid: Part 2
The elevation, and the north to south location of this pyramid's main chamber, has been accounted for in Part 1 of this essay. We will therefore now be looking at the lateral location of the Upper and Lower Descending Passages; the east to west location of the Main Chamber; a comprehensive rationale for the location of the Lower Chamber; a more complete analysis of the Upper Descending Passage's precise location; and an explanation of the short and unfinished corridor (labeled as "Corridor X" by Maragioglio and Rinaldi) which is located just above the Lower Ascending Passage.
The Lateral Location of the Two Descending Passage Entrances
Had the derivation of the interior layout of the Khafre pyramid been exactly like that seen for the Red and Bent Pyramids, it would have had an initial diagram very much like that shown in Figure 8 above. Although not used for the main rationale for the Descending Passage elevations, this is the diagram that was apparently used to determine the various lateral locations we are now about to seek. Note that the derivation for the Khafre Pyramid was based upon both a design pyramid having an exterior angle with a tangent of 1.333, and an inner pyramid having an exterior angle with a tangent of 1.2727 - with both of these having the same base length. (A further and more detailed explanation of this scenario is given in Part 1 of this essay.)
Petrie, in his survey of the Khafre pyramid, found the central "axis" of the Upper Descending Passage to be 490.3 inches (= 12.454 m) east of the pyramid's north side midline.14 A variation on the theme that has been employed in the previous pyramids to determine lateral locations was perhaps put to use here. Instead of using a perpendicular dropped from the pyramid's side at a point at which a radial line from the diagram's center passes through, a perpendicular appears to instead have been dropped from the point at which the top horizontal line of the 'S/2 = .7857' square passes through the inner pyramid's north side. (The inner pyramid being the 'tangent = 1.2727' pyramid. Refer to Figure 8). This perpendicular then meets the pyramid's base, at actual scale, 12.62 m ( = 497 inches) to the left of the midpoint of the pyramid's half-side. Superimposing the midpoint of this half-side over the midpoint of the pyramid's full northern side will, then, provide the design basis for what I believe to have been the intended location of the axis of the Upper Descending Passage.15
Although this is quite close to Petrie's finding of 490.3 inches, it is about 7 inches further to the east, which in this instance is a bit of a surprise considering the accuracies of the rest of the correlations which follow. I suggest that it may be possible that Petrie's published finding of 490.3 inches was in fact a misread of an actul 496.3 inch finding recorded in the field.16
Maragioglio and Rinaldi make clear that the line of the Lower Entrance Passage is "exactly below the upper entrance", and that pyramid's "whole system of internal passages is on the same vertical plane".17 Therefore, the above derivation will speak for the Lower Passage's east to west lateral location as well.
The Lateral Locations of The Main Chamber
As given in Part 1 of this essay, the south wall of the Main Chamber lies 1.17 m north of the pyramid's east to west axis. Since this chamber measures 5 meters north to south, its north wall lies 6.17 m north of this axis.18 A perpendicular dropped from the point at which the radial line to intersection point # 16 crosses the inner (i.e., the tangent = 1.2727) pyramid's side will meet the pyramid's base 6.18m to the left of the half-side midpoint. Align this midpoint over the midpoint to the full western side of the pyramid and you will have derived the Main Chamber's north wall location. (See Computations section. Also, note that this is precisely the same technique used in the derivations for the north walls of the interior chambers of the Red and Bent pyramids.) This positioning therefore places the center of the main chamber 3.68m north of the pyramid's central axis, and as pointed out in the Part 1 essay, this distance equals 7 royal cubits. As a result, this alignment may be the initiating factor behind the 7 royal cubit northward shift of the entire Lower Passage seen proposed in the earlier section.)
According to Maragioglio and Rinaldi's diagrams, the Main Chamber's west wall lies 1.19 m east of the pyramid's north to south axis, and its east wall sits 15.34 m east of the same axis.19 (See Figure 9 below.) Apparently, the chamber's west wall location was derived from Figure 8 by dropping a perpendicular from the point at which the radial line to Intersection Point # 12 crosses the pyramid's outer (tangent = 1.333) side. This perpendicular meets the pyramid's base at a point that is 1.31 m to the right of the midpoint of the half-side. With this half-side midpoint scenario superimposed over the midpoint of the pyramid's full southern side, one then has the intended west wall location. (Should the descending passage actually be 497 inches east of the pyramid's north/south axis rather than the 490.3 inches given by Petrie, then the surveyed location of the west wall would sit 1.36m from this axis, i.e., making it an even closer correlation with the diagram's prediction.)
The Main Chamber was evidently designed to be 27 royal cubits long and 9.5 royal cubits wide. As we've seen before, a chamber's dimensions were apparently based upon a desire to use uncomplicated multiples of the royal cubit. This essay will not attempt to explore further the basis for the specifics of a chamber's dimensions, only the rationale by which it was positioned relative to the pyramid's base and central axis.
The Lower Chamber
Reference to Figures 5 and 6 in Part 1 of this essay may here be found helpful. In these two diagrams it was seen that the initial design pyramid was shifted upwards and to the right in order to determine the layout of the Upper and Lower Descending Entrance Passages. The horizontal portion of the Lower Passage, Point L to Point T in Part 1's Figure 6 (which is reproduced below in Figure 11), can be seen to extend from the 1.185185 line to the 1.0476 line - these two lines being respectively the sides to the squares having the same area, and the same perimeter length, as the 1.333 unit circle.
At the middle of the Lower Horizontal Passage in the actual pyramid, and in that passage's west wall, there is an opening to another passage which descends down to meet the Lower Chamber itself.20 We will begin this part of the analysis by deriving the basis for the north/south location of this opening in the Lower Passage west wall down to the Lower Chamber.
According to the derivation diagram for this pyramid, the midpoint location between Points L and T lies half way between the 1.185185 and 1.0476 lines, meaning that this midpoint is 1.1164 units from the diagram's center. However, the diagram's 'center' in this case is the central axis of the initial diagram, and therefore also of the pre-shifted pyramid. The rightward component of the shift moves the entire pyramid .4373065 units to the right (north) relative to both its original pre-shift central axis, and relative to its associated circle and square scaffolding. As a result, the 1.1164 point winds up being (1.1164 - .4373065 =) .67909 units from the post-shifted pyramid's central axis. At actual scale, this translates to being 73.07 meters north of this post-shifted axis. To this is then added the 3.7 meters shift to the north that was imposed only to the Lower Passage and its continuation along the Upper Horizontal Passage, thereby leading to a final location that is 76.77 m north of the post-shifted pyramid's center. 76.77 m north of the pyramid's center is within inches of Legon's survey findings (i.e., 76.58 m north of center) for the location of the east to west axis of the opening to the Lower Chamber.21 We have thus shown that the diagram's predicted center of this Lower Horizontal Passage does appear to correlate with the same relative position in the actual pyramid, but why was this center point chosen for a chamber location in the first place?
As explained in the Part 1 essay, the inner pyramid in the above Figure 8 diagram has the same relative proportions as the Khufu Pyramid (it has a 14/11 = 1.2727/1 height to base ratio). A perpendicular dropped from the point at which the radial line to Points # 2, 3, 4 crosses the side of this inner pyramid will meet the pyramid's base - at the scale of the actual pyramid - 76.58 meters north of the pyramid's midpoint . This exactly matches Legon's measurements, and is, I believe, the intended rationale for the middle of the opening to the passage to the Lower Chamber.22
The findings of Legon, and of Maragioglio and Rinaldi, show that from the floor of the Lower Horizontal Passage to the floor of the Lower Chamber is a drop in elevation of 2.44 meters, and that the west wall of the Lower Chamber sits about 16.7 m west of the Lower Horizontal Passage's west wall.23 Figure 10 below is a sectional view of this setting.
The north/south midline of the Lower Horizontal Passage, being in the same plane as the midline of the Upper Descending Passage, therefore sits 12.454 m east of the pyramid's north/south axis. The width of the Lower Passage is given as 1.04 m, and so the Lower Chamber's west wall works out to be about (.52 m + 6.25m+10.43m =) 17.22 m west of the Lower Horizontal Passage's midline. This means that the Lower Chamber west wall is located (17.22 m - 12.454 =) 4.77 m to the west of this north to south axis. In Figure 10 above, if a perpendicular is dropped from the point at which the radial line to Intersection Point # 16 crosses the outer pyramid's side, this perpendicular will meet the pyramid's base at a point that is 4.94 m to the left of the midpoint of the half-side. As was the case in locating the west wall of the Main Chamber, if this arrangement is now superimposed over the midpoint of the pyramid's full southern side, it will provide the Lower Chamber west wall's intended east to west location.24
As with many of the other chambers we have looked at, the size of this chamber was designed in terms of uncomplicated royal cubit amounts, in this case measuring 6 by 20 royal cubits. As a result, only one determinative is needed for its north/south location, and only one determinative is needed for it east/west location. The sloping passage into the chamber meets the chamber's east wall directly at its midpoint, and so the chamber itself is symmetrical north to south about the north to south 76.58 m line identified and derived above.
The floor of the Lower Chamber, according to Legon's survey, lies (12.9 m + 2.44 m =) 15.34 m below the level of the pyramid's base.25 If the perpendicular from the point at which the radial line to Point # 20 crosses the inner pyramid's side is slid down to now be at the point at which the radial line to Point # 16 crosses the inner pyramid's side, it will then extend 15.06 m below the pyramid's base level. This was possibly the rationale used for the Lower Chamber's depth, and is a mechanism which is entirely consistent with the method that was used to determine Chamber elevations in the Red and Bent Pyramids.26
The Upper Descending Passage, The Upper Horizontal Passage, and "Corridor X"
In the original derivation diagram for this pyramid (Figure 5 in Part 1 of this essay), the elevation of the entrance to the Upper Descending Passage - converted to actual scale - is 12.24 m above the pyramid's base level, and the end point for this passage is at 4.64 m below the pyramid's base level. Legon determined the actual entrance to have been at an elevation of 12.9 m, and the endpoint to be at a depth below base level of 3.55 m.27 He also found the length of this passage to be 36.95 m, which is within a decimeter of the diagram's 36.85 m prediction. Although the discrepancies between the diagram and survey measures for these elevation and depth findings are relatively small, it is curious that the predicted passage length is then nearly exactly the same as the surveyed actual. Further analysis of this situation would therefore seem to be warranted, and in fact, a deeper look does provide a number of new insights into the pyramid's possible design history. As seen in Figure 11 below, the lower endpoint of the Upper Descending Passage (Point V) is determined by the 1.0476 line in the diagram, and as stated above, this endpoint sits at the relative depth of 4.64 m below the base level of the shifted pyramid.
With the pyramid being shifted upwards and to the right as part of the design process, the axis of the shifted pyramid winds up being .4373065 units north of its original location (Point C lies on this shifted axis). Therefore, Point V - the point at which the Descending Passage makes its turn southward to become the Upper Horizontal Passage - will then be (1.0476 - .4373065 =) .610295 units to the right (north) of the shifted pyramid's central axis. Another way of saying this is that the end of the Descending Passage is located (1 - .610295 =) .389705 units south of the pyramid's northern base edge (.389705 units at actual scale = 41.93 m).
Although Legon found the length of the Upper Descending Passage to be the same as that predicted by this theory, he found that it feeds into the Upper Horizontal Passage 42.77 m south of the pyramid's northern base edge - that is, it is .84 m further south than the diagram's prediction of 41.93 m.28 Does the diagram offer some form of explanation for this .84 m difference? The answer is that it does.
In the Figure 8 diagram, if a perpendicular is dropped from the point at which the radial line to Point # 10 crosses the outer (tangent = 1.333) pyramid's side this perpendicular line will meet the pyramid's base at a point that is (.601898 units x 107.6 m =) 64.76 m north of the pyramid's center. Therefore, this point is (107.6m - 64.76m =) 42.83 m south of the pyramid's north edge - a correlation that almost exactly matches Legon's findings for the north to south placement of the endpoint of the Upper Descending Passage. It appears, then, that the architects decided to have the Descending Passage end at the # 10 perpendicular line instead of its original location in the diagram. That this was the rationale for the as-built placement of this endpoint will be reinforced by the sequential correlations now to be described. We will look at these correlations before proceeding on to examine how this slight change in endpoint location also had the effect of changing the elevation of the entrance to the Upper Passageway on the pyramid's north side.
Located along the Upper Horizontal Passage, just a short distance south of the Descending Passage endpoint, is a large opening in the passage floor. Maragioglio and Rinaldi describe the northern edge of this opening as being the edge of a never completed "Corridor X ", which they surmise to have been a passage that was begun in error at this juncture during the pyramid's construction.29 (See Figure 12 below.) This unfinished "corridor" was then completely filled in and walled off by the pyramid's builders, and Ascending Passage "A" (so named by Maragioglio and Rinaldi), which rises from the south end of the Lower Horizontal Passage, was then built instead.
According to Maragioglio and Rinaldi's scaled drawings, the northern edge of the opening to Corridor X is located about 6.84 m south of the point where the Upper Descending Passage ends and turns southward to the horizontal.30 With this endpoint being 64.76 m north of the pyramid's center (64.76 m being the point defined above by the perpendicular associated with Intersection Point 10), then 6.84 m to the south of this will be (64.76 m - 6.84m =) 57.92 m north of the pyramid's center. This, then, is the proposed relative placement of the north edge of this north edge opening to "Corridor X" along the Upper Horizontal Passage.
If, in Figure 8, a perpendicular is dropped from the point at which the radial line to Point # 11 crosses the outer (tangent = 1.333) pyramid's side, it will meet the pyramid's base at a point that is 58.15 m north of the pyramid's center. This is only .23 m greater than the 57.92 m distance we have just inferred from Maragioglio and Rinaldi's findings, and is therefore the likely design rationale for the location of the north edge of this opening in the floor of the Upper Horizontal Passage. We will be looking a bit further into the placement of Corridor X after first identifying one last alignment along the Upper Horizontal Passage.
With the filling in and walling up of Corridor X, and with the subsequent construction of Ascending Passage "A", the north edge for the opening in the floor for Ascending Passage "A" was then positioned at a point that Maragioglio and Rinaldi measured to be about 2.73 m yet further south than the north edge for the opening to Corridor X.31 (Refer to Figure 12 above.) This means that this Ascending Passage point is (57.92m - 2.73m =) 55.19 m north of the pyramid's center. A perpendicular that is dropped from the point at which the radial line to Point #12 crosses the outer (tangent = 1.333) pyramid's side in Figure 8 will meet the pyramid's base at a point that is 55.11 m north of the pyramid's center. This perpendicular was then the basis for the placement of the north edge of the Ascending Passage opening.
We can infer from the foregoing sequential correlations that, at some point in the design process, a decision was made to create alignments along the Upper Horizontal Passage on the basis of the perpendicular protocol process being described here. In so doing, three consecutive angle intersections were employed - these being those labeled as Points #10, 11, and 12. It is not clear what the design basis was for the location of the sloping floor to Corridor X, although it is worth noting that by measuring from Maragioglio and Rinaldi's drawings one finds that the floor of Corridor X appears to meet the Upper Horizontal Passage at a point that is quite close to the midpoint of the half-side. The diagrammatic rationale for the location of the endpoint to the floor of the Ascending Passage "A", at the point where it meets the Upper Horizontal Passage, has already been discussed in Part 1 of this essay.
This usage here of consecutive Intersection Points is interestingly reminiscent of what was seen in regard to the design parameters of the Bent Pyramid's Upper Chamber.
The Upper Descending Passage - A Closer Look
As detailed above, the decision to reposition the location of the bottom end of the Upper Descending Passage at the perpendicular associated with Intersection Point #10 evidently included the desire to retain the derivation diagram's original relative length for the Upper Descending Passage (which at actual scale = 36.85 m).
In Figure 13 below, Line BA is the derivation diagram's Descending Passage line (seen as Line DV in Figure 11), and Line EP is the line of the Descending Passage as it was actually constructed in the pyramid.
The decision to move the endpoint of the Descending Passage to the perpendicular associated with the radial line to Intersection Point # 10 was accompanied by yet one other decision, one which was made necessary by this endpoint adjustment. Since the length of the Descending Passage was to remain unchanged from that given by the diagram, a determination was still needed as to where up or down along the 'Point # 10 perpendicular' the endpoint for the passage should be placed. I believe that the architects decided to use the depth of Point S, as it is seen in the diagram of Figure 11, for this purpose.
Figure 11's Point S identifies the intersection of line OF (which is the same line as the radial line to Point # 5 in Figure 8) with the .8888 line (which is the side of the square whose area is the same as the 1 unit circle). Computation will show that Point S is 3.082 m below the pyramid's base level, and I propose that this was the depth used to decide the elevation of the Upper Horizontal Passage floor.32
Once the decision is made to establish the endpoint of the Descending Passage along the Point # 10 perpendicular at a depth of 3.082 m (shown above as Point P), it turns out that there are then two, and only two, locations at which a Descending Passage with a length of 36.95 m will precisely end at the pyramid's north side.33 One of these is at an elevation of 25.42 m (Point E2 in Figure 13 above), and the other is at an elevation of 12.64 m (Point E above). This latter elevation is quite close to both the diagram's elevation (= 12.24m) and the surveyed elevation (= 12.9m) for the entrance location. It would be interesting to learn whether anything was built into the pyramid that might in any way mark the location of the 'Point P to Point E2' line identified in this analysis.
An unexpected, and possibly significant, correlation develops when one carefully charts out a scaled drawing of the derivation diagram's position for the Upper Descending Passage in relation to the surveyed positions of Ascending Passage "A" and Corridor X. Figure 13 gives a broad view of this situation, revealing that a continuation of the derivation diagram's orginal line of the Upper Descending Passage (Line BA) will apparently meet with one of the horizontal features located at the lower end of Corridor X. A closer look at the particulars of this arrangement is actually given in Figure 12, where we see that the line of the diagram's Upper Descending Passage meets the lower of the two ledge-like features at the bottom of Corridor X just at that point where the ledge is bordered by a short north wall
One possible scenario that this suggests is that the architects may have originally planned to have the Upper Descending Passage end at this lower ledge-like feature, and then have it turn south to continue horizontally for a short distance before entering a chamber whose north wall was aligned with the perpendicular line associated with Intersection Point # 11. We have seen similar such arrangements in the analyses presented earlier for the Red and Bent Pyramids. The lower ledge-like feature in question here appears to be about 7.1 m below the pyramid's base level.34 Point S in Figure 11 computes to be 7.28 m below the pyramid's base, and may well have been the rationale used for the depth location in this instance. Furthermore, the short wall bordering the north side of the lower ledge measures to be precisely in line with the vertical 'S/2 = 1' line seen in Figure 11, meaning that the Descending Passage (in the early plans) could have been intended to end at this 'S/2 = 1' line.35
Another possible scenario shown in Figure 12 is that the original plan may have been to have the Upper Descending Passage intersect with a northward horizontal extension of the uppermost ledge-like feature, thus forming a short horizontal passage which then would have proceeded southward either to a chamber such as described above, or to the ascending floor of Corridor X. Although this arrangement may have been the original intent, I do not see a rationale within the context of the derivation diagram which would independently account for the choice of a horizontal passageway feature at this elevation.
One additional possibility is that the earliest plans may have called for Corridor X to begin its ascent at the southern end of the Lower Horizontal Passage, with this southern end then being at the point at which the vertical 'S/2 = 1.1313' line passes through. (Refer again to Figure 11). It would appear, however, that further clarification of any of the above conjectures may have to wait until a more detailed survey of Corridor X and its immediate surroundings becomes available. For instance, the present survey material of which I am aware does not include a precise finding for the angle of inclination taken by the intended floor of Corridor X.
The derivation of the design of the Khafre pyramid shows clear evidence of being evolutionarily descended from the design derivations of the Red and Bent pyramids. By the time of the Khafre construction the method for determining entrance passage elevations was now no longer the sliding down of a diagrammatic line segment from one intersection point to an intersection point immediately below. In fact, this change in approach had already occurred by the time of the derivation of the Khufu pyramid, a slightly earlier derivation which also demands that there be an initial shift involving the entire design pyramid.
Furthermore, in the Khafre plans we learn that the design was based on the existence of a veiled pyramid element which has the same 14 to 11 ratio as the Khufu structure. Unlike what was seen with the Bent Pyramid, however, there is no known overt visual clue that a second design parameter is involved.
The algorithm presented in this and the accompanying essays does not profess to explain all of the aspects of Egyptian Pyramid design, but rather is focused on describing the foremost imperatives driving the design of interior passageway and chamber locations, at least as can be seen in the four pyramids discussed. A remaining question is then whether the concerns and methods of the proposed algorithm were indeed those of the architects of these pyramids. I submit that the ability of the protocols to simultaneously account for the angle, length, and elevation of each passage, and to do this for not only one but for a series of structures, argues against the accuracy of the results being due to some form of amplified coincidence.
An important factor acknowledged to have been left unresolved thus far is the issue of why there was a shift, or transfer, of a prominent line from one intersection point to another such point in the diagram for each pyramid. The answer, I believe, likely lies in one, or two, not necessarily contradictory directions. Either these shifts were done in an attempt to have the pyramid embody as much of the 'circle and square' convergence information as possible, and/or they were done in an attempt to preserve the secrecy of what was at the time deemed to be privileged knowledge.
If an intentional discontinuity had been placed by the architects into the original sectional plans prior to these plans being given to a pyramid builder, it would have meant that any builder - or any unauthorized person - who gained access to the plans for more than one section of the pyramid would have found it nearly impossible to work the geometry backwards to the causative knowledge. The relatively minor, or secondary, shifts (such as were touched upon in the analysis of the location of the Khafre Pyramid's lower passageway and main chamber) were likely also to have been intentionally worked into the builder's plans for the very same reason.
As mentioned earlier in footnote 1, there is surviving written evidence which shows that the 8/9ths correlation was being used by scribes during the Middle Kingdom period to compute a circle's area in 'square' units. It is assumed, therefore, that during the even earlier Old Kingdom period the 'area' squaring of the circle knowledge had already been discovered and was not of a privileged nature. However, these same surviving texts do not appear to touch upon the perimeter to circumference squaring of the circle correlation.36 This latter 'squaring' may in fact have been the knowledge that was to be kept restricted, if indeed it was secrecy that was the major concern.
The Royal Cubit Rod
A few words need to be said about the means by which the scribes could have determined the relative lengths of the various lines, and the slopes of the various angles, that are seen in the diagrams. A knowledge of trigonometry is helpful nowadays in computing these factors, but it is likely that the ancient Egyptians made their determinations through direct measurement. If the proposed diagrams are drawn carefully and of workable size, then surprisingly accurate relative lengths can be determined empirically.37
We know that the royal cubit was divided into seven 'palms' with each palm containing four 'fingers', and that each of these resulting twenty-eight 'fingers' could then be further subdivided into from two to sixteen finer subdivisions.38 Using a cubit rod that is ruled in this way, the relative base to height measure of an angle can be taken directly from the diagram. For instance, in regard to the 27°16' angles seen in the diagrams, if the royal cubit's 28 'fingers' are each divided into 9 sub-units, then for each horizontal run of 1 full cubit (containing 28 x 9 = 252 sub-units) a vertical rise of 130 of these sub-units can be found by direct measurement. Alternately, for each vertical rise of 252 sub-units (= 1 royal cubit), a horizontal run of 489 units will be found.39 The builders may have been given their construction directives in terms of such a 'rise and run' format, although in some circumstances they may have simply been given a premarked template of appropriate size with which to guide the laying out of a particular angle.40 The angle given to the builder may also have been a version of the angle from the diagram that was slightly rounded off to facilitate construction. In cases where such a rounding off may have been used, the constructed angle can not then be expected to exactly match the design angle as it is derived in the parent diagram. Whatever the methods that were used, the Egyptians were clearly capable of doing exacting work, and were inventive enough to find ways to implement that which their designs suggested.
The theory that has been proposed in this essay develops a rationale which successfully determines the interior layout designs of three of the most important pyramids of the Old Kingdom period, showing how these layouts are an integral part of, and in fact derive from, each pyramid's height to base design ratio. An accompanying essay shows that this same rationale also succeeds in explaining the interior layout of the Khufu Pyramid. Given the consistent level of accuracy of the theory's predictions, and given its ability to explain much of what has previously been notably obscure, it is hoped that a broadened consideration will soon be given to the theory's further predictive capabilities and implications.
1. Problem 50 of the Rhind Mathematical Papyrus (RMP) finds the area of a circle by squaring 8/9ths of the circle's diameter. Problem 48 of the RMP is generally interpreted as showing that this 8/9ths understanding was arrived at diagrammatically. See R. J. Gillings, Mathematics In the Time of the Pharaohs (New York, 1982), 139 - 46. It is worth noting that Gillings specifically refers to the scribe who wrote the RMP as being "the first authentic circle-squarer in recorded history" (p. 145). An interesting relationship exists between the proposed 'S/2' value of 8/9ths of the radius and the proposed 11/14ths value, in that by squaring 8/9ths one gets a near numerical equivalent to 11/14ths. This connection between 8/9ths and 11/14ths may very well be the relationship that Archimedes is referring to in Proposition 2 of his treatise on the measurement of a circle. See T. L. Heath, The Works of Archimedes, (Cambridge, 1897), 93.
2. See Legon, "The 14:11 Proportion at Meydum", p. 22, and also, Robins and Shute, "Irrational Numbers and Pyramids", p. 44. A "formula" for finding the area of a circle was a relationship much needed by scribes in computing amounts of grain stored in cylindrical containers. The extant papyri do not reveal immediate evidence that the scribes knew how to compute the circumference of a circle. However, there has been suggestion that perhaps Problem 10 of the Moscow Mathematical Papyrus may provide an instance of this capability. See T. E. Peet, "A Problem in Egyptian Geometry", J.E.A. 17, pp. 104-107.
3. V. Maragioglio and C. A. Rinaldi, L'Architettura Delle Piramidi Menfite Parte V (Rapallo, 1966), 102. Because the height is 2/3rds the length of the full base length, it will be 4/3rds the length of the half base length.
4. Maragioglio and Rinaldi, L'Architettura Parte V, 52, 62.
5. J. A. R. Legon, 'The Design of the Pyramid of Khaefre', GM 110 (1989), 30 Table I.
6. Maragioglio and Rinaldi, L'Architettura Parte V , 52, report the entrance elevation "with reasonable approximation" to be at 12.9m. Legon, in GM 110, 31 Table II, reports the same. The upward shift of the design pyramid to Point P computes to be a rise of .583074 units with an associated lateral shift to the right of .583074/1.3333 = .4373065 units. (Refer to Computations addendum). At the scale of the actual pyramid, these amounts become 62.74m and 47.05m respectively. It is possible, if not likely, that these amounts were slightly adjusted so as to be 120 royal cubits (i.e., .5238m x 120 = 62.86m) and 90 royal cubits (i.e. .5238m x 90 = 47.14m).
7. Legon, GM 110, 30-32, Table I and Figure 2.
8. Legon, GM 110, 30-32, Table I and Figure 2.
9. Maragioglio and Rinaldi, L'Architettura Parte V, 60-62.
10. Maragioglio and Rinaldi, L'Architettura Parte V, 60.
11. The chamber's width is as computed from Legon's findings. See Legon, GM 110, 30-32, Tables. I, II, and Figure 2.
12. Legon, GM 110, 30-32, Table I and Figure 2.
13. Maragioglio and Rinaldi, L'Architettura Parte V, 112-118.
14. W.M.F. Petrie, Pyramids and Temples of Gizeh, Chapter 9, Section 73. Note that Petrie gives his measurements in inches, which are here converted into meters. (Referenced here is the original and unabridged printing of Petrie's work.)
15. A very similar method to that shown here for the lateral location of the Khafre entrance passage was used to determine the lateral location of the entrance passage in the Khufu Pyramid. (Refer to the author's essay on this pyramid.) In both instances, the intersection of the pyramid's side with the topmost line of a pivotal design square was employed (as opposed to the earlier mode of using a radial line intersection with the pyramid's side).
16. As of this writing I have not been able to locate Petrie's relevant field notes from his 1880 -1882 seasons in Egypt, and it may be that these records have not survived. (The Petrie Library in London does not have them, nor do they know where they may be.) I have been unable to find, and am thus unaware of, any other survey of the Khafre pyramid which may have undertaken a more recent and detailed measure of the entrance passage's distance east from the north side's midline. As I have pointed out in my essay on the Khufu Pyramid (refer to footnotes 5 and 12 in that work), there are occasional transcription (or other) errors in the published text of Petrie's Pyramids and Temples of Gizeh. Such may be the case here.
17. V. Maragioglio and C. A. Rinaldi, L'Architettura Parte V, p.60.
18. V. Maragioglio and C. A. Rinaldi, ibid.
19. Maragioglio and Rinaldi, L'Architettura Parte V, Plate 6 Figure 11; Plate 10 Figure 4.
20. J.R. Legon, "The Design of the Pyramid of Khaefre", GM 110 (1989), 33. Also see, Maragioglio and Rinaldi, L'Architettura Parte V, Plate 9 Figure 1.
21. Legon, "Khaefre", Table II and Figure 2, pp. 31-32. Legon lists the center of the passage to the Lower Chamber as Point D, which in Table II he has as being 31.02 m from the pyramid's north base edge. His measures imply that this center point is therefore (107.6 m - 31.02 m) = 76.58 m from the pyramid's center. Note that Legon lists all of his north to south findings relative to the pyramid's northern base edge.
22. Having this perpendicular define the midpoint of the Lower Passage may be the reason that the length of the Lower Horizontal Passage was evidently lengthened by .96m from the diagram's 14.8 m predicted length for this component. As explained along with Table 1 in Part 1 of this essay, the lateral length of the diagram's Ascending Passage is less than the surveyed length by almost this same amount, meaning that the surveyed horizontal combined length for the Lower and Ascending Passages is almost exactly that predicted by the diagram.
23. Maragioglio and Rinaldi, L'Architettura Parte V, Plate 6 Figures 11 and 13. Also see Legon, "Khaefre", Tables I and II and Figure 2, pp. 30 -32.
24. As seen earlier with the Main Chamber, if the axis of the entrance passage is 497 inches from the pyramid's north to south axis rather than the 490.3 inches given by Petrie, then the Lower Chamber's west wall will be (17.22m - 12.62m =) 4.6m west of the pyramid's north to south axis. In this possibility, then perhaps used was the perpendicular associated with intersection point # 11. When this perpendicular is dropped from the tangent 1.333 pyramid side, it will meet the pyramid's base 4.35m to the right of the half-side midpoint. Superimposing this over the midpoint to the pyramid's northern side may then have been the actual rationale used for the west wall location.
25. Legon, "Khaefre", Tables I and II, pp.30 - 31.
26. As with the rationale for the Lower Chamber's north/south location, the derivation for the chamber's floor elevation is also based on the perpendicular dropped from the side of the inner (tangent = 1.2727) pyramid. This appears to imply that there was an attempt, for reasons unknown, to more closely connect this chamber with the 14/11 design pyramid. As shown in the text, however, the chamber's west wall was apparently derived via the radial line intersection with the outer (tangent = 1.333) pyramid. (It may be possible - although, I feel less likely - that the floor level of this chamber was determined by the point labeled "w" in Figure 11. In this diagram, Point w is located 15.09 m below the base of the pyramid.)
27. Legon, "Khaefre", Tables I and II, pp.30 - 31.
28. Legon, ibid.
29. Maragioglio and Rinaldi, L'Architettura Parte V, pp. 60, 112-114. Note that the 3.7m shift which was imposed on the Lower Horizontal Passage and its extension to the Main Chamber was not imposed on this portion of the Upper Horizontal Passage.
30. Maragioglio and Rinaldi, L'Architettura Parte V, Plate 7 Figures 2, 3; Plate 9 Figure 1.
31. Maragioglio and Rinaldi, L'Architettura Parte V, ibid.
32. Maragioglio and Rinaldi, L'Architettura Parte V, p. 54, state that the Upper Horizontal Passage is "about three meters below the level of the courtyard". All of their elevation measurements are given relative to the level of the courtyard pavement, and I accept their assumption that this is the intended base level of the pyramid. Legon does the same in his essay, although he gives a depth of 3.55 m for the Upper Horizontal Passage (see "Khaefre", Table II, p. 31.). The Computation section of this essay provides the derivation of the depth of Point S.
33. Legon's measurement for the intended length of this passage is 36.95 m. (See "Khaefre", Table I, p.30.) The derivation diagram gives 36.85 m for this passage length. Although this difference is small, it is large enough to lead to a surprising disparity when one computes the two possible resulting entrance locations for this smaller length. For a passage length of 36.85 m, the two entrance elevations will be at 13.05 m and 24.99 m. (rather than the 12.64 m and 25.42 m amounts given in the text for a 36.95 m passage length. Refer to the Computation section for these derivations). The 13.05 m elevation very nearly matches Legon's 12.9 m finding.
34. As measured from Maragioglio and Rinaldi, L'Architettura Parte V, Plate 9, Figure 1 and Plate 6, Figure 10.
35. Again measuring from Maragioglio and Rinaldi's Plate 9, Figure 1. This north wall lies about 4.3m south of the endpoint of the Descending Passage, meaning that it is about (64.76m - 4.3m =) 60.46m north of the pyramid's center. The S/2 = 1 unit line seen in Figure 11 computes to be 60.55m north of the pyramid's center.
36. Although I am not aware of anything in the written record that directly relates to either the perimeter mode of 'squaring the circle', or to its associated 11/7ths and 11/14ths relationships, Problem 38 of the Rhind Mathematical Papyrus is not without interest. This problem essentially asks the scribe to divide the number one by 22/7. No mention is made of circles or squares, yet the situation involves numbers which recall those seen in a perimeter 'squaring' instance. See Gillings, Mathematics In the Time of the Pharaohs, 205.
37. Anyone wishing to prove this fact for themselves should have little difficulty in achieving the required accuracy if the initial radius of the circle in the diagram is made, at minimum, a royal cubit in length.
38. N. E. Scott, 'Egyptian Cubit Rods', Bulletin of the Metropolitan Museum of Art 1 (1942), 70.
39. The Egyptians referred to angular measure in terms of 'sekeds', with a seked being the number of cubits and/or 'palms' and 'fingers' of horizontal run required by an angle for each vertical rise of one royal cubit. The 27°16' angle, with the stated horizontal run of 489 of the 1/9th units for each rise of 1 royal cubit (= 252 of the 1/9th units), therefore has a seked of 1 royal cubit, 6 palms, 2 1/3rd fingers (which is the same as 1 royal cubit, 6 and 7/12th palms). In a similar manner, the seked for the design angle of the Red Pyramid, with its height to half-base ratio of 17/18, has a seked of 1 royal cubit and 5/12th palms. The design of the royal cubit rod permits its user to choose in each instance a 'finger' subdivision which best suits ease of measurement and numerical manipulation. Interestingly, when each finger is divided into 9 sub-units, thereby giving the royal cubit a total of 252 of these sub-divisions, we find that 8/9ths of 252 is exactly 224, and 11/14ths of 252 is exactly 198. It is possible that this is the subdivision that was the one most often used by the architects both when setting out the initial diagram, and in the subsequent determination of the various seked relationships. See the Computation section for a further discussion regarding the use of sekeds.
40. An issue of ongoing debate is whether, during construction, slopes were measured in terms of ratios (such as has been presented in this essay), in terms of sekeds, or as sekeds that have been 'rounded off' to the nearest finger or palm. For a discussion of this issue see Roger Herz-Fischler, The Shape of the Great Pyramid (Waterloo, 2000), 34-45; and also see Corinna Rossi, Architecture and Mathematics In Ancient Egypt, (Cambridge, 2004), 203-214. Rossi includes mention of the use of wooden triangular templates to guide slope construction. Legon also discusses this issue in his article "On Pyramid Dimensions and Proportions".
Chace, A. B. 1929. The Rhind Mathematical Papyrus. Oberlin: The Mathematical Association of America.
Gillings, R. J. 1982. Mathematics in the Time of the Pharaohs. New York: Dover.
Heath, T. L. 1897. The Works of Archimedes. Cambridge.
Legon, J. A. R., 1989. "The Design of the Pyramid of Khaefre". GM 110, 27 - 34.
----------- 1990. "The 14 to 11 Proportion at Meydum". DE 17, 15 - 22.
----------- 1991. "On Pyramid Dimensions and Proportions". DE 20, 25 - 34. This paper is available on line at http://www.legon.demon.co.uk/pyrprop/propde.htm
Lehner, M. 1997. The Complete Pyramids. London: Thames and Hudson.
Maragioglio, V. & Rinaldi, C.A. 1966. L'Architettura Delle Piramidi Menfite Parte V. Rapallo.
Peet, T. E. 1931. "A Problem in Egyptian Geometry", J.E.A. 17.
Petrie, W. M. F. 1883. The Pyramids and Temples of Gizeh. London. The text of Petrie's original 1883 publication is currently available on line at http://www.ronaldbirdsall.com/gizeh/index.htm.
Robins, G. & Shute, C. C. D. 1990. The Rhind Mathematical Papyrus. New York: Dover.
----------- 1985. "Mathematical Bases of Ancient Egyptian Architecture and Graphic Art".
Hist. Math. 12, 107 - 122.
----------- 1990. "The 14 to 11 Proportion in Egyptian Architecture". DE 16, 75 - 80.
----------- 1990. "Irrational Numbers and Pyramids". DE 18, 43 - 53. This paper is available on line at http://www.hallofmaat.com/modules.php?name=Articles&file=article&sid=39
Rossi, Corinna. 2004. Architecture and Mathematics In Ancient Egypt. Cambridge.
Scott, N. E. 1942. 'Egyptian Cubit Rods', Bulletin of the Metropolitan Museum of Art 1
Vyse, H. 1842. Operations Carried On at the Pyramids of Gizeh in 1837, V. 3. London
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