The Architect's Plan

 

Ancient Egyptian Pyramid Design:

The Bent Pyramid

Leon Cooper

 

The Bent Pyramid: Part I

The Basic Derivation Algorithm

This presentation examines the method by which the enigmatic interior layout design of the Bent Pyramid can be accurately derived and accounted for. The accompanying essays in this series detail how this method was apparently implemented in order to derive the interior layout designs of the Red, the Khafre, 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 are almost all 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 Bent Pyramid Part 1

 

The design and construction history of the Bent, or South, pyramid at Dahshur is a fairly complex one, as is made evident by the apparent evolution of its development through three distinct stages. Not only was its exterior slope angle altered nearly half the way into construction, but the finished pyramid encases a possibly earlier structure which had an entirely different base length and slope.3

The present pyramid exterior rises from grade level with a general angle of about 54°28'. At a height of about 47.25 m this angle abruptly changes to near 43°22'.4 Leaving aside why an initial angle of 54°28' was used, or why it was then reduced during construction, we will look at how the 'squaring the circle' design protocol provides rationales for:

1) the elevation at which the the bend occurs;

2) the elevation above the base for the pyramid's lower descending passage entrance (north side);

3) the general, if not intended, angle of this lower passage;

4) the length along the floor of this lower descending passage;

5) the elevation above the base for the upper descending passage entrance (west side);

6) the two separate angles of slope exhibited by this upper passage, and

7) the length along the floor of this passage.

 

Location of The 'Bend' and Location of The Lower Descending Passage

 

The diagrammatic protocol for the interior design layout of the Bent Pyramid is essentially the same as that seen in the Red Pyramid. It is somewhat more complicated, however, since there are three circles involved instead of two, and also because there are two entrance passages to this pyramid rather than one. As a result, there are more factors being put into play, but the methodology remains the same. One begins by using the diagram seen in Figure 3 above, with there being added to this the 'circle and square arrangement' which we have seen follows from the triangular cross section of the pyramid being superimposed upon the diagram. In the case of the Bent Pyramid, there are three (rather than two) of these additional 'circle and square arrangements'. We will now see why, and how, this comes to be.

In Figure 4 below, three circles have been drawn: 1) one on the length OP - which is the height that the pyramid would have reached had the 54°28' angle been continued; 2) one on the length OA - which is the actual as constructed height of the pyramid; and 3) one on OS - which is half the pyramid's base length.

 
Figure 4

As is the case with the Red Pyramid, squares having equal perimeters - and squares having equal areas - could have been drawn for each of these circles. For the Bent Pyramid, this would mean a total of three circles, six squares, and twenty two intersection points. (This diagram is included in Part 2 of this essay). For now, however, all that will be needed is the square whose perimeter equals the circumference of the circle on OP (this being the outer square, which has the same perimeter length as the outer circle), and the square having the same area as the circle on OA (i.e., the inner square, which has the same area as the middle circle).5

Again, the pyramid's half-base length (OS in this instance) is made equal to 1 unit. A line from the center of the diagram to the intersection at Point B therefore creates a right triangle having a base of 1.1 and a hypotenuse of 1.4. If OS, half the pyramid's base, is now made equal to the actual pyramid's half-base length of 94.8 m, then the corresponding elevation of Point N will be 47.78 m, a very close match with the elevation at which the pyramid's bend has been surveyed to occur.6 Knowing this measurement and the angles involved, the relative height of OA (the height of the constructed pyramid) can be computed to be 1.1084 units, here rounded off to 1.11. Therefore, the square which has the same area as a circle of radius 1.11 (the inner square of Figure 4) can be found to have a half- side length of 1.11 x .8888 = .9866.

Next, we have that a line from the center of the circle to the intersection at Point E will create a right triangle with a base of 1.1 and a hypotenuse of 1.11. With OS equal to the 'as built' length of 94.8 m, Point K will compute to have an elevation of 11.7 m, which is almost exactly the surveyed elevation (11.9m) for the entrance to the Bent Pyramid's lower descending passage on the pyramid's northern side.7

The intended slope angle for the lower passage is now supplied by the angle of line OD, and the actual (as surveyed) length of this passage is supplied by the part of line OD that extends from Point O to where OD intersects the side of the pyramid at Point H. The angle that OD makes with the horizontal is 27°16', and the relative length of line segment OH is 78 m. Line OH is then next moved to Point K to become the lower descending passage (Line KZ), whose length Maragioglio and Rinaldi found to be 78.6 m.8

Due to damage suffered by this pyramid, either from earthquake or by subsidence - or both, the lower passage is now broken into three distinct sections with each resting at slightly different angles. According to Petrie, these angles range from 26°20' at the bottom of the descending passageway, to 28°22' at the top.9 Averaging these two extremes (if they may be called that) yields a result very near to the 27°16' angle suggested by this theory.

Table 1 summarizes the findings for the Bent Pyramid's lower entrance passage.

The Upper Descending Passage

 

Leaving aside speculation as to why a second descending passage was designed into this pyramid, and why it was placed on the pyramid's western side, we will move directly to a look at the derivation of its elevation, slope, and length.

Due to the fact that the determinations made for the location of the pyramid's bend and the location of the lower descending passage had by this point in the design process utilized three ''circle and square'' intersections, it was evidently thought that the two potentially usable lines still left available in the initial diagram had factors which made them either problematic or otherwise undesirable. Therefore, new diagrammatic components were apparently deemed necessary in order to provide suitable diagrammatic scaffolding for an additional descending passage.

This next derivation may at first seem a bit complex, but in fact it is not - especially when these manipulations are carried out empirically in their original diagrammatic context. As a result of the need for additional intersection points, and as shown below in Figure 5, there was added: 1) a circumscribed square on the R = 1 unit circle such that half of this square's side (S/2) equals 1 unit; and 2) a circle circumscribed within the 1.1 unit square such that its radius (R) = 1.1 units; and 3) a square having a perimeter equal to the circumference of this latter circle - such that half of this square's side (S/2) is equal to 1.1 x 11/14 = .86428 units.

The 'circle and square' intersection at Point Y in Figure 5 allows for the creation of a right triangle with a base of 1 and a hypotenuse of 1.1. The resulting angle that the line OY therefore makes with the pyramid's base is 24°38'. Point W - the point at which line OY crosses the pyramid's side - has a relative elevation above the pyramid's base line of 32.74 m. This elevation for Point W is in exact agreement with Dorner's survey results for the elevation of the Bent Pyramid's upper passage entrance.10

The upper passage begins its descent into the pyramid at a reported angle of 30°09'.11 The 'circle and square' intersection at Point J creates a right triangle with a base of .86428 and a hypotenuse of 1, and therefore the angle made by line OJ with the horizontal is 30°12'. Just as we've seen before, the segment of the second highest 'circle and square' intersection line that lies within the body of the pyramid (here, line OG) is once again transferred down to the point at which the lower-most intersection line (line OY) crosses the pyramid's side (at Point W).

However, the full length of line segment OG (at its 30°12' angle) was apparently not used in this instance, perhaps in order to avoid any possible spatial or structural conflicts with the northern passage's chamber complex, or perhaps simply to incorporate further elements of the pyramid's diagrammatic design history.

In Figure 5, Line OG computes to have a relative length of 77.47 m. Maragioglio and Rinaldi report the full length of the actual western descending passage to be 67.66 m.12 The difference here may be explained by looking at the finding, as alluded to earlier, that the originally planned pyramid for this site was substantially smaller, having been measured to have had a half base length of about 78.6 m, and an exterior slope angle of between 57 and 60 degrees.13 Point X in Figure 5 identifies the point at which the line OJ crosses the exterior side of this vestigial pyramid, and computation will show that line OX is of a length (= 68m) well suited to have been the diagrammatic rationale for the length of the as-built descending passage.

As mentioned above, the upper passage begins its descent at an angle of about 30°09'. After 22 meters, this angle changes to a shallower pitch of about 24°17'.14 Assuming that this change is not due to a major settling of the superstructure, it is possible that this second angle was originally planned to be at the 24°38' angle of the diagram's lower intersection line (line OY in Figure 5), meaning that the floor of the passage has suffered only relatively minor damage due to subsidence.

Table 2 summarizes the findings given in the text for the Bent Pyramid's upper entrance passage.

 

The Bent Pyramid: Part 2

As discussed above, an analysis of the design of the Bent Pyramid is made difficult by two major factors. The first difficulty is that the pyramid's exterior has one slope angle at its lower levels, and a different and more steep slope angle at its upper levels. As already shown, this aspect of the pyramid's design was a part of the plan at an early stage, and was a contributing factor to the derivation of the layout of the two entrance passages.

The second issue that one runs into is that the available survey information for the Bent Pyramid at times lacks the level of detail, precision, and completeness that would be preferred. The reasons for this are varied, but stem largely from damage done to the pyramid's interior by both earthquakes and thieves over the millenia. The reader can find the design rationales for this pyramid's two entrance passages given in Part 1 of this essay, and reference to that discussion is suggested as we now cover many of the pyramid's other design features.

 

 

The Lower Chamber

 

From the Part 1 discussion we saw that, in the derivation diagram, the relative height of the entrance to the Lower Descending Passage is 11.68 meters above the pyramid's base. (1 unit in the diagram is made equal to the actual pyramid's 94.8 meter half the base length). The lateral location of this entrance is directly on the east to west midline of the pyramid's north side.15 If a perpendicular is dropped from the point at which the radial line to Intersection Point # 13 in Figure 6 crosses the pyramid's side, it will meet the pyramid's base exactly on the midpoint of the half-side. When the half-side midpoint is superimposed over the pyramid's full north side midpoint, the derivation rationale for the central axis location of the Lower Passage is accomplished.

The diagram's length for the Lower Descending Passage has earlier been shown to be 77.95 m, meaning that it ends below the base level of the pyramid. (See Figure 4 in Part 1, and the analysis in the Computation section). Figure 7 (below) shows the particulars of this situation.

As can be seen in Figure 7, the derivation diagram's Descending Passage ends 24.03 m below the base level of the pyramid at a point that is 17.17 m north of the pyramid's center. From this juncture, the passageway turns to the horizontal and, according to Maragioglio and Rinaldi's scaled drawings, it proceeds southward about 5.7 m until it ends at the south wall of an antechamber.16 (See Figure 8 below.) The entrance to the main Lower Chamber is located in this same south wall, at a point that is 7.25 m above the antechamber floor.17 Adding this fact to the derivation diagram's findings will place the Lower Chamber floor at a point that is (24.03m - 7.25 m =) 16.78 m below the pyramid's base level.

As a result of these placements, the north wall of the Lower Chamber (which is also the upper south wall of the antechamber) would then appear to be 17.17m - 5.7m = 11.44 m north of the pyramid's center. If in Figure 6 a perpendicular is dropped to the pyramid's base from the point at which the radial line to Point # 18 crosses the pyramid's side, it will meet the pyramid's base at a point that is 10.97 m to the left of the half-side's midpoint. If this midpoint is then superimposed over the midpoint of the pyramid's full western side, it will provide the apparent intended location of the Lower Chamber's north wall. Again, note that this radial line crosses the pyramid's side above the bend. (Refer to the Computations section to see how these intersections above the bend can be calculated.)

That this is the correct method for determining the intended north wall location is supported by the fact that the same radial line is involved in successfully providing the elevation below grade at which this chamber sits. In Figure 6, if the perpendicular dropped from the point at which the radial line to Point #18 crosses the pyramid's side is shifted down to the point at which the radial line to Point #11 crosses the pyramid's side, this perpendicular will then extend 16.7 m below the pyramid's base. (Note that this is the same basic procedure as was seen in Figure 13 of the Red Pyramid essay). This 16.7 m depth is in nearly exact agreement with the diagram's earlier prediction which - in concordance with Maragioglio and Rinaldi's measurements - found that the floor of the 1st Chamber lies 7.25 m above the elevation of the end of the Descending Passage, and hence is (24.03 m - 7.25 m =) 16.78 m below the pyramid's base.

Remembering that the North / South axis of the pyramid is also the axis of the Lower Entrance Passage, we can learn from Maragioglio and Rinaldi's drawings that the east wall of the Lower Chamber is about .75 m east of the north / south axis.18 If a perpendicular is dropped from the point at which the radial line to Intersection Point # 16 in Figure 6 crosses the the side of the "tangent = 1.4" pyramid (the pyramid created by extending upwards the slope angle of the lower section - see the dotted line pyramid side in Figure 6), it will meet the pyramid's base at a point that is 1.09 m to the right of the half-side's midline. Super-positioning this half-side midpoint over the pyramid's full south side midpoint will then provide the likely derivation for the positioning of the Lower Chamber's east wall location.

This Lower Chamber was apparently specifically designed to be 8 royal cubits east to west, and 12 royal cubits north to south.19 It would therefore seem that, as with the chambers of the Red Pyramid, once the derivation protocol had supplied the location of one of a chamber's walls, the distance to the wall sitting opposite to it was then determined by a separate desire to utilize a specific royal cubit length for that distance. This being said, such was apparently not the case with the Bent Pyramid's Upper Chamber.

 

The Upper Chamber

 

The diagrammatic derivation for the elevation of the entrance to, and the length of, the Upper Descending Passage was given in Part 1 of this essay. Not covered was the derivation of the lateral placement of this entrance along the pyramid's western side, and also the particulars for the Upper Chamber. Maragioglio and Rinaldi state that the entrance is 13.7 m south of the the pyramid's western face midline, and that the passage itself is about 1 m wide.20 Given that the 13.7 m distance is from the midline of the western side to the midline (axis) of the passage, it will then be (13.7 m - .5 m =) 13.2 m from the western side's midline to the north wall of the passage. If a perpendicular is dropped to the pyramid's base from the point at which the radial line to Intersection Point # 8 crosses the pyramid's side, it will meet the pyramid's base at a point that is 13.3 m to the right of (i.e., in this case south of) the half-side's midpoint. Superimposing this scenario over the midpoint of the pyramid's full western side will then provide the apparent rationale for the north to south location of the Upper Entrance and Passage. (Note that this is the same radial line that was used to determine the location of the bend in this pyramid. See Part 1 of this essay.)

As it turns out, the locations for all of the Upper Chamber's four walls can be determined via the lateral location protocol. To begin with, survey findings show that the east wall of the Upper Chamber is 13.3 m east of the pyramid's north to south axis, and therefore this location can also be determined by employing the same perpendicular associated with the Point # 8 line used above, but this time by aligning the half-side midpoint with the midpoint of the pyramid's full southern side.21 (See Figure 9 below).

The east to west measure of this chamber is 5.26 m, and so the chamber's west wall sits (13.3 m - 5.26 m =) 8.04 m east of the pyramid's north to south axis. If a perpendicular is dropped from the point at which the radial line to Intersection Point # 11 crosses the side of the 'tangent = 1.4' pyramid (the pyramid created by extending upwards the slope angle of the lower section - as shown by the dotted line pyramid side in Figure 6 above), it will meet the pyramid's base at a point that is 7.94 m to the right of the half-side's midline. If the midline of this scenario is superimposed over the midline of the pyramid's full southern side, the west wall location will be identified.

As seen in Figure 9, the south wall of this chamber is located about .2 m further south than the south wall of the horizontal passage leading into it. Since this passage is a continuation of the Upper Descending Passage, its axis is 13.7 m south of the east / west axis of the pyramid. And since the passage is 1m wide at this point, the chamber's south wall will then be (13.7m + .5m + .2m =) 14.4 m south of the pyramid's east / west axis. A perpendicular dropped from the point at which the radial line to Intersection Point # 7 crosses the pyramid's side (this is below the bend, and so the tangent of the pyramid's side equals 1.4), will meet the pyramid's base at a point that is 14.39 m to the right of the half-side's midline. If one superimposes this scenario over the midline of the pyramid's full western side, the south wall location will be derived.

With the north to south length of the chamber having been measured to be 7.97 m, the chamber's north wall is then (14.4 m - 7.97 m =) 6.43 m south of the pyramid's east / west axis. If a perpendicular is dropped from the point at which the radial line to Intersection Point # 12 crosses the pyramid's side (we're back now above the bend, and back to a 'pyramid side' tangent of .9444), it will meet the pyramid's base at a point that is 6.58 m to the right of the half-side's midline. If one superimposes this scenario over the midline of the pyramid's full western side, the intended north wall location will be derived.

It is quite curious that the four walls of the Upper Chamber are derived from perpendiculars associated with the radial lines to Intersection Points # 7, 8, 11, and 12. Since the radial line to Intersection Point # 8 also passes through Points # 9 and 10, it appears that there was a clear intent here by the architect to utilize these intersections sequentially.

Unfortunately, there is apparent conflicting survey information regarding the elevation of the Upper Chamber. In their diagrams, Maragioglio and Rinaldi seem to imply that the elevation of the floor of both the Upper Horizontal Passage and the Upper Chamber varies from being directly at the pyramid's base level to being 3.2 m above it.22 I propose that the protocol which provides the Upper Chamber elevation is similar to that which has been shown to have been used for the derivation of the 3rd Chamber of the Red Pyramid. Referring again to Figure 6, if a perpendicular is dropped from the point at which the radial line to Point # 21 crosses the pyramid's side, and if this perpendicular is then shifted up to the point at which the radial line to Point # 22 crosses the pyramid's side, it will end 2.27 m above the pyramid's base level. This is the same procedure as was shown in Figure 13 in regard to the Third Chamber of the Red Pyramid, and in both cases the determination is made by the lines which lead to the final two intersection points. It of course remains possible that the original intent here was to have this chamber's elevation be at the pyramid's base level, although this would be a departure from what is seen in the analyses for the interior-most chambers in the Red, Khafre, and Khufu pyramids.

 

Further Observations

 

This analysis has detailed a significant number of similarities between the proposed derivation methodologies of the Red and Bent Pyramids. As a result, and as previously mentioned, it would be extremely interesting to see if these developments can be used to determine the existence of a construction feature which marks a possible Descending Passage line to the Red Pyramid's uppermost (i.e., Third) Chamber.23

The interior layout derivations shown here for the Bent Pyramid strongly suggest that the design rationale was fully dependent upon the final, and as-built, exterior shape and dimensions of the structure. This finding would appear to be in contradiction with the evidence that suggests that the original intent at this site was to build a pyramid with a half-base length of about 78.6 m, and having an exterior whose sides were inclined at an angle of about 60°.24 It is perhaps not a coincidence that a fully realized pyramid of this exterior slope angle and 78.6m half-base length would have had a height almost exactly equal to that of the actual as-constructed Bent Pyramid. This convergence opens the door to the possibility - if not likelihood - that the "precursor" pyramid was part of the plans for the Bent Pyramid from the beginning. A forthcoming essay by the present author will further explore the design relationship of the precursor pyramid with the finished structure.

I do not know what significance, if any, can be attached to the fact that the location of the Lower Chamber is largely dependent upon the radial line to intersection point # 18, while the Upper Chamber seems similarly dependent upon the radial line to intersection point # 8. Similarly focused connections can also be seen in regard to aspects the Khafre Pyramid's design.

 

Concluding Thoughts

 

The algorithm presented in this essay 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 passageways and chamber locations, at least as can be seen in the three pyramids discussed here. An accompanying paper details how these same principles are in evidence in the interior layout of the Khufu Pyramid. 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.25 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.26

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.27 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.28 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.29 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 four 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. 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.

 

Notes

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. Maragioglio and Rinaldi, L'Architettura Delle Piramidi Menfite Parte III, (1964), 98-100. The precursor pyramid is discussed again in Part 2 of this essay.

4. For the angles quoted here see H. Mustapha, 'The Surveying of the Bent Pyramid at Dahshur', ASAE 52 (1952), 597. There has been ongoing discussion regarding the slight variations in some of these angles as detected by survey, and what these variations might signify. For an inroad into this issue see J. A. R. Legon, 'The Problem of the Bent Pyramid', GM 130 (1992), 49-56. As with the Red Pyramid, there is the distinct possibility that the design of the interior features of the Bent Pyramid was based on one specific pyramid height to base ratio, while the actual height to base ratio used in construction was allowed to vary ever so slightly from the ideal in order to incorporate other desired numeric or geometric relationships for the pyramid's exterior. This latter point is made by Legon in his article referenced above (p. 55), and is also implicit in some of Rossi's presentation in Architecture and Mathematics in Ancient Egypt (see especially pp. 210-221). A forthcoming essay by the present author will fuirther explore the details of this issue. Note that Mustapha's finding for the elevation at which the bend occurs is about 2 m. higher than the 47 m. given by J. Dorner in 'Form und Ausmasse der Knickpyramide', MDAIK 42 (1986), 56. In addition, Petrie, A Season in Egypt, p. 30, reported this elevation to be at 47.25 m.

5. With OS being made equal to 1 unit, OP will be 1.4 units long. 11/14ths x 1.4 yields an S/2 of 1.1 units for the square having a perimeter equal to the circumference of the circle on OP. The actual pyramid height (OA in Figure 5 = 1.11 units) multiplied by 8/9ths yields an S/2 of .9866 for the square having the same area as the circle on OA. As shown in the text, once the height of NT in Figure 5 is found, the length of OT is easily determined and thence so is the height AM. OA follows as NT+ AM.

6. As previously stated, Petrie in A Season in Egypt, 30, gives 47.25 m. for the elevation of the bend. Here, he also gives his finding for the length of half of the pyramid's base to be 94.8 m. The diagrammatic derivation of the elevation of the bend can be made solely via the 1.4 pyramid's partculars, and so is fully independent of any exterior angle that might have been chosen for the upper part of the pyramid.

7. Petrie, A Season in Egypt, 30, gives 11.96 m. for the elevation of the entrance. Mustapha, ASAE 52, 599 gives 11.8 m., while Dorner in MDAIK 42, gives 11.9 m.

8. Maragioglio and Rinaldi, L'Architettura Parte III, 60. Mustapha, ASAE 52, 599, interpreted this lower passage to be 79.53 m.

9. Petrie, A Season in Egypt, 30.

10. Dorner, MDAIK 42, 56, gives 32.76 m; Mustapha, ASAE 52, 599, gives a height of 33.3 meters.

11. Maragioglio and Rinaldi, L'Architettura Parte III, 66. Dorner in MDAIK 42, 56, gives 30°15'.

12. Maragioglio and Rinaldi, L'Architettura Parte III, 66.

13. Maragioglio and Rinaldi, L'Architettura Parte III, 98 -100.

14. Maragioglio and Rinaldi, L'Architettura Parte III, 66.

15. Maragioglio and Rinaldi, L'Architettura Parte III, p. 58.

16. Maragioglio and Rinaldi, L'Architettura Parte III. Refer to Plate 10, Figure 2 and to Plate 11, Figure 2.

17. Maragioglio and Rinaldi, L'Architettura Parte III, p. 103.

18. Maragioglio and Rinaldi, L'Architettura Parte III, Plate 11, Figure 2. Also see Plate 10, Figure 2.

19. Maragioglio and Rinaldi, L'Architettura Parte III, Plate 10, Figure 2. The dimensions of the Lower Chamber are listed as being 4.96 m east to west and 6.3 m north to south.

20. Maragioglio and Rinaldi, L'Architettura Parte III, p. 66. Maragioglio and Rinaldi do not make explicitly clear, either in the text or in their diagrams, if it is the axis of the Upper Descending Passage that is in fact 13.7 m south of the midline of the side. The published results of Mustapha (p. 599) and Fakhry (pp. 87 - 94) are also not specific on this point. I have assumed that it is the axis that is meant, and the Upper Chamber's analysis correlations appear to bear this out.

21. Maragioglio and Rinaldi, L'Architettura Parte III, Plate 10 Figure 2. All of the following measurements for this chamber have been taken from this same Plate 10, Figure 2. Note that even though the Upper Chamber's east and west walls can be derived via the protocol, the surveyed east / west 5.26 m width for this chamber coincides with a 10 royal cubit length. The chamber's north / south length is about .1 m longer than 15 royal cubits.

22. Maragioglio and Rinaldi, L'Architettura Parte III, Plate 10 Figure 2; Plate 13, Figure 1.

23. See this analysis in the Red Pyramid Computations section. A similar possibility exists for the King's Chamber of the Khufu Pyramid. See the Computations section of the Khufu essay.

24. Maragioglio and Rinaldi, L'Architettura Parte III, p. 98. Note that they refer to the base length of this proto-pyramid in terms of royal cubits.

 25. 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.

26. 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.

27. N. E. Scott, 'Egyptian Cubit Rods', Bulletin of the Metropolitan Museum of Art 1 (1942), 70.

28. 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.

29. 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".

 

References

Chace, A. B. 1929. The Rhind Mathematical Papyrus. Oberlin: The Mathematical Association of America.

Dorner, J. 1986. "Form und Ausmasse der Knickpyramide". MDAIK 42, 43 - 58.

----------- 1998. "Neue Messungen an der Roten Pyramide" in: Stationen: Beiträge zur Kulturgeschichte Ägyptens. R. Stadelmann gewidmet. Mainz.

Gillings, R. J. 1982. Mathematics in the Time of the Pharaohs. New York: Dover.

Heath, T. L. 1897. The Works of Archimedes. Cambridge.

Herz-Fischler, R. 2000. The Shape of the Great Pyramid. Waterloo: Wilfrid Laurier

Johnson, G. B. 1997. "The Red Pyramid of Sneferu: Inside and Out". KMT 8:1, 18 - 27.

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

----------- 1992. "The Problem of the Bent Pyramid". GM 130, 49 - 56. This paper is available on line at http://www.legon.demon.co.uk/bentprob.htm

Lehner, M. 1997. The Complete Pyramids. London: Thames and Hudson.

Maragioglio, V. & Rinaldi, C.A. 1964. L'Architettura Delle Piramidi Menfite Parte III . Rapallo.

----------- 1966. L'Architettura Delle Piramidi Menfite Parte V. Rapallo.

Mustapha, H. 1952. "The Surveying of the Bent Pyramid at Dahshur". ASAE 52, 595 - 601.

Peet, T. E. 1931. "A Problem in Egyptian Geometry", J.E.A. 17.

Petrie, W. M. F. 1888. A Season in Egypt, 1887. London. Petrie's text is currently available on-line at: http://digi.ub.uni-heidelberg.de/sammlung6/allg/buch.xml?docname=Petrie1887

----------- 1883. The Pyramids and Temples of Gizeh. London. The text of Petrie's original 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

 

Abbreviations:

MDAIK = Mitteilungen des Deutschen Archaologischen Instituts, Kairo

KMT = KMT: A Modern Journal of Ancient Egypt

GM = Göttinger Miszellen

ASAE = Annales du Service des Antiquités de l'Égypte

Hist. Math. = Historia Mathematica

DE = Discussions in Egyptology

 

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