The Architect's Plan

The Derivation Diagram of The Red Pyramid: Part I

The Basic Derivation Algorithm

This presentation examines the method by which the enigmatic interior layout design of the Red 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 Bent, 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 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, 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, take 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 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 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.

Implementing The Algorithm
 

The Red, or North, Pyramid at Dahshur is variously described in the literature as having an exterior slope angle of between 43 and 45 degrees.³ This pyramid's interior passageway design parameters will now be shown to be derivable from a right triangle having a 43°22' base angle, thus implying a pyramid height to half-base ratio of 17 to18. Although there is good reason to believe that the Red Pyramid was actually built with a slightly steeper exterior angle that was nearer to 44°40', the following analysis gives every appearance of explaining the original design process used to derive the pyramid's interior layout. It would seem that the steepening of the exterior angle was an overlay that was added to an already existing scheme.⁴

In Figure 4 below, we see a repeat of Figure 3, but with the addition of a pyramid with the same 17 to 18 height to half-base ratio that I propose was used by the architects as the design basis for the interior layout of the Red Pyramid. This pyramid has been entered into the diagram in such a way that the circle's horizontal diameter (DB) also serves as the pyramid's full base length. As a result, the radius (line OB) of the diagram's beginning circle will be half of the pyramid's base length.

The next step in the derivation is to then enter a circle which has the pyramid's height (Line OC, which is 17/18ths the length of OB) as its radius, and following this, to then enter the squares for the area and circumference for this radius OC circle. Figure 5 shows these steps.

As can be noticed in Figure 5, there are a number of points at which the two circles and four squares intersect with one another. In Figure 6, there have been lines drawn from the diagram's center to each of these intersection points (in the circle's upper right quadrant) - with there being 16 in all. Highlighted are the two lowest of these lines, and these will now be used to derive the Red Pyramid's descending passage parameters.

Since the bottom two intersections are determined by the points at which each of the two circles intersect with the outermost square, Figure 7 below is simply Figure 6 having been pared down to the necessary essentials, and then labeled.

The outer square in Figure 7 has been drawn such that OP (i.e., half the length of the outer square's side) is 8/9ths the length of OB, meaning that this outer square has the same area as the circle of radius OB. Also, OA has been drawn to be 11/14ths of OC (that is, it is 11/14 x 17/18 = .7857 x .94444 = .74205 units), and so the full perimeter of the inner square will be very nearly the same distance as the circumference of the inner circle (of radius OC).

With lines drawn from Point O (the diagram's center) to the two points at which each of these circles intersect the right-hand side of the outer square we establish Points D and E. Since OP equals .8888 (= 8/9ths) of a unit, and OE (which is the same as OC) equals .9444 (= 17/18ths) of a unit, angle POE can be easily found (it is 19°45'). The cosine of angle POD is simply .8888 divided by 1, and so this angle turns out to be 27°16'.

Point K identifies the point at which the line OE intersects the pyramid's side. As we are in possession of both the angle made by the line OE with the horizontal - and the angle of the pyramid's slope at B - the elevation of Point K above the pyramid's base is easily computable. If OB is made equal to 359.625 feet, which is the actual length of half the base of the Red Pyramid, then the diagram's elevation of K above this base works out to be 93.55 feet.⁵ The angle that the line OE makes with the horizontal (19°45') is not, however, the same angle as is made by the actual entrance passage. Instead, the angle made by the intersection line OD (27°16') was apparently used for this purpose.⁶

In Figure 8 above, if the part of line OD that extends from Point O to the pyramid's north side (line segment OL) is allowed to slide downwards along the pyramid's side so that OL now begins at Point K, it will create line KN, and the following agreements with the surveyed pyramid emerge. At the scale of the pyramid, line KG is of the same 204.2 foot length found by Dorner for the original length of the descending passage.⁷

In addition, if a perpendicular is drawn upwards from Point N to meet the base at T, then GT will equal 51.15 feet - and this is almost precisely the uninterrupted length of the pyramid's horizontal passage as it runs from the end of the descending passage on through to the south end of the pyramid's first chamber.⁸

Table 1 below provides a comparative summary of the findings given thus far for the Red Pyramid's entrance passage. Note these same relative findings can be extracted from the diagram through direct empirical measurement, given only that the diagram is drawn carefully and of appropriate size. I have found a minimum radius of half a meter to be sufficient for a surprising accuracy of results. I will detail at the end of this essay how I believe the different divisions of the royal cubit rod were used to advantage for this purpose.

 

* See footnote 5 below.

** See footnote 7 below

The shift, or transference, of line OL to the intersection point of Line OK with the pyramid's side is a device that we will see used again in the design of the other Old Kingdom pyramids to be discussed, and again having the effect of providing accurate correlations. Speculation as to the possible rationales for this maneuver will be explored at the end of this paper. An analysis deriving the locations of the interior chambers of the Red Pyramid will next be pursued in Part 2 of this essay.

Thoughts To This Point

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 locations of the various interior passageways and chambers. The papers linked to below detail how these same principles are in evidence in the interior layouts of the Khufu, Khafre and Bent Pyramids. 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 and with remarkable accuracy provide an explanation 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 are 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 can be assumed, therefore, that at the time of the even earlier Old Kingdom period the technique for squaring of the circle in terms of area had already been discovered and was not of a privileged nature. However, these same surviving texts do not appear to touch upon the squaring of the circle in terms of circumference correlation.⁹ 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

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. When the proposed diagrams are drawn carefully and of workable size, surprisingly accurate relative lengths can be determined empirically.¹⁰

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.¹¹ 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.¹² 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.¹³ 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 Derivation Diagram of The Red Pyramid: Part 2

Part 1 of this essay provides the derivation methods by which the design parameters of the entrance passage of the Red Pyramid appear to have been initially conceived.

We will next proceed to a look at how the same basic protocol would seem to have been used in determining the lateral location along a pyramid's side for the pyramid's entrance passage, and for the rather enigmatic positioning of the three chambers within this pyramid. It will be helpful to refer as often as necessary to Part 1 of this essay, and to the computation section.

Although I am here using both modern terminology and modern methods of mathematical computation, note that all of the diagrammatic and mathematical manipulations are achievable within the context of the ancient Egyptian 'royal cubit' measuring rod, and within the context of the mathematical - and angle measurement - systems, that current scholarship broadly assumes to have been in use during the Old Kingdom period.

Figure 9 is a reprise of Figure 6, and is the progenitive diagram in the derivation process for the Red Pyramid.

Numbered 1 through 16 in the diagram above are all of the circle and square intersection points for a pyramid of the 17/18ths proportions. Lines have been drawn from the diagram's center to each of these points.

From this diagram we will now proceed to derive the lateral location of the entrance passage to the Red Pyramid, as well as the relative lateral location of, and the elevation of, this pyramid's 3rd Chamber.

The Lateral Location of the Descending Passage Entrance

The derivation of this location will be used to serve as a template as we move forward through this essay. This same basic procedure appears to have been used to derive the various lateral locations in this, and the Bent, Khafre, and Khufu pyramids.

Radial line # 9 in Figure 9 leads to the first of the intersection points located above a 45 degree angle as measured from the diagram's center (Point O). Intersection Point # 9 marks where the 'radius 1 unit' circle intersects with the square having the same perimeter length as the circumference of the 'radius .9444 unit' circle (i.e., the innermost square). Therefore, the sine of the angle formed by this radial line is .74205 ÷ 1, and the angle will then be 47°54.4' . The Tangent of 47°54.4' is 1.107.

Figure 10 is a closer look at the radial line on its way to Intersection Point # 9, meeting the pyramid's north side at a Point P on its way. Line ON here is the line ON from Figure 1, which is half the length of the pyramid's full base length, (and so is given a relative length of 1 unit in the diagram). Line PD is the perpendicular dropped from the intersection of the radial line with the pyramid's side down to the pyramid's base. With OD equal to 'A' and DN equal to 'B', we have:

1) A + B = 1; (therefore A = 1 - B) and

2) H/A = the tangent of 47°54.4' = 1.107, and so H = 1.107 x A; and

3) H/B = the tangent of 43°21.8' = .9444, and so H = .9444 x B

And so, we have, A = (1 - B); and also from equalities 2) and 3) above we have, 1.107 x A = H = .9444 x B. Substituting in (1 - B) for 'A', we next have:

1.107 (1 - B) = .9444 x B; and so 1.107 = 1.107xB + .9444xB = 2.05144xB; and so

B = .53962; A = .46038; and H = .50964

From these findings we see that the length of 'A' ends (.5 - .46038 =) .03962 units to the left of Line ON's midpoint. The Red Pyramid's actual full base length is 719.25 feet, and so Line ON at actual scale can be taken to be 359.625 feet. This multiplied by .03962 places Point D at 14.25 feet to the left of the midpoint of the pyramid's half-side. This point almost exactly agrees with the relative location of the east (i.e., left-hand) wall of the Red Pyramid's Entrance Passage, which survey shows to be located 14.2 feet to the left of the midpoint of the pyramid's full northern side.¹⁴

What is being proposed here, then, is that the derivation protocol for lateral locations within a pyramid calls for the midpoint of the half-side, when using the above scenario, to be superimposed over the mid-point (Point O in Figure 10) of a pyramid's full base length side. However odd this may seem, this same technique was apparently the method used to determine other locations within this pyramid, and within the other pyramids we will be looking at.

The North to South Location, and Elevation, of the 3rd Chamber

In Figure 11 below we see an east to west side view of the layout of the interior chambers and passages of the Red Pyramid. The Entrance Passage descends to what Perring proposes to be the pyramid's base level, and then the passage proceeds horizontally through two almost identically sized chambers. From there, a passage to the 3rd Chamber has its opening high up in the 2nd Chamber's south wall.

The derivation diagram for the Red Pyramid places the north end of the Descending Passage and the beginning of the Horizontal Passage at a point that is 79.06 feet north (here, to the right) of the center of the pyramid.¹⁵ The passage and chamber length measurements seen in Figure 11 are from survey data, and using these specifics one can plot where the 3rd Chamber's north wall sits relative to the diagram's prediction for the end of the Descending Passage (and hence from the diagram's pyramid center).¹⁶ Using this method we find that in Figure 11 the Third Chamber's north wall lies 34.2 feet south of the derivation diagram's central axis (refer to Figure 8).

 

In the same way that the lateral location protocol was used to derive the east to west lateral location of the Descending Passage, it can now be used to derive the north to south lateral location of the north wall of the 3rd Chamber. The radial line # 4 in Figure 9 has a cosine equal to .8395 / 1, which means that the angle of this line will be 32°54.8', and the tangent .64726. Plug these facts into the algorithm shown with Figure 10 and you arrive at an "A" value of .59335, meaning that this length ends at a point that is .09335 units past (to the right of) the half-side midpoint. When these relative units are multiplied by the actual pyramid's half-side length of 359.625 feet (which here has been equated to 1 unit), .09335 units becomes 33.6 feet. Therefore, by next aligning the midpoint of the half-side with the midpoint of the full western side of the pyramid, the 33.6 foot location of the perpendicular will fall within inches of the surveyed lateral placement for the 3rd Chamber's north wall. By this I mean that Dorner's survey data show this 3rd Chamber north wall sits 113.6 feet south of the bottom endpoint of the Descending Passage, Perring's data has it at 113.34 feet, and the derivation diagram's prediction has it at 33.6' + 79.06' = 112.7' feet south of this endpoint.¹⁷

There is reason to believe that, as is the case with the Bent Pyramid, there may be a Descending Passage design element which slopes down to the 3rd Chamber from the Red Pyramid's west face. We saw earlier how the proposed algorithm of Figure 10 defined the location of the east wall of the red pyramid's Descending Passage, and as seen in Figure 12 (below) this east wall continues on to become the east wall of the pyramid's First Chamber. Since (as has just been shown) the placement of the 3rd Chamber's north wall would seem to be determined using the same basic protocol, it is possible that there may be a passage, or a symbolic designed in memory of such a passage, similarly associated with the north wall of the 3rd Chamber. (See Computations section for a possible scenario for the derivation of such a passage.)

 

As shown in Figure 8 of this essay, the diagram's line for the lower part of the Descending Passage continues downwards to a Point N which lies below the base level of the diagram's pyramid. Rather than have the passageway that leads to interior chambers begin at this "true" end of the Descending Passage as given in the diagram (i.e., from the diagram's Point N), the pyramid's architects instead decided to have the horizontal passage begin at the point at which the line of the Descending Passage crosses the pyramid's base level as registered in the derivation diagram.¹⁸ As a result of this decision the first two chambers were also designed to sit at this 'base' level. However, the 3rd Chamber was placed an unexpected 25.6 feet higher up. (See Figure 11). Why the architects decided to have a 3rd Chamber - and have its floor at a different elevation - I do not know, but why it was placed at the elevation that it was is apparently due to the following rationale.

In order to derive, from Figure 9, elevation positions which are other than at the diagram's pyramid base level one needs to employ a minor variation of the "shift" mechanism that has thus far been utilized. If one now takes the 'pyramid part' of the radial line which goes from the pyramid's center to Intersection Point # 15 in Figure 9, and then slides this line segment up (instead of down) to the point at which the radial line from the pyramid's center to Intersection Point # 16 crosses the pyramid's side, the bottom end of this line segment will end at 25.2 feet above the pyramid's base, - and this is within a few inches of the surveyed relative elevation of the original location of the 3rd Chamber floor.¹⁹ Figure 13 below shows this implementation. Another way of visualizing this is to drop a perpendicular to the pyramid's base from the point at which the radial line to Point # 15 crosses the pyramid's side, and then shift this perpendicular line up to the point at which the radial line to Point # 16 crosses the pyramid's side. The resulting elevation difference is the same.

 

The Lateral Locations of the Red Pyramid's First and Second Chambers, And of The 3rd Chamber's West Wall

According to Maragioglio and Rinaldi, the Red Pyramid's first two chambers extend 8.37 m (= 27.46 feet) and 8.34 m (= 27.36 feet) in length north to south, while the 3rd Chamber is 8.35 m (= 27.4 feet) in length east to west.²⁰ Whatever the reason that this basically 27.4 foot length - apparently meant to be 16 royal cubits - was deemed important, it is clear that it was of central concern. Once one chamber wall location was derived via the lateral protocol, then the opposite wall location was apparently spoken for by the desire to have the chamber have this specific 16 royal cubit length.

It has earlier been shown (in Figure 8) that the derivation diagram's line for the Descending Passage ends at a below grade "Point N", and that this Point N falls at almost exactly the same distance south of the end of the Descending Passage as does the surveyed location of the south wall of the 1st Chamber.²¹ This, then, is the rationale that I believe was used to implement the lateral north to south placement of the south wall of the 1st Chamber.

The proposed derivation for the lateral placement of the 2nd Chamber is slightly different. In this case, it appears that the intention was to have the center of the Second Chamber align with the perpendicular associated with the # 8 intersection point (refer to Figure 9). If a radial line is drawn to Intersection Point # 8, a perpendicular line dropped from the point at which this radial line crosses the pyramid's side will meet the pyramid's base (at the scale of the actual pyramid) about 4 feet north of the pyramid's center. Overlay the midpoint of the half-side over the midpoint of the full east side of the pyramid, and you will have derived a point that is 4 feet to the north of the pyramid's east/west axis. Given the analysis of the entrance passage as seen in the Computations section, and adding to this the survey findings that it is about 75.8 feet from the bottom end of the entrance passage to the middle of the pyramid's Second Chamber, one arrives at a finding that the middle of the Second Chamber sits about 3.3 feet shy of the pyramid's central axis.

I believe it likely, therefore, that in construction, the angle of the descending passage was steepened ever so slightly from the original 27°16' angle in order to retain the 75.8 foot distance from the bottom end of the descending passage to the middle of the Second Chamber, and to now allow the middle of the Second Chamber to be 4 feet to the north of the pyramid's central axis.²² A further reason to believe that this is the case can be seen below in Figure 14, which is identical to what was seen in Figure 11 except that here all horizontal distances are also given in royal cubits.

Clearly there was a desire on the part of the architects to have the north to south distances of the pyramid's passages and chambers be arranged in royal cubit multiples. By making the slight adjustment mentioned above, the center of the Second Chamber then aligns perfectly into the royal cubit protocol. Had the center of this Chamber been placed at the pyramid's center, as has been assumed in the Dorner and Perring surveys, lost is the orderliness of having the center of the Second Chamber sit 44 royal cubits from the end of the Descending Passage and 30 cubits from the south wall of the Third Chamber.

Continuing on to other locations, the east wall of the 1st Chamber is a continuation of the east wall of the Descending Passage, and therefore lies 4.34 m (= 14.24 feet) east of the pyramid's north/south axis. The east wall of the 2nd Chamber sits 2.6 meters (= 8.53 feet) to the west of this, and 2.6 meters (= 8.53 feet) seems to have been an intended 5 royal cubits.²³

The east wall of the passage to the 3rd Chamber, and the east wall of the 3rd Chamber itself, are both located .5238 meters (= 1.718 feet) - or 1 royal cubit - east of the pyramid's central north/south axis.²⁴ The 3rd Chamber has an east to west length of 8.35 m (= 27.4 feet), which - as stated before - equals 16 royal cubits. The west wall of this chamber, therefore, lies (8.35 m - .5238 m =) 7.826 m (= 25.68 feet) west of the pyramid's north/south axis. (See Figure 12). Even though there is an apparent 10 inch discrepancy in the following suggestion, this wall may have been intended to be aligned with the perpendicular associated with the radial line to Intersection Point # 10, which falls 26.4 feet to the left of the half-side midpoint. (With some hesitancy I add that it is interesting to note that the location of this wall 25.68 feet to the west of the pyramid's axis is - as we've just seen - all but exactly the same distance that this chamber is elevated above the floor level of the Second Chamber. It is therefore an alternate possibility that the rationale used to determine the Third Chamber's elevation was also used as the rationale for the west wall lateral location. Although this latter justification is not seen in the other pyramid design derivations, it nevertheless presents a nearly exact correlation. It may also be that the designers simply wanted the west wall to be 15 royal cubits west of the pyramid's axis, and so made a slight shortening of the 26.4 foot line spoken of above in association with Point # 10 in order to make this accommodation.)

Concluding Thoughts

One of the initial steps in the architect's design process appears to have been to choose a relative height to base dimension for the planned pyramid, and then to create a diagram (such as is seen in Figure 1) which is based on the 'squaring' of each of the resulting two circles in terms of their area and circumference. From this diagram were then chosen the constituent lines from which to implement an interior design for the pyramid. For the Descending Passage, the choice was apparently limited by the need to not have it be overly steep, thus leading to the use of the lower radial lines..

In each of the pyramids addressed in this, and the accompanying essays, there appears to have been an overall willingness on the part of the architects to at times make very minor placement adjustments relative to what is found in the initial derived diagram. There is a sense that the architects were trying to incorporate and accommodate a range of concerns, and were not bound by a strict or blind adherence to an inflexible doctrine. All this is not to say that extreme precision was beyond their reach when such was their aim. In the Red Pyramid we see this in the slight change in exterior slope angle, and slight change in the angle of the Descending Passage. That there can be found more than one theme at play in the design of these pyramids would appear to be consistent with the overall world view in evidence in ancient Egyptian thought. Henri Frankfort refers to an Egyptian willingness to entertain a "multiplicity of approaches" in the search of a "multiplicity of answers" in regard to their religious concepts.²⁵ John A. Wilson speaks of this modality in Egyptian thought as being a willingness to embrace dualities, "that these concepts are essentially alternatives did not seem to bother the Egyptian.......one concept could be taken as complementing another instead of contradicting it".²⁶ This is precisely the approach that I believe is in evidence in the proposed derivation process.

The analysis given here proposes which of the diagrammatic components were chosen for final inclusion in the constructed pyramid, and which of these components were apparently set aside. However, there is reason to speculate that some of these other "set aside" options from the diagram might have in fact have been utilized in a manner that has not as yet been identified. As previously mentioned, the diagram's line for the Descending Passage extends below the pyramid's base level, and this surely suggests that there may be some feature located at the below grade terminus of this line. Also, as previously mentioned, it is possible that there is some form of a Descending Passage, symbolic or otherwise, which meets the 3rd Chamber from the pyramid's western side.

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. For an overview of these slope angle measurements see C. Rossi, Architecture and Mathematics in Ancient Egypt, pp. 203, 219, and 243. (Also see further discussions in Part 2 of this essay, and in footnote 16 below.) Although the design of the interior of this pyramid seems certain to rest on the 17/18 relationship, it also appears that the exterior angle of the pyramid was constructed to conform to a slightly steeper pitch in order to include other desired correlations within the structure. For instance, an exterior angle of about 44°30' implies a height to half-base ratio of 49 to 50, a notably useful ratio which allows the hypotenuse of the right triangle to be the easily remembered number of 70. (This same 49-50-70 ratio for the Red pyramid exterior is suggested by W. M. Flinders Petrie in his A Season in Egypt, p. 27.) It is not inconsistent with what is known about other aspects of Egyptian thought that such kinds of dualities would have been brought into the design plans for the pyramids by the architects. An essay by the present author further exploring the possible rationales for the exterior design of this pyramid will be made available in the not too distant future.

4. A 17/18 relationship results more precisely in an angle of 43°21.8', which is the angle that will be used here. That the pyramid may have been built with a slightly steeper exterior angle is discussed above in footnote 3. The empirical and mathematical means by which these angular measures may have been dealt with by the Egyptians is proposed later in this essay.

5. J.S. Perring, in H. Vyse, Operations Carried On at the Pyramids of Gizeh in 1837, V. 3, p. 61 states an elevation of 94 feet. In an accompanying essay which compares the Perring and J. Dorner surveys of the Red Pyramid, I propose an analysis of Perring's data which reveals that - had he understood the pyramid to have an exrterior slope nearer to 44°40' - he (Perring) would have predicted that the original location of the entrance would have been about 99 feet south of the pyramid's north base edge and at an elevation of about 98 feet. Note that Point K in Figure 7, at the pyramid's actual scale, lies about 6.3 feet to the south of the side of the inner (S/2 =.74205) square. Also note that J. Dorner, 'Neue Messungen an der Roten Pyramide', in Heike Guks (ed.), Stationen: Beiträge zur Kultur- geschichte Ägyptens. R. Stadelmann gewidmet (Mainz, 1998) claims an entrance height of 101 feet. An analysis of this differing result can be found in the author's essay on the Perring and Dorner surveys. As the present analysis places the entrance at 99.06' south of the pyramid's north base edge in the original design (see the analysis of the first diagram in the Computations section), it is my belief that the entrance was kept at this distance south of the north edge but simply raised straight upwards to meet the 44°40' exterior side. Much more on this is to be found in the Perring/Dorner analysis mentioned above.

6. G. B. Johnson, 'The Red Pyramid of Sneferu: Inside and Out', KMT 8:3, (1997), 21, refers to this angle as being 27°. Dorner, "Neue Messungen", gives 26°34', while Perring, Operations Carried On, found 27°56'.

7. J. Dorner, 'Neue Messungen', 28. However, Perring claimed this distance to have been 205.5 feet. The author's essay on the Perring and Dorner surveys explains how Perring likely erred in arriving at this slightly too large amount, and how he would have arrived at nearly 204 feet had he understood the pyramid's exterior slope to have been nearer to 44°40'. The derivation of the 204.2 foot length of line segment KG can be found in the Computation section.

8. See Dorner, "Neue Messungen", 28.

9. 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.

10. Those 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.

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

12. The Egyptians referred to angular measure in terms of 'sekeds', with the 'seked' being the number of cubits, palms and fingers of horizontal run required by any given 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 vertical 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.

13. 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".

14. V. Maragioglio and C.A. Rinaldi, L'Architettura delle Piramidi Menfite Parte III, p.128. Maragioglio and Rinaldi state that "the axis of the descending corridor is 3.81 m ( = 12.5') east of the N-S symmetry axis of the building". In Plate 19, Figure 2 accompanying this work they show the width of this passage to be 1.04 m (= 3.41'). Therefore, to find the relative position of the east wall of this passage add 3.81 m + .52 m = 4.33 m (= 14.2 feet).

15. Maragioglio and Rinaldi, L'Architettura Parte III. Plate 18, figure 5, shows the beginning of the Horizontal Passage to be 75.56 feet (= 23.03 m) from the center of the Second Chamber, which they note (p. 130) had been found by Perring (as reported in Vyse, H., Operations Carried On at the Pyramids of Gizeh in 1837, V. 3,) to be on the central axis of the pyramid. As will be shown, the diagrammatic derivation presented here predicts otherwise.

16. These interior measures are again from Maragioglio and Rinaldi, L'Architettura Parte III, Plate 18, figure 5 (although I have taken the liberty of converting all data into "feet" and "inches"). Bear in mind that the derivation diagram shows the center of the pyramid sitting further south than in Maragioglio and Rinaldi's findings, and that Figure 11 in the text is using the diagram's prediction for this center location and then superimposing surveyed measurements on top of it. Maragioglio and Rinaldi (p. 128) give Perring's measures for: a) the elevation of the entrance to the Descending Passage (94 feet), b) the length of the Descending Passage (205.5 feet), and c) the passage's angle of inclination (27°56' - this latter measurement being one which they themselves confirmed). However, as the Perring and Dorner surveys essay by the present author shows, Perring's data for the entrance passage parameters - as usually interpreted - do not add up. Sense can be made out of this data if one assumes that his 94 foot elevation of the descending passage entrance refers not to the entrance's original location, but to its location as he then found it. Analysis from this understanding then yields the finding that the actual lower end of the Descending Passage sits further north (and hence further from the pyramid's center) than where Maragioglio and Rinaldi, and others, have assumed it to be - and this agrees with the diagram's predictions. Dorner, in "Neue Messungen an der Roten Pyramide", measured an exterior angle of 44°44' (which he then rounds off to 45°) and a descending passage angle of 26°30' (an angle significantly less than that found by both Perring and Maragioglio & Rindaldi), from which he proposes that the descending passage ends not at the pyramid's base level, but rather almost 10 feet above it. If his measurements are correct, it would mean that an upward shift of this amount was uniformly imposed upon the descending and interior passages - in relation to their positionings as given by the theory's diagrammatic derivation, (which indeed accurately provides the elevations and lateral locations of the interior chambers and passages as they sit relative to each other.) Such uniform shifts are seen in the later pyramids. I strongly concur with Dr. Rossi's 2004 assessment that "unfortunately, there is no agreement among scholars about the actual dimensions of this (i.e., the Red) pyramid". (p. 219).

17. Dorner, "Neue Messungen", 28, Table 3 and Figure 3. Vyse, op. cit., 64-65. See the Computations section for the derivation of the 79.06 foot amount. As discussed in the Perring/Dorner essay, Maragioglio and Rinaldi detail that the floor of the Descending Passage turns to the horizontal about 5 inches south of the point where its ceiling does. Uncertainty as to which point measurements were taken from may help explain some of the difference here between the 113.4 and 112.7 foot amounts.

18. Maragioglio and Rinaldi, L'Architettura Parte III, p. 128, follow Perring's statement that this passage is at the pyramid's base level. Although this is clearly the case in the proposed original design derivation, the steepening of the pyramid's exterior angle from the original design's 43°21.8' appears to have resulted in an increase in elevation of all of the pyramid's interior features. See the author's essay comparing the Perring and Dorner surveys for more on this issue.

19. Maragioglio and Rinaldi, L'Architettura Parte III, Plate 19, Figure 1, and p. 130. There is room for a little confusion here in that some portions of the floor of the southward portion of the second Chamber have been long since removed. Maragioglio and Rinaldi 's Plate 19 makes clear that the depth of this removal totals about 4 feet, and that their 12.6 foot measurement was made from what appeared to be the originally constructed floor level of the Second Chamber. Perring (in Vyse, Operations Carried On, 64-65) gives the elevation of the passage to the Third Chamber from "the original floor" of the Second Chamber as 25 feet 3.5 inches, while Dorner, "Neue Messungen", Table 3, p. 28 has it as it at about 8.4 meters (= 27.6 feet) above the Second Chamber floor. It is not clear how it is that Dorner's measure is about 2 feet greater than either Perring's or Maragioglio & Rinaldi's. Note that plunderers have long since removed the flooring to the Third Chamber and of the passage to it, and Maragioglio & Rinaldi clarify that their 25.6 foot elevation reading was made to the original floor level of the elevated passage to the Third Chamber, stating (on p. 130) that "its original position is clearly seen".

20. Maragioglio and Rinaldi, L'Architettura Parte III, Plate 18, Figure 5.

21. Maragioglio and Rinaldi, L'Architettura Parte III, Plate 18, Figure 5.

22. The entrance passage slope angle would need to be about 27°22' to account for this small adjustment.

23. The measures given here are perhaps best presented in Dorner, "Neue Messungen", Table 3 and Figure 3 on p. 28. Also see Maragioglio and Rinaldi, L'Architettura Parte III, Plate 18, Figure 5.

24. See Dorner, "Neue Messungen", Table 3 and Figure 3 on p. 28. Also see Maragioglio and Rinaldi, L'Architettura Parte III, Plate 18, Figure 5.

25. Henri Frankfort, Ancient Egyptian Religion (New York 1961), 18-20.

26. John Wilson, Henri Frankfort, et. al., Before Philosophy (Baltimore 1973), 55-56.

References

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

Dorner, J. 1998. "Neue Messungen an der Roten Pyramide" in Heike Guks (ed.), Stationen: Beiträge zur Kultur- geschichte Ägyptens. R. Stadelmann gewidmet.

Frankfort, H. 1961. Ancient Egyptian Religion. New York. Harper and Row.

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

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

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

Legon, J. A. R., 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

-------------------- 1990. "The 14:11 Proportion at Meydum". D.E.17, 15 - 22.

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

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

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 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.

Wilson, J. A., Frankfort H., et. al. 1973. Before Philosophy, Baltimore: Penguin.

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