The Architect's Plan

 

The Derivation Diagram of the Khufu

Pyramid

By Leon Cooper

 

The diagrammatic derivation methods for the interior passage and chamber locations of the Red, Bent, and Khafre Pyramids, have been detailed by the author in separate essays.1 These derivations show that the interior designs of these three pyramids appear to each be based upon a diagrammatic 'squaring' of the circle, a squaring that is done both in terms of the circle's area and the circle's circumference. However, the proposed derivation for the original interior design of the Khufu Pyramid that is presented below - although having its basis in these same imperatives - is achieved in a slightly different manner from that seen in the other pyramids. The end result is that many of the Khufu Pyramid's most puzzling features can be derived within a single unified diagrammatic context.

Before beginning, I want to make it clearly understood that although much of what follows is presented in terms of modern understandings (for example, decimal unit notation is used and angles are often spoken of in terms of degree measurement), 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 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 in the following diagrams, these can be found on the Computations page.

* * * * * * * * * * * *

Introduction

The diagrammatic means by which a circle can be 'squared' both in terms of its area and its circumference is shown in the Red, Bent, and Khafre essays. It may be helpful to the reader if I first provide a review of this material before proceeding on to the Khufu discussion.

 

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 for all intents and purposes 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 this 8/9ths amount.2 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 on 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 clear 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 square which appeared to have 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 to have 28 subdivisions - which they 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 (i.e., 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., line OB below) will be 14 fingers in length.

By centering the square we can see that half the side (i.e., OT) of this "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.3 (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 as seen in the Khafre, Red, and Bent Pyramids, then 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., OR) will be 8/9ths the length of radius OB, and half the side of the inner square (i.e., OT) will be 11/14ths the length of radius OB. This diagram is the derivation beginning point for the above mentioned pyramids, but it is not the beginning point for the Khufu Pyramid.

Had it been, then the next diagram in the derivation would have looked like the construction below.

Here we have entered a triangle - on the base line (Line DB) of the previous diagram - which has the same proportions as the Khufu Pyramid. Next, a circle has been drawn using this triangle's height as its radius. Following this, the squares having the same area - and the same perimeter length - as this new circle have also been entered. Lastly, the points of intersection between these circles and squares have been noted, and lines have been drawn to them from the diagram's center. If the Khufu interior had been designed in the same way that is seen in the Khafre, Red and Bent Pyramids, then the derivation of the Khufu interior design would have proceeded from this diagram. However, the original design process of the Khufu interior was apparently achieved in a somewhat more sophisticated, and more unified, manner - while yet achieving these same two-fold 'squaring of the circle' objectives.

Note also that in the previous methodology the architects had to first decide upon a pyramid height to base proportion, and then plug this triangle into the circle and square diagrammatic format. In the Khufu derivation, the exterior and interior spatial relationships are both derived together in one flowing context - a distinct departure from the earlier algorithmic modality.

 

The Exterior Proportions of The Khufu Pyramid

 

The Khufu pyramid, in cross-section, can be seen as being two equivalent back to back right triangles, each having a height to base ratio of 14 to 11. The sequence of diagrams which follows will derive a pyramid having these proportions while also - in the same context - deriving a "squaring of the circle" in terms of the circle's circumference. There are a number of diagrams to this derivation, so please bear with. Not one of these is difficult, and the results are well worth the small amount of time involved.

Figure 1 shows the first steps to be taken. A circle of random size is drawn, its radius is considered as being 1 unit long, and the circle's horizontal diameter is entered. Since a circle's diameter is, in essence, two radii placed end to end in a straight line, the next step is to identify the mid-points of each of these two radii. Circles are now drawn on these two mid-points, and each of these inner circles will consequently have a radius that is .5 units in length.

Also in Figure 1, perpendicular lines are entered through each of the midpoints, and these new lines are extended top and bottom to meet the rim of the outer circle.

Figure 2 moves things along considerably. From the top and bottom ends of the vertical diameter (Points A and B), lines are drawn to the mid-points of the two initial radii - and continued on through to the further edge of each of the two interior circles (refer to lines AE, AT, BD, and BF). Horizontal lines (lines DF and ET) are then drawn connecting the top - and bottom - endpoints of AE, AT, BD, and BF, creating an interior six-pointed star.

Also in Figure 2, line AE is pivoted from Point A creating an arc which crosses the plane of the original circle's horizontal diameter both left and right (as at Point C). Likewise, BF is pivoted from Point B. These arcs will then lie tangent, top and bottom, to the two smaller interior circles. They also intersect each other left and right exactly at the plane of the original horizontal diameter (again, as at Point C).

If the original diameter is extended to Point C, then the line OC can be used as a radius to create yet a new outer circle. Since OA is a radius of the diagram's beginning circle, it has a length of 1 unit. As ON is equal to .5 units, line AN can then be found to be 1.118034 units long. Therefore, AE (and also AT, BF, and BD) will equal 1.1618034 units. This means that if AE is swung around to rest at Point C, it will create the right triangle AOC where the lengths of AO and AC are known, and so the length of OC can be found, and it is 1.272 units long. With these developments, this initial derivation sequence can now be completed. (As mentioned in the introductory remarks, dimensions are given here in modern decimal terminolgy. The Egyptian scribe or architect would have expressed these numbers in terms of cubits and cubit subdivisions.)

In Figure 3, the original vertical diameter (OA) is extended upwards to meet the new outer circle. Lines are drawn to this point from the end points of the original circle's horizontal diameter, thus creating the larger central triangle (here in red). Following this, a square is drawn about the original circle such that each side of this square has the same length as the initial horizontal diameter (that is, each side equals 2 units).

The height of the larger central triangle is 1.272 units (it is equal to length OC in Figure 2), and the triangle's full base length is 2 units. This turns out to be exactly the same height to base ratio as is found in the Khufu pyramid. Furthermore, the circumference of the outer circle, as computed in modern terms, is equal to 1.272 X 2Pi, or 7.992 units. Each side of the circumscribed square is 2 units long, and so the square's full perimeter measures to be 8 units in length - almost exactly the true circumference length of the outer circle. This diagram, therefore, not only produces the Khufu Pyramid proportion, but essentially squares the circle in terms of the circle's circumference length.

 

The Khufu Pyramid's Interior

 

The previous diagram (Figure 3), and its derivation, have appeared in print before.4 What follows next, to my knowledge, has not. We will begin by taking another look at the diagram of Figure 3 to see how it can lead to a derivation of the the Khufu pyramid's interior layout.

In Figure 4, lines are drawn from the center of the diagram to two of the points (Points M and L) at which the circumscribed square intersects with the outer circle. Perpendicular lines are then dropped to the pyramid's base from the points at which lines OM and OL intersect with the pyramid's side. Knowing from the previous discussion that lines OL and OM must each have a relative length of 1.272 units, we find that all of the lengths and locations in Figure 4 can be straightforwardly computed (or indeed, directly measured) from the diagram.

Line OM works out to cross the pyramid's flank at an intersection point that is almost exactly directly above Point Z, the midpoint of the half-side. Line OL crosses the pyramid's flank almost exactly at the point where the rim of the .5 unit radius circle also passes through the pyramid's side. This intersection point computes to lie directly above a point (here shown as Point P) that lies .618 units to the right of the diagram's center (at Point O). This latter finding will be discussed first, while the line OM correlation will become important in regard to the Queen's Chamber discussions.

 

Squaring the Circle in Terms of Area

Having seen the exterior proportions of the Khufu pyramid fully generated in the previous derivation, which was a derivation that squares the circle in terms of 'circumference', there now arises the related question of whether there might also be a way - within the context of this same derivation - to square a circle in terms of the circle's 'area'. The interior square seen in Figure 5, generated via what has come before, provides a surprising answer to this question.

The interior square seen above (highlited in red) is produced by first connecting - with vertical lines - the ends of (what I shall call) the 1.618 unit lines which extend from Points A and B, (these points being the top and bottom of the original circle's vertical diameter). The ends of these 1.618 lines are labeled Points D, E, F, and T, in Figure 2 and in Figure 5.

Through comparison of the similar triangles AON and AVE (see Computations section), it can be shown that Line VE, and Line OH, are each .7236 units long, and so Lines DE and FT will each cross the diagram's horizontal diameter this same .7236 units distance from the diagram's center. The inner square can then be completed by: 1) extending the vertical lines DE and FT until they extend this same .7236 distance both above and below the horizontal diameter; and 2) by joining the top and bottom ends of these vertical lines with a horizontal line. Each side of this inner square will then have a length equal to .7236 x 2 = 1.4472 units. Therefore, the area of the square will be 2.0944 square units. This number, 2.0944, turns out to be exactly 2/3rds of Pi, and so this means that the area of the inner square is exactly 2/3rds of the area of the initial circle. (Since the radius of the initial circle was 1 unit, the initial circle's area will be Pi x Rsquared = Pi x 1squared = Pi). The Egyptian scribe would have measured these areas in his own way, but the result he found would have essentially been the same. This inner square does indeed contain an area that is 2/3rds that of the intial circle.

To provide an area that is equal to the full area of the original circle, the inner square (which I will hereafter anachronistically refer to as the Pi square) can be extended above and below the horizontal diameter by half of the .7236 amount (that is, by adding .3618 units to the .7236 units to make a total of 1.0854 units), in order to create a rectangle (hereafter referred to as the Pi rectangle) that measures 2.1708 x 1.4472 units. This Pi rectangle is shown in the next diagram, along with the beginnings of the layout for the pyramid's interior passageways.

In Figure 6, the line which determines the pyramid's Descending Passage is derived by connecting the point where lines BD and ET intersect (at the lower edge of the left-side inner circle) with the .618 locator on the horizontal diameter (Point P in Figures 4 and 6); and then lastly, continuing this line through Point P upwards to the pyramid's northern flank. The angle that this Descending Passage line makes with the horizontal diameter turns out to be 26°33.9', an angle whose tangent is 1/2. These factors allow the diagram's elevation of the Entrance to this passage to be easily computed, and it is at a height of .13713 units above the pyramid's base. Also determinable is the distance south from the pyramid's north base edge that this Entrance is located. It works out to be .107745 units to the left of the pyramid's north base edge. We will soon compare these units with the actual survey measurements.

The line of the Ascending Passage is determined by the 'lower right'-to-'upper left' ascending diagonal of the rectangle that forms the top half of the Pi square (see Figure 6). The point of beginning for this diagonal is therefore located on the diagram's horizontal diameter (which is also the base of the diagram's pyramid), at a distance which is .7236 units from the diagram's center (due to the fact that it is where the side of the Pi rectangle crosses the horizontal diameter). The angle that the ascending diagonal makes with the horizontal is, again, 26°33.9'.

As a result, this diagonal crosses the central vertical axis of the pyramid at a point (Point K) that computes to be .3618 units above the pyramid's base, and this is the diagram's elevation of the King's Chamber Passage (.3618 = .7236 divided by 2). The King's Chamber itself is located a soon to be derived amount to the left (south) of the central axis. Just above the .3618 intersection point in Figure 6 is the horizontal line that forms the base of the inverted interior triangle (line DF in Figures 5 and 2). This line computes to be .4472 units above the pyramid's base, and is therefore located .4472 - .3618 = .0854 units above the King's Passage level. (In Figure 5, Line NH is .7236 - .5 = .2236. Line DN = .5, and so DH can be computed to be .4472).

Figure 7 below is the completed Derivation Diagram, and before beginning a comparative analysis, it will first be helpful to see how the above .0854 spacing appears to have been used to determine the Queen's Passage location. Note that in this, and in the previous views of the pyramid, we are looking east to west, meaning that north is to the right.

As can be seen in Figure 7, the diagram's roof of the Grand Gallery follows a line that begins on the central axis at the .0854 point above the King's Passage level. This line runs downward to the right, running parallel to the line of Ascending Passage, until it meets the 1.618 unit line (line AT in Figures 5 and 2). From this meeting point, a perpendicular is dropped to the Ascending Passage, and the line of the Queen's Passage proceeds to the left from this juncture. The Queen's Passage line then meets, and ends, at the rim of .5 radius circle that is just to the left of the central axis. Computation will show that the diagram's elevation of Queen's Passage above the pyramid's base is .177545 units. (Refer to Computations section.) Since there will be other variables that come into play regarding the exact location of the Queen's Chamber itself, the chamber location will be discussed in more detail in a later section of this essay.

The Descending Passage in Figure 7 continues below the base of the pyramid until it meets, and ends at, the edge of the right-hand side .5 radius circle. The line of the Subterranean Passage then extends horizontally to the left (that is, to the south), ending at the rim of .5 radius circle which lies to the left of the central axis.

To summarize, we have thus far developed:

1) the elevation of the Entrance to the Descending Passage above the pyramid's base;

2) the distance to the south of this Entrance relative to the pyramid's north base edge;

3) the elevation of the King's Passage;

4) the elevation of the Queen's Passage; and

5) the below-grade elevation of the Subterranean Passage.

It is time to see how closely these findings compare with measurements taken from the actual pyramid. For this purpose, I will use the survey data of W. M. F. Petrie.5

Petrie found the average base length of a side of the Khufu pyramid to be 9068.8 +/- .5 inches (see p. 43 of Petrie's work referenced in footnote 5, and note that all further Petrie references will be to this volume). In the diagram just developed, the pyramid's full base length is 2 units long. Therefore, for the diagram to be predictive, each unit from the diagram should correspond to a length of 4534.4 inches in the actual pyramid. As will be shown, the relative positionings as computed from the diagram will be found to be remarkably accurate given but one integral difference - all positions inside the diagram's pyramid are about 48 inches (= 14/6 cubits) lower than they exist in the actual pyramid, and they are all about 37.7 inches (= 11/6 cubits) further to the north than they are in the actual pyramid.

In other words, it appears that all interior positions were 'shifted' upwards and to the south along the same 14/11ths (= 1.2727) tangent gradient exhibited by the pyramid's exterior slope. The following list and subsequent Table will detail a few of the salient features which provide evidence of this shift: 

1) The diagram has the Entrance to the Descending Passage at an elevation of 621.8 inches. Petrie (p. 55) measured it to be 668.2 inches. Difference: 46.4 inches.

2) The diagram places the entrance 488.56 inches to the south of the north base edge. Petrie (p. 55) has it 524.1 inches south of the north base edge. Lateral Difference: 35.54 inches.

3) The diagram has the Ascending Passage meeting the King's Passage at 1640.5 inches above the base. Petrie (p. 75) measured the line of the Ascending Passage to end at an elevation of 1689 inches. Difference: 48.5 inches.

4) The diagram places the beginning of the Queen's Passage at an elevation of 805.06 inches. Petrie (p. 65) lists the point of beginning of this horizontal passage to be at an elevation of 852.6 inches. Difference: 47 inches.

5) The diagram has beginning of the Subterranean Passage at a depth below the pyramid's base of 1222.24 inches. Petrie (p. 59) has this point at a depth of 1181 inches. Difference 41.24 inches.

 From these findings, and from the findings which follow, it is evident that - as was the case in the author's aforementioned analysis of the Khafre Pyramid - all interior features of the Khufu pyramid undergo a unified shift relative to the pyramid's exterior when being translated from the original design diagram (as given in Figure 7) to the as-constructed building. In the case of the Khafre Pyramid, this initial unified shift was fairly large. Here, it appears that the shift of the pyramid's interior features, following the tangent of the pyramid's slope, has them move upwards by 48 inches (i.e., 14/6ths royal cubits), and then southwards (i.e., to the left) by 37.8 inches (= 11/6ths royal cubits), relative to their computed positionings in the Derivation Diagram. Table 1 provides a comparative summary of the findings given thus far.6

With the proposed shift taken into account, the correlations up to this point are strikingly accurate. Although the concept of there having been a shift involved in the design strategy for the pyramid may admittedly at first appear somewhat strange, the present author's work involving the Red, Bent and Khafre Pyramids has shown that shifts prove to be a consistent factor of pyramid design during the Old Kingdom period. Speculation as to the reasons for such unexpected protocols will be considered later, for now we will first move forward to a fairly detailed look at the Ascending Passage's design history.

 

The Ascending Passage

 

As explained in regard to the Derivation Diagram of Figure 7, the initial rationale for the placement of, and the angle of, the Ascending Passage was that it should follow the line of the diagonal of the rectangle that forms the top half of the Pi square, with the passage itself ending at the pyramid's central vertical axis (and thereby defining the King's Passage level). In the actual pyramid, Petrie's measurements for the length along the slope of the Ascending Passage, and his findings for the elevations of its beginning and end points, imply that the passage makes an average angle with the horizontal of 26°13.7'.7 He further found that the upper end of this passage is at a distance that is 61.7 inches south of the central axis (Petrie, pp. 72, 74, 95). The shift mentioned above does account for these two departures from the initial diagram, but it does so with a bit of revealing complexity - as will now be derived.

In Figure 8, line AO is the central vertical axis of the pyramid as seen in Figure 7. Rectangle LHZG forms the upper half of the Pi square, and line GH is the original diagonal to this rectangle, and hence it is the diagram's original line of the Ascending Passage. With the first stage of the implementation of the shift, all interior features are lifted upwards by 48 inches relative to the pyramid's exterior sides and base. Although obviously not drawn to scale, this first part of the shift in Figure 8 is reflected by the line BT, which is simply line GH moved up 48 inches from the base of the pyramid. Since the the tangent of the angle of the original diagonal (line GH) is .5, and since line segment GB is 48 inches, then the resulting line segment GD must be 96 inches. Therefore, if the intent was to have the (theoretical) line of the Ascending Passage continue to have its point of beginning be at the pyramid's base level, it must now have its starting point repositioned to Point D, 96 inches further north than Point G. Since Point G is .7236 units from Point O (the pyramid's center), then Point D will be .7236 + .02117 = .74477 units from this center. (96"÷ half the pyramid's base equals 96"/4534.4" = .02117 units).

When the shift is implemented, not only does the exterior of the pyramid in the diagram stay put, but so too does all of the diagrammatic scaffolding. Therefore, the upper half of the Pi square remains exactly where it was, and if the line of the Ascending Passage is still intended to go to the upper left-hand corner of this rectangle, it must then still go to Point H (and not to Point T). The tangent that line DH makes with the horizontal is findable from HL÷LD = .7236 ÷ 1.46837 = .49279. This turns out to be the tangent of 26°14' - which is almost exactly the same angle that is implied by Petrie's measurements. (Note that 1.46837 results from .7236+.74477). We will now see that this line of reasoning then offers explanations for many of the other conundrums regarding the Ascending Passage.

 

The King's Passage Level

 

Figure 9 looks at how all of the preceding effects the resultant positionings at the King's Passage level - and here things get even more revealing - and hopefully not too complicated. I admit that this can get confusing, so please refer freely to Figures 8 and 9.

Line EM marks the plane of the vertical axis of the pyramid - which, according to Petrie (p. 74), is directly in line with the vertical face of the Great Step. Line 1 in Figure 9 is the line DH from the previous diagram, ascending at an angle of 26°14'. Had DH instead been at the 26°33.9' angle of line DT in Figure 8, it would then have passed directly through the Point E of Figure 9, this being Point K of Figure 8 raised (shifted upwards) 48 inches to the 1,688.6 inch elevation level. Because of Line DH's lowered angle, however, it instead meets this 1688.6 inch level at a Point N, some 49.5 inches south of Point E. However, the Ascending Passage, as actually built, was not built directly along Line DH (Line 1 above).

Petrie takes pains to mention that although the top surface of the Great Step lies at an elevation of 1694.6 inches (p. 75), the line of the as-built Ascending Passage (here represented by Line 3 - not Line 1) actually ends at an elevation of 1689 inches (see Point F above) - some 5.6 inches below the as-built surface of the horizontal passageway (Petrie p. 75. Also see Petrie's chart on p. 95.) Petrie refers to this point as being the "virtual" end of the Ascending Passage. Furthermore, Petrie makes it clear (pp. 74-5, 95) that this ending point for the actual Ascending Passage (i.e., Point F), at an elevation of 1689 inches, is the point that is 61.7 (+ or - .8) inches south of the pyramid's central axis. (This means that the line of the actual Ascending Passage - i.e., Line 3 -, if extended up to the 1694 inch elevation of the King's Passage floor, would have ended about 11.3 inches further south of this 61.7 inch point, at a juncture - Point J above - that is 73 inches south of the pyramid's center.) I warned the reader that this might get a bit confusing.

However, Figure 9 explains what is happening here. If Line 1 (DH) is continued to the 1694.6 elevation (to Point S), due to its angle of inclination it would be 12.18 inches further south than it is at Point N (which is at the diagram's 1688.6 inch elevation). This means that Point S would be 61.7 inches south of the pyramid's central axis. Now, 61.7 inches can be seen as being the length of 3 royal cubits, and so it would appear that the elevation of the King's Passage was intentionally raised from 1688.6 to the 1694 inch level specifically to create this implied 3 cubit 'ideal design' situation for the line I am calling Line 1 - the Line DH from Figure 8. Reinforcement for this supposition follows.

Let us look at what happens when we implement the second part of the initial shift. As the reader may recall, only the vertical component of the interior shift has so far been enacted. With the 37.8 inch southward component of this shift, the line of the Ascending Passage (Line 1) is moved this 37.8 inch amount southward to Point W (Line 2), well past its actual surveyed location. (Point W is therefore 49.5" + 37.8" = 87.3" south of the pyramid's central axis.

The line of the Ascending Passage, therefore, was apparently moved back to the north to a point (Point F) which lies directly beneath the Point S of Line 1. This seems to have been done so that the line of the Ascending Passage would again end at the 3 royal cubit 'ideal design' distance south of the pyramid's center, but this time not at the 1694.6 inch level - but rather at the diagram's original 1688.6 inch level.8 The magnitude of this second - and final - shift therefore works out to have been 25.6 inches.

I propose, then, two possible reasons why the added shift back to the north may have been deemed necessary. As described above, it may have been to maintain the 'ideal' 3 cubit correlation of the "DH" Ascending Passage line (Line 1). And, it may also have had to do with making the Great Step present a less steep obstacle. If the Ascending Passage had been built to follow Line 2, the Great Step would have presented a 55 inch hurdle at its face rather than its - as actually built - 36 inch barrier.

Whatever the reasoning for this extra added northward shift, the foregoing discussions explain:

1) Why the Ascending Passage is inclined at the general angle of 24°14';

2) Why the Ascending Passage ends at the elevation that it does;

3) Why it ends south of the pyramid's central axis, and why it ends 61.7 inches south of this axis;

4) Why there is even such a feature as the Great Step;

5) Why the Grand Gallery has the height that it does;

6) Why it is that the Grand Gallery ends south of the pyramid's mid-line, and

7) Why the Grand Gallery begins and ends precisely where it does.

We will now look at one further line of reasoning that corroborates all of what has thus far been proposed.

 

The Intersection Point Between the Ascending and Descending Passages

 

What has been detailed above also leads to the precise location for the intersection point of the Descending Passage with the Ascending Passage. We start this part of the analysis by first computing these locations as found in the Derivation Diagram of Figure 7. In Figure 10 below we see an enlargement of this situation from that diagram.

In the Derivation Diagram, the Descending Passage crosses the pyramid's base at a .618 unit distance north of the pyramid's center, and the line of the Ascending Passage crosses the base at a .7236 unit distance north of the center. With both lines in the initial diagram being at an angle of 26°33.9', the situation is a symmetrical one, and the lengths relative to the scale of the actual pyramid compute to be those shown in Figure 10.

Figure 11 depicts this same intersection after: 1) the implementation of both components of the the first shift - a shift which effects all interior positions; 2) after the lowering of the Ascending Passage's angle from 26°33.9' to 26°14' (for the reasons stated above); and, 3) after the additional shift back to the north of 25.6 inches - a shift which effects only the Ascending Passage and not the Descending Passage.

As one applies the vertical 48 inch (upward) first part of the shift to the situation in Figure 10, due to the tangent of both initial angles being .5, the points of beginning for both the Ascending and Descending Passages - at the pyramid's baseline - will each be shifted 'outward' by 96 inches. (In Figure 10, the .618 point of beginning shifts to the left (south), and the .7236 point of beginning shifts to the right (north). Applying next the horizontal component of the shift, both of these points of beginning will now be moved 37.8 inches southward. And finally, after the lowering of the Ascending Passage's angle, the Ascending Passage - and the Ascending Passage only - then receives the added shift bringing it back again to the north by 25.6 inches - apparently in response to the effect engendered by the Ascending Passage's lowered angle at the King's Passage level (as discussed above in relation to Figure 9).

In Figure 11, we see the net result of these various manipulations. The Descending Passage now crosses the pyramid's baseline at a .58844 unit distance north of the pyramid's central axis, and the point of beginning for the line of the Ascending Passage now crosses the baseline at a distance of .74208 units from this central axis. When these unit amounts are translated to the scale of the surveyed pyramid, the predicted lengths are as they are labeled in Figure 11.

Petrie found that the line of the floor of the Ascending Passage meets the floor of the Descending Passage at an elevation of 172.9 inches above the pyramid's base (p. 65). Additionally, he found that this intersection point is 1517.8 inches south of the pyramid's north base edge (pp. 65, 95), and that it is 1110.64 inches down along the slope of the Descending Passage from the passage's entrance on the pyramid's north face (pp. 61, 62). The post-shift diagram predicts all of these positions with astounding accuracy. (See the Computation section for a full diagrammatic explanation of these derivations.) And so, in summary:

1) Petrie (p. 65) found exactly same 172.9 inch elevation for the intersection point of the two passages as arrived at by the post-shift diagram;

2) Petrie (pp. 65, 95) found this intersection point to be 1517.8 inches south of the pyramid's north base edge. The diagram predicts 1520.4 inches;

3) Petrie (pp. 61, 62) found the intersection point to be to be 1110.64 inches down along the slope of the Descending Passage from the passage's entrance on the pyramid's north side, the diagram predicts 1111.3 inches.

That the proposed derivation scenario can simultaneously account for both the upper and lower particulars of the Ascending Passage, and do so with such extreme accuracy, is a combined outcome which is unlikely to have been the case unless the original set of assumptions were essentially correct. I believe that having both ends of this analysis arrive at such accurate results must be seen as arguing strongly in favor of the validity of the overall theory.

 

Chamber and Entrance Locations

 

When first working out the details of the derivation process, I had expected to find that the King's Chamber had been designed, in the Pre-shift Diagram, to be located at the edge of the left-hand side .5 radius circle (Refer to Figure 7). For this to have been so, given the elevation of the King's Passage, the chamber's south wall would have needed to have been situated 702.3 inches from the pyramid's center. However, Petrie found that the inside of the King's Chamber's south wall is actually located at 537 inches south of the pyramid's center (p. 83). The chamber's south wall is 67 inches thick, and so the south wall's south-facing exterior must then be located 537 + 67 = 604 inches south of the pyramid's center.9 Clearly, this distance is well short of the 702.3 inches that I had anticipated.

Instead, the rationale that appears to have been used to determine the King's Chamber location is the same basic rationale possibly used to determine where along a pyramid's side, in horizontal terms, the pyramid's entrance passage was to be located. Although I will be soon discussing this procedure only as it relates to the Khufu pyramid, it can be shown that the same principle can be successfully applied to locate lateral (i.e., horizontal) entrance passage locations in at least three other major pyramids of the period.10

Figure 12 is a reprise of Figure 4, except that here has been added the Pi Square, along with a perpendicular that has been dropped down to a Point E on the pyramid's baseline from the point at which the top of the Pi Square crosses the pyramid's northern flank. For the moment, our interest will be with Point P.

As described in the discussion accompanying Figure 4, Point P is situated .618 units north of the pyramid's center, and it is one of the two locator points for the Descending Passage in the Pre-Shift Diagram. It obviously lies .118 units to the right of the midpoint (Point Z, in Figure 12) of the 'right-side half' of the pyramid's baseline. In considering where, laterally, along a pyramid's side an entrance passage should be located, the designers of the Khufu Pyramid apparently now made the following connection.

The architects decided, for the purposes at hand, to have the midpoint of the 'half-side' be considered as correlating to the midpoint of the pyramid's full side. If the mid-point of the right 'half-side' of the pyramid's base (i.e., Point Z) is superimposed over the mid-point of the pyramid's entire western side, then Point P, at its .118 unit distance to the right of Point Z, will correlate to being (.118 units x 4534.4" =) 535 inches south, of the full west side's midpoint.11 As we saw earlier, Petrie found the interior of the south wall of the King's Chamber to be located 537 inches to the south of the pyramid's center, or 2 inches from this prediction. (It is a possibility that the King's Passage length was increased ever so slightly to 536.25 inches from an originally intended 535 inches in order to allow this length to become exactly 26 royal cubits long - i.e., 26 x 20.625" = 536.25". Such dualities of purpose appear to have been a common occurrence in the pyramid design process, as many such instances can seen.) Although the over-all explanation proposed here may at first seem strange, it is in fact a method which can be used to accurately account for many of the interior lateral locations considered in this series of essays.

Turning next to Point E in Figure 12, the entrance to the Descending Passage on the pyramid's north side can be found in much the same way just used for the King's Chamber. Point E, in Figure 12, works out to be .068555 units to the left of the half-side midpoint (Point Z). Superimposing this midline over the midline of the pyramid's full north side, Point E would then lie (.068555 x 4534.4 inches =) 310.86 inches to the left (east) of this side's mid-point. Petrie says that the central axis of the Entrance Passage lies 287 inches east of the midline of the north side (pp. 55, 66). With the east to west width of this entrance being 41.5 inches, the east wall of the entrance is then 307.8 inches east of the middle of the north side - a point which is within 3 inches of the Point E prediction.12 Again, we see that a slight adjustment may have been made in order to accomodate a whole number of royal cubits. 15 royal cubits of 20.625" will equal 309.4 inches, this being but 1.6 inches from the surveyed actual. I submit that this, then, may have been the process and rationale for the intended siting of the Entrance Passage's east wall position.

The same technique may have been at work in deciding the ultimate north/south location of the Queen's Chamber. As seen in Figure 4 (and again in Figure 12 above), line OM crosses the pyramid's north flank very nearly in line with the midpoint of the half-side. (Refer to the Computations Page for this derivation) This factor may have had some bearing in the apparent siting of the center of the Queen's Chamber on the central axis of the pyramid, I do not exclude this as a possibility, although I feel that what follows offers an intriguing explanation as well.

 

The Queen's Chamber Location

 

The derivation for the elevation of the passage to the Queen's Chamber has already been presented in the discussion accompanying Figure 7, Detailed there is the finding that the Derivation Diagram's line for the Queen's Passage extends from the Ascending Passage southward to the rim of the .5 radius circle which lies to the left of the pyramid's central axis. In that diagram, this distance breaks down into being 1671 inches from the Ascending Passage to the pyramid's central axis, and another 147.8 inches from the central axis to the circle's edge - for a total of 1818.8 inches. Following the implementation of the two shifts previously described, and the associated lowering of the Ascending Passage's angle, the length of the first part of the Queen's Passage is lessened such that it extends only a 1633.8 inch amount from the pyramid's central axis to the Ascending Passage (and not 1671 inches).13

Although the initial shift 'pushes' the Queen's Passage - and the Queen's Chamber with it - 37.8 inches to the south, note that the passage and the chamber are are not moved still further south by the lowering of the angle of the Ascending Passage - and the Chamber location is not in any way effected by the secondary 25.6 inch shift back to the north, which is applied to the Ascending Passage only.

Figure 13 shows the pre-shift situation for the Queen's Chamber and Passage. The exterior of the chamber's south wall is 147.8 inches to the south of the pyramid's central axis, where the wall meets the .5 circle. With the shift, the entire Chamber and Passage are relocated 37.8 inches to the south, meaning that the exterior of the Chamber's south wall will now be at (147.8 + 37.8 =) 185.6 inches south of the central axis. This wall was measured by Gantenbrink to be 77 inches thick, and so the inside surface of the south wall would (by this scenario) be only 108.6 inches south of the pyramid's central axis. Petrie found this actual separation distance to be 103 inches (p. 67), a difference of 5.6 inches.14

However, Petrie measured the thickness of the chamber's south wall to be 80 inches (p. 70), and so if this figure is more accurate than Gantenbrink's, the difference between prediction and the actual placement lessens to only 2.6 inches. In either event, support for the overall scenario given in this essay is made available through the curious occurrence of the step in the Queen's Passage.

 

The Queen's Passage Step

 

The floor of the Queen's Passage is straight and relatively level for its entire length except for its one mysterious 'step', a drop of about 20 inches (one royal cubit perhaps), which Petrie measured to be located 319.8 inches north of the pyramid's central axis (p. 66). Since the present theory considers the constructed pyramid to be a "post-shift" implementation of the original design, it would then stand to reason that the pre-shift location of the step in the Queen's Passage would have been 37.8 inches further north than its present position. This would then place it at 357.6 inches north of the central axis in the original diagram, and this is almost exactly the diagram's location, deep below the pyramid's base, for the end-point of the Descending Passage. (As mentioned along with Figure 7, the Descending Passage ends where it meets the edge of the right-hand side .5 radius circle. (See the Computation section for the derivation of the distance to this location from the central axis.)

It is credible, therefore, that in the Pre-shift Diagram the Queen's Passage Step was intended to mark the meridian which passes through the diagram's end-point of the Descending Passage. In a symmetric fashion, if in the Pre-shift Diagram another such line were to be drawn at a 26°33.9' angle - and passing through the same .618 marker that is used by the Descending Passage - yet heading upwards and to the left (south), it would intersect the upper edge of this very same .5 circle on the very same meridian 357.8 inches north of the central axis. In addition, the upper intersection of this theoretical line at the upper edge of the .5 circle turns out to be at an elevation that is exactly half the way between the Pre-shift Diagram's elevation of the Queen's and the King's Passages.

The above, however, does not explain why the proposed meridian was marked with a "step", rather than some other marking feature. One possibility is that it was done in order to fascilitate a continuous upward line of travel of the Queen's Chamber northern "air-shaft" under and around the Grand Gallery assemblage, since it has the effect of lowering the floor level of the Queen's Chamber.

 

The Subterranean Passage

Turning now to the Subterranean Passage, Petrie notes (pp. 59, 95) that from its point of beginning at the lower end of the Descending Passage, to the pyramid's central axis, is a distance of 306 inches. Within the context of the Derivation Diagram, and with the implementation of the shift, the expectation would be for this distance to have been 320 inches (that is, 357.8 - 37.8 inches). That the actual location of the end of the Descending Passage turns out to be 14 inches further south than the 320 inch prediction from the post-shift diagram is apparently due to the fact that the length of the Descending Passage, as measured by Petrie (p. 57), is 4,143 inches instead of the diagram's predicted 4,123 inches. This 20 inch extra length is also the reason that the elevation of the Descending Passage's endpoint is about 7 inches lower than the 1174 inch depth predicted by the post-shift diagram (see Table 1). (It should be noted here that whole sections of the descending passage were clogged with rubble at the time that Petrie attempted his measurements. He admits, on p. 57, to being able to only make "rough measurement" through these areas. Hence, it is possible that the measurement which he gives of the overall slope of this passage may be unduly weighted to the angles taken at the upper regions of the passage - regions which are apparently about 5 minutes of arc steeper than those below. See Petrie, p. 58. Such a shallowing of the angle of descent could account for at least a part of the 7 inch discrepancy pointed out here.)

It is also possible that the above factors are perhaps due to a scenario that is quite similar to that which was seen in the reasoning for the lowered angle of the Ascending Passage.

Figure 14 is an enlargement from the Pre-shift Diagram of Figure 7. From earlier discussions we know that the diagram's entrance to the Descending Passage (at the scale of the Khufu Pyramid) is 621.8 inches above the pyramid's base, and that this passage - in the pre-shift diagram - ends at 1222.24 inches below the pyramid's base (see Table 1).

Therefore, the height of the right triangle to which the Descending Passage forms the hypotenuse is (621.8 + 1222.2 = ) 1844 inches. With the Tangent of the passage's angle being 1/2, the base of this right triangle will then be 3688 inches, which leads to a computed diagram passage length of 4123 inches. If, however, the first part of the shift is enacted such that the entrance - but not the passage end point - is raised by the shift's 48 inches (and not yet moved the shift's 37.8 inches southward), then the height of this new right triangle will increase to 1892 inches. The base will remain 3688 inches, and the hypotenuse will increase to 4,145 inches - which is almost exactly the passage length found by Petrie (p. 57). The 37.8 inch southward component of the shift would then seem to have been implemented at this juncture, and done in such a way - unlike as was seen in the case of the Ascending Passage - that the angle of the Descending Passage was reverted back to the angle which it had in the original diagram, while the 20 inch increase in the overall passage length was retained.15

Concluding this look at the pyramid's below-ground features, we note that Petrie (p. 59) found the interior side of the north wall of the Large Subterranean Chamber to be located 40 inches south of the pyramid's central axis. It is then possible that the interior side of this north wall was positioned directly in line with the pyramid's central axis in the original diagram, prior to it then being shifted 37.8 inches southward - thus placing this interior point 2.2 inches from Petrie's finding.

 

The Queen's Chamber Air Shafts

 

As shown in Figure 7, the "air shafts" of the Queen's Chamber extend from the base of the chamber's north and south walls, angling upward to the upper corners of the Pi Square. A detailed analysis of these shaft angles reveals a number of closely related possible scenarios, two of which will be explored here. The first is a look at what the angles for the north and south shafts work out to be, given the Pre-shift Diagram's .177545 unit elevation for the Queen's Passage and Chamber. (See Computation section for the derivation of this elevation. The drop in elevation caused by the Queen's Passage Step apparently did not effect the mechanics of the derivation - only the actual as built elevation of the chamber.)

In the pre-shift Derivation Diagram, it was proposed that the exterior of the Queen's Chamber south wall might have been designed to sit against the .5 radius circle, with this being 147.8 inches to the south of the pyramid's central axis. The air shafts to this chamber were built such that they travel horizontally through the chamber's north and south walls before bending upward. They each make this bend at a height above the chamber floor that is consistent with the intended points of beginning for the line of each shaft having been located near the inside base location of each wall.16 (See Figure 15).

Since, in the Pre-shift Diagram, the pyramid's central axis does not align with the center of the Queen's Chamber - as it does, according to Petrie (p. 66), in the actual pyramid, - one would expect that - based on this theory - the angle taken by the north shaft will be just a bit steeper than the angle taken by the south shaft. Note that in the Pre-shift configuration, as shown in Figure 15, the east to west centerline of the Queen's Chamber lies 32.2 inches north (to the right) of the pyramid's central axis. (As stated earlier, and using Gantenbrink's measurement of the thickness of the chamber walls, this means that, after the shift, the center of the chamber will be 37.8 - 32.3 = 5.6 inches south of the pyramid's central axis.)

Figure 16 (below) presents a look at the Queen's Chamber in the Pre-shift Diagram from a more distant vantage, showing the air shafts as they head on their way to the corners of the Pi Square. As before, the diagram's elevation of the Queen's Passage is .177545 units, and the elevation of the upper corners of the Pi square is .7236 units. Putting to use the above discussed lateral parameters, the tangent of the angle for the north shaft will work out to be .54605/.69378 = .78707, meaning that this angle will be 38°12.3'. Gantenbrink found the angle of the north shaft to be 38°13.3'.

At the same diagrammatic Queen's Passage elevation of .177545 units, the tangent of the south shaft works out to be .546055/.70799 = .77128. This indicates an expected angle of 37°38.5'. However, Gantenbrink found the angle of the south shaft to be a slightly greater 39°36.5' - an angle that is even greater than that for the north shaft.

Curiously, if the elevation for the Queen's Chamber in the above computations is instead considered to be identical to the Pre-shift Diagram's elevation for the entrance to the Descending Passage (that is, at .13713 units), then the angle for the south shaft computed from this lowered level will be 39°38.2', which is almost exactly the angle (39°36.5') found by Gantenbrink. Assuming that Gantenbrink's findings are correct, a possibility that is then worth considering is that the Queen's Chamber Passage was positioned at the same elevation as the pyramid's entrance in an earlier incarnation of the Khufu Pyramid's design process. If this was the case, then either by intention or by an admittedly unlikely 'holdover' error, this former angle for the south shaft was retained in the finalized plans.17 Also very much possible, of course, is that other considerations may have been at play causing a minor adjustment to this angle in order to accomodate a duality of purpose. For example, possible stellar alignments for these channels have been proposed.18

Figure 16 also suggests a rationale for the locations of the recently discovered "doors" in the Queen's Chamber shafts. In the Pre-shift Diagram these locations appear to coincide with the points at which each shaft crosses the upper rim of its respective .5 radius circle. For the south shaft, computations done with the parameters of the .177545 unit elevation, and which assume an exterior south wall location of 147.8 inches south of the pyramid's center (see Figure 15), will find the distance from the south wall exterior to the "door" to be 57.3 meters. Gantenbrink found this actual distance to be 57.5 meters.19

Working from the same elevation, and with the lateral locations as given in Figure 15, the "door" intersection point of the north shaft can be shown to be 56.5 meters from the exterior side of the north wall. I am not aware of a definitive published straight line measurement for this northern shaft, although a "National Geographic News" web site contains information which states that this length is the same as that which had been found by Gantenbrink for the south shaft.20

If, as stated earlier, the intended positioning of the Queen's Chamber was to have it centered on the Pyramid's east/west axis (see the end of the discussion regarding Figure 12 above), then both shaft angles will compute to be 37°55.3' and the distance to the "door" intersection point will be 57.036 meters. See Computation Section for this analysis.

Included in the Computation Section (under Further Further Notes in Sections 15 and 16) is an analysis which incorporates the royal cubit drop in elevation of the Queen's Chamber due to the Queen's Passage Step. This analysis leads to a result that exactly matches Gantenbrink's 57.5 meter finding for the length of the shaft to the shaft's first "door", and also includes a testable prediction that both the north and south shafts extend another 9.2 meters beyond the location of this first "door". As new robotic explorations have been planned for these shafts, it is hoped that such planning includes the capability to explore out to this distance.

 

King's Chamber Air Shafts 

 

Once again it will be helpful to refer to Figure 7, where it will be seen that the shafts which emanate from the King's Chamber extend upwards to the upper horizontal line of the original square - the square which was circumscribed around the diagram's original circle. Figure 17 is a closer look at this situation, showing that the south shaft from the King's Chamber meets this top line at the point where the side of the Pi Rectangle passes through. The north shaft extends to the north corner of the original square - a point which is directly above the pyramid's northern base edge.

The rationale for the elevation of the King's Passage was given earlier, as was the rationale for the 535 inch southward placement for the interior face of the King's Chamber south wall (refer to the lateral location discussion accompanying Figure 12). With this 535 inch south placement evidently having been imposed after the implementation of the shift, it is possible that the King's Chamber position - within the original context of the Pre-shift Diagram - was considered to be 37.8 inches north of the position it was slated to be given by the lateral location protocol. This would place the interior face of the south wall at (535" - 37.8" =) 497.2 inches south of the central axis, meaning that the interior side of the chamber's north wall - which will be 206 inches (i.e., the length of the chamber's N/S width) to the north of the south wall - is at a point that is therefore (497.2 - 206) = 291.2 inches south of the central axis.

If we again propose that, in the diagram, the angles of these shafts had their points of beginning at the inside base of these walls, then computations using the parameters given above will find that the angle made by the south shaft will be 46°6.5', and the angle made by the north shaft will be 30°57.5'. Gantenbrink found the south shaft to rise at 45°, and the north shaft at 32°36'.21

The above theoretical findings are quite close to the presumed actuals, but differ by just enough to spur further supposition. An alternative possibility is that, in the original plans for the pyramid, the King's Chamber was positioned such that the inside of its north wall was directly in line with the pyramid's central axis - much the same as was earlier proposed for the Large Subterranean Chamber's north wall. We have seen that Gantenbrink's detailed findings for the Queen's Chamber walls led to the assumption that the points of beginning for the Queen's Chamber shafts were nearly at the base of the inside parts of those walls (see Figure 15). However, in the case of the King's Chamber, Gantenbrink's findings make it clear that the shafts have their points of beginning very near (or at) the outside portions of the King's Chamber walls. If the shaft angles are recomputed taking into account both this factor and the diagrammatic alignment of the inside of the north wall with the pyramid's central axis, then one finds a north shaft angle of 32°56' (Gantenbrink measured 32°36'), and a south shaft angle of 43°53.5' (Gantenbrink measured 45°). These latter assumptions may therefore indicate the more correct alignment rationale. As stated above in regard to the Queen's Chamber shafts, it is also possible that a stellar alignment consideration was also a part of the design rationale, causing an adjustment to be made to effect a duality of purpose.

 Concluding Thoughts The derivation of the design for the Khufu Pyramid is closely related to the derivations for the design diagrams of at least three other pyramids of the Old Kingdom period, as is evidenced in the author's accompanying essays. The thematic principle underlying each of these structures is that they were designed to incorporate various diagrammatic elements which result from a geometric 'squaring' of a circle - a squaring both in terms of circumference and in terms of area. Seen in the Khufu Pyramid, as well as in the other pyramids discussed, is a complete willingness on the part of the architects, if not a direct imperative, to implement what may have been a ritualized shift in the placement of certain of these diagrammatic components. The rationales for these shifts may have included an attempt to utilize more segments of the diagram in the actual interior layout design for each pyramid, and/or it may simply have been an attempt to keep the causative geometric knowledge in the hands of only those who were authorized to possess it.The theory detailed in this essay stands as a unified treatment, offering an explanation for the design of the Khufu Pyramid not only on its larger scale, but also in consideration of much of its specific detail. As was shown in regard to the location of the entrance to the Descending Passage, the theory's prediction is within one or two inches of Petrie's survey. Even greater accuracy is seen in the prediction for the intersection point between the Descending and Ascending Passages, and perhaps even more dramatically in the prediction of the location of the upper end-point of the Ascending Passage, where the theory accurately explains the why and wherefore of this passage having an endpoint that is five or so inches below the top surface of the Great Step.A listing of the correlations provided by the theory includes:

1) the elevation and east-to-west location of the entrance to the Descending Passage;
2) the length and angle of the Descending Passage;
3) the location and elevation of the Ascending Passage;
4) the angle and length of the Ascending Passage;
5) the location of the intersection between the Ascending and Descending Passages;
6) the location of the Queen's Passage;
7) the length of the Queen's Passage;
8) the reason for the location of the "step" in this passage;
9) the reason for the height of the Grand Gallery;
10) the reason for the length of the Grand Gallery and
11) why the Gallery ends south of the pyramid's central axis;
12) the reason for the Great Step - and for its dimensions;
13) the reason for the location of the upper end of the Ascending Passage - and why this is below the level of the top of the Great Step;
14) the reason why the King's Chamber is located where it is;
15) the reason why the walls of the King's Chamber are not only below the level of the Chamber floor, but why the walls are based at the same elevation as the end of the Ascending Passage;
16) the reasons for the locations and angles of the air shafts emanating from both King's and Queen's Chambers; and
17) the reason for the locations of the "doors" in the Queen's Chamber air shafts (and hence why there are no such " doors" in the King's Chamber air shafts).
Should the theory proposed here be proven to be more or less correct, then the implications that it presents regarding the mathematical, religious, architectural, and other cultural concerns of the Old Kingdom period will have admittedly profound repercussions. In addition, the ability of the Derivation Diagram to provide extremely accurate predictions for the locations of the pyramid's known interior features opens up the intriguing question of whether the diagram can be used to point the way towards the location of other features which are as yet unknown. There are a number of such possibilities which suggest themselves, and I will here propose three that I believe may be the easiest to confirm:

1) Noting that, in the Pre-shift Diagram, the "doors" in the Queen's Chamber shafts appear to coincide with the points at which each shaft crosses the upper rim of its associated .5 radius circle, it is reasonable to suspect that there might be another marker in these shafts at the point at which they each then intersect with the vesica piscis of the original diagram (see Figure 7). The distance along the shaft to this intersection computes to be nearly 67 meters when measured from the exterior of each Chamber wall.22

2) In the same way that the Step in the Queen's Passage marks the Pre-shift Diagram's 357.8" meridian, there may also be a marker in the Ascending Passage on this same meridian. Unlike the Step in the Queen's Passage, which was shifted southward by just the 37.8" of the first shift, the marker in the Ascending Passage -assuming that it too was made part of the plans at the Pre-shift Diagram stage - would have been: a) moved southward about 8 inches due to the lowering of the passage's angle from 26°33.9' to 26° 14'; b) moved southward by another 37.8" due to this component of the first shift; and c) moved back to the north by 25.6" due to the second shift. The marker may therefore be located in the Grand Gallery at about a 376 inch distance down from the Great Step (and so about 1440 inches up from the Queen's Passage entrance). I can not say what form this marker may take, only that if the architects decided to mark the 357.8" meridian in the Ascending Passage, then this is the likely lateral location of its placement.

3) Noting that the lateral location of the King's Chamber may have been determined using the same protocol that was used to determine the lateral entrance passage location in this pyramid (and which was also used to determine the lateral entrance locations in at least three other Old Kingdom pyramids), it is possible that there may be a descending passage to the King's Chamber, or perhaps some form of a symbolic vestige of such a passage. This feature, should it exist, would likely descend to the King's Chamber from the pyramid's western side at an angle of about 27°16', with this descending feature's south wall (or side) in a continuing line with the interior side of the chamber's southern wall, The Bent Pyramid has a descending passage leading to an interior chamber from its western side, and it can be shown that the same protocols were used in determining that passage's lateral location as are being suggested in this instance.23

Applying these protocols, this proposed descending passage feature to the King's Chamber would end at the southern end Chamber's west wall, and have a possible width extending from 494 inches to 535 inches south of the pyramid's central axis (that is, it would theoretically have the same width as the pyramid's entrance passage). The line of the floor of this descending passage may then meet the interior surface of the Chamber's west wall at a point as high as 74 inches above the Chamber's floor. (It is probable, however, that the angle of this passage may have been steepened slightly to allow it to meet the west wall precisely at the King's Chamber floor level.)

The entrance to this passage, or some marker to acknowledge its symbolic location, would be located on the pyramid's western flank (at the exterior of the remaining core masonry) at an elevation in the range of about 2860 inches above the pyramid's base (depending on the specific height and width of the casing stones), and again, some 494 to 535 inches south of the pyramid's central axis. (See the Computation section for a more detailed treatment of this proposal.)

As has been detailed in this essay, the architects of the Old Kingdom pyramids employed the 'point to point' design opportunities made available by the 'squaring of the circle' diagram derived for each pyramid. However, as we've seen, other design objectives - such as (in certain instances) the utilization of specific royal cubit values - were also intentionally brought into the mix.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.24 John A. Wilson speaks of the modality of 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".25 This is precisely the approach that I believe is in evidence in the pyramid derivation process.

Clearly, a number of design objectives were simultaneously of concern in the drafting of these monuments, and while each objective by itself may involve a fairly simple design technique, taken together they reflect a complex turn of mind not unlike that which is implicit in the Egyptian hieroglyphic writing system and, as mentioned above, in the religious thought of the period. By acquainting ourselves with these objectives, and with the techniques that were used to achieve them, I believe that a treasure trove of insights into the high culture of the period now awaits the informed researcher. The tools which have been developed in these few essays have thus far been used to retrieve only a part of the information which I am certain they can help make attainable.

 Notes

1. Link to the author's essays on the Red, Bent, and Khafre Pyramids.

2. 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, 139 - 146. It is interesting to note 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, 93.

3. 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.

4. For instance, see Robert Lawlor's work, Sacred Geometry.

5. Petrie, The Pyramids and Temples of Gizeh (1883). Note that this is the first printing of this work, and not the1885 abridged version. All further references to Petrie's findings will be from this earlier volume, and if not otherwise footnoted, these will be given as page numbers in parentheses in the text.

6. It is a noted curiosity that a horizontal line drawn at the elevation level of the King's Chamber Passage cuts the cross-sectional area of the pyramid exactly in half. (See Petrie, pp. 186 -187, 221). Achieving this unique correlation may have been a part of the rationale for the specific magnitude of the shift.

7. On page 65 Petrie gives the elevation of the beginning of the line for the Ascending Passage as being 172.9 inches above the pyramid's base. On this same page he states that the "sloping length" of the first part of the passage - up to the beginning of the Grand Gallery section - is 1546.8 inches. On page 71 he adds that the "length of the slope" of the Grand Gallery section is 1883.6 inches, and then on p. 75 that the endpoint of the entire Ascending Passage is 1689 inches above the pyramid's base. This all means that he found the total increase in elevation for the Ascending Passage to be 1689" - 172.9" = 1516.1 inches, and that he found the total length of run along the passage slope to be 1546.8 " + 1883.6" = 3430.4 inches. By dividing 1516.1 by 3430.4 we can arrive at the average sine of the angle for this slope, and it is .44196. This is the sine of an angle of 26°13.7'. (Note that in Petrie's table given on p. 95, the elevation of the beginning of the Ascending Passage is incorrectly given as 179.9 inches, instead of the clearly intended 172.9 inches. 172.9 inches is indeed the elevation arrived at when one computes this point based on Petrie's data for Descending Passage entrance elevation, slope angle, and distance to the beginning of the Ascending Passage.)

8. That the diagram's 1688.6 inches is the 'true' elevation of the constructed pyramid's King's Passage is supported by Petrie's finding (pp. 83 and 95) that the walls of the King's Chamber end not at the 1694 inch elevation of the surface of the Chamber's floor, but beneath this surface at the lower elevation of 1689 inches.

9. For the thickness of this wall see Rudolf Gantenbrink's architectural drawing at his "Cheops Shafts" web page. All future references to Gantenbrink's work will be to his architectural drawings at this same web site.

10. See links below to the author's essays on the Red, Bent, and Khafre Pyramids.

11. At the scale of the actual pyramid, .118 units will equal 535.04 inches.

12. Petrie (p.50) lists (in conjunction with Plate ii from his text) the east-to-west width of the Descending Passage to be 41.53 inches.

13. Petrie lists 1626.5 inches for this distance (p.66). The 7.3 inch difference here is due to the fact that Petrie measured the angle of the beginning stretch of the Ascending Passage (from its intersection with the Descending Passage) to be slightly less than the average angle for the entire length of the passage. It is not clear why the angle of the Ascending Passage was not maintained throughout at the 26°14' average value.

14. The Queen's Chamber measures 206 inches north to south, and Petrie (pp. 66, 67) found the center of the chamber to be directly in line with the pyramid's central axis. As to the east and west wall locations of the Queen's Chamber, the east wall is simply a continuation of the east wall of the Queen's Passage, meaning that this wall is in the same plane as the east wall of the pyramid's Entrance Passage. Since Petrie found (p. 67) the chamber to be 206 inches north to south and 226.5 inches east to west, the implication is that the intended design was to have the chamber interior measure 10 royal cubits by 11 royal cubits. (This would then mean that the Queen's Chamber west wall would be about 308" - 226.5" = 81.5" east of the pyramid's north / south axis. In his summary table on p. 95, Petrie apparently incorrectly listed this distance as 72". I believe 82" was the figure intended here.) A similar arrangement to the Queen's Chamber is seen with the King's Chamber, where that chamber's east wall is a continuation of the east wall of the passage leading into it, and so this east wall also lies in the same plane as the east wall of the Entrance Passage. Petrie (pp. 80, 81) found the King's Chamber to be have the King's Chamber measure 10 royal cubits by 20 royal cubits. Note that the overall process appears to have been to locate one chamber wall via the proposed derivation protocol, and then locate the wall opposite based on a need to have it be a certain specific number of royal cubits distant. This same methodology is seen used again in the design rationales for the Red, Bent, and Khafre Pyramids.

15. Petrie (p. 58) measured the angle of the Descending Passage to be 26°31.3'. Note that the angle whose cosine is 3688 / 4145 is 27°9.5'. It is not clear why this wasn't the angle used in the finalized plans. As some have suggested, the passage may have been given its 26°31.3' angle in order to allow an alignment with the lower transit of the nearest (to the celestial north pole) circumpolar star of the period (see Piazzi Smyth, The Great Pyramid, pp. 367-373). An alternate explanation may be had from the following comparison: In the Ascending Passage - the angle of that passage became lowered from the implementation of the vertical portion of the shift because the end-point of the diagram's line for the passage remained anchored to the corner of the Pi Square - a corner which is part of the diagram's scaffolding and which therefore remained unshifted as all of the diagram's interior passageway lines were shifted. The as-built Ascending Passage wound up being lengthened by about 30 inches due to this lowered angle because the upper terminus of the actual passage stayed true to the pre-shift diagram's directive to have it end at the King's Passage level - even after this level went through the shift (i.e., going from a pre-shift elevation of 1640.5" to the post-shift elevation of 1688.7"). However, with the Descending Passage, it may have been decided to implement things in a conceptually opposite way, meaning that the length of the passage was allowed to change directly as a result of the vertical implementation of the shift, but the passage's angle after the shift stayed true to the amount it had been in the Pre-shift diagram.

16. As stated, I am here using the Pre-Shift diagram's .177535 unit elevation for the floor of the Queen's Passage. The step in this passage lowers the elevation of this floor by about 20 inches, and - according to Gantenbrink's drawings - the point of beginning of these shafts (as they were built) was evidently designed to be at the inside surface base of these walls at this lower level. It would appear that the angles of the shafts, and the positioning of the "doors" were determined in the context of the Pre-Shift diagram. I have presented here what I consider to be a core scenario for the original diagrammatics of these pyramid features. There are justifiable minor adjustments that one can make to this scenario which would result in slight changes to the angles for these shafts given in the text. The challenge is to explore the various possibilities offered by the diagram in order to try to understand the thinking behind the choices made by the pyramid's architects.

17. There is precedent for the inclusion in the finalized plans of some features from an earlier design stage. See the discussion regarding the length of the Bent Pyramid's Upper Descending Passage in the Bent Pyramid essay.

18. Such alignments were earlier proposed by physicist Virginia Trimble and Egyptologist Alexander Badawy in 1964, and later expanded upon by Robert Bauval in 1994. See Bauval and Gilbert's The Orion Mystery for their theory, and for specific references to the earlier theories. Trimble's 1964 paper appears as an appendix in The Orion Mystery.

19. Metric units are used here because Gantenbrink gives all of his measurements in meters. Note also that in his drawing, Gantenbrink begins measuring from the wall's inside surface. As is the case with a number of other other pyramid locations, lengths as they exist in the Pre-shift Diagram were here retained in the constructed pyramid. See the Computations section for these analyses. Also given there are the computed shaft lengths for a Queen's Chamber positioned such that it is centered directly on the pyramid's axis in the derivation diagram.

20. National Geographic.Com. Update : Third "Door" Found in The Great Pyramid. Sept. 23, 2002. (http://news.nationalgeographic.com/news/2002/09/0923_020923_egypt.html) This article states that the measured distance to the door in the north shaft is 65 meters, and then goes on to say that this is the same distance as was found by Gantenbrink for the south shaft 'door'. However, Gantenbrink's web posting shows in specific detail that the distance he found from the south wall's interior to the 'door' is 59.5 meters. (He measured it to be 57.5 meters from the wall's exterior, to which is then added the 2 meters for the wall's horizontal thickness. Refer to Figure 15). It is possible that the more than 6 meter difference between the south and north shaft findings may be due to the fact that the actual north shaft does not angle upward in a straight line, but instead travels extra distance as it curves around the Grand Gallery feature.The south shaft has no such impediment, and so does angle upward in a relatively straight line. In other words, the statement in the National geographic article may be in error as regards the southern shaft length.

21. Petrie (p. 83) was able to obtain measurements for only parts of the north and south shafts. He found the south shaft to vary in angle from 44°26' to 45°30', and the north shaft to vary from 30°43 to 32°4'.

22. It is about 67 meters distant for the south shaft, and about 66.5 meters for the north shaft. (These distances being in 'line of sight' terms. See Computations section for a more complete analysis.)

23. Refer to a west passage analysis in the Khufu Computations section. Also refer to such discussions in the Red and Bent Pyramid essays.

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

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

References

Bauval, R. and Gilbert, A. 1995. The Orion Mystery. Random House, New York.

Cooper, L. 2004. See the author's Red, Bent, and Khafre Pyramid essays linked to below.

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

Gantenbrink, R. <http://www.cheops.org>. From the main page, link to "Cyber Drawings", and then lastly to the "Cheops Shafts" page.

Gillings, R. J. 1982. Mathematics in the Time of the Pharaohs. Dover, New York.Heath, T. L. 1897. The Works of Archimedes. Cambridge.

Lawlor, R. 1982. Sacred Geometry. Thames and Hudson. London.

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.

National Geographic.Com. Update: Third "Door" Found in Great Pyramid. <http://news.nationalgeographic.com/news/2002/09/0923_020923_egypt.html>Peet, T. E. 1931. "A Problem in Egyptian Geometry", J.E.A. 17.

Petrie, W.M.F. 1883. The Pyramids and Temples of Gizeh. London. This is the original unabridged printing, and is currently available on line at http://www.ronaldbirdsall.com/gizeh/index.htm

Robins, G. and Shute, C. C. D. 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

Smyth, P. 1994. The Great Pyramid: Its Secrets and Mysteries Revealed. Gramercy, New York. (Originally Published in 1880 as Our Inheritance in the Great Pyramid.)

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

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