The above is the Derivation Diagram of the Great Pyramid. This is how it is derived.
(Note: Parts of this work are still in progress, so please excuse the fact that there will be periodical updates.)

The Great Pyramid at Giza was designed, in large part, to embody one of the holy grails of ancient mathematics, namely the "squaring" of the circle. As discussed in a previous section, this was accomplished by having the pyramid's four sides incline upwards at an angle of 5150'. The result is a situation where a circle that is drawn using the pyramid's height as its radius will have the same circumference length as the pyramid's perimeter as measured at its base.

As shown here, the relative lengths of the height and sides in the Great Pyramid are .78615 to 1.236. This relative height (.78615) when multiplied buy 2 Pi (2PiR = circumference), is an extremely close equivalent to the perimeter of the base (4 x 1.236).

As with all preceding diagrams in the previous sections of the Deep Secrets work, the diagrams to follow can straightforwardly be produced using a compass and straight-edge only. I refer readers not already familiar with the various techniques discussed below to peruse earlier sections for explanation. These diagrams are not complicated, but there are a few of them. Bear with.

We begin in Diagram 1 with a circle that has been quartered by means of two diameters passing through the circle's midpoint at right angles to each other. Furthermore, the midpoints of the two radii (these radii here shown red and green, and which together constitute the circle's horizontal diameter) have been determined, and a vertical line has been drawn through each midpoint meeting the circle top and bottom. For the purpose of ongoing comparisons, each radius of this initial circle will be considered as having a relative length of 1 unit.

In Diagram 3, lines have been drawn from the apex (and nadir) of the circle's vertical diameter on through each of the midpoints of the two horizontal radii, to meet the furthest rim of each smaller circle. The first segment of this line (one of which is seen as the red line above) is the diagonal of a rectangle which has sides of 1 unit and .5 unit. This diagonal therefore computes to have a relative length equal to 5 divided by 2, or 1.118034. The second part of this line (in blue above) equals .5 of the initial radius, and so the whole line (red plus blue) has a relative length of 1.618034. The ends of these 2 pairs of 1.618034 lines are joined to complete their respective triangles. (I will continue to give these computed relative lengths not because they are necessary to the immediate derivation at hand, but because they will be helpful in later discussions. Remember that all lengths are given relative to the radius of the initial circle of Diagram 1, with this radius being considered to have a length of 1 unit.)

Now starts the fun and games. A compass is next set to the length of one of these new 1.618034 lines, and pivoting from the top and bottom of the circle's vertical diameter, arcs are drawn (top and bottom) tangent to the inner circles. These arcs are drawn to their intersection points outside of the main circle, and the circle's horizontal diameter is extended left and right to these points of intersection. (Such intersecting arcs create a shape known to geometers as a "vesica piscis", Latin for "fish bladder".)

All that has been added in Diagram 5 is the outer circle - which has a radius that extends from the original circle's center to the arc intersection points (such a radius is here shown in red). With one of the 1.618034 lines (shown in dotted blue) pivoted to rest at this intersection point, a right triangle can be created having its vertical side equal to 1 (it is a radius of the initial circle) and a hypotenuse equal to 1.618034 (again, the dotted blue line). The radius of this new outer circle therefore can be computed (using the Pythagorean Theorem), and can be found to have the relative length of 1.27202. We are now ready to complete the derivation.

In this last step, a square with sides equal in length to the original circle's diameter is inscribed around the original circle. This square then has a perimeter length of 8 units. The outer circle has a radius of 1.27202, and so this outer circle's circumference is 1.27202 multiplied by 2Pi. The result is 7.9923, an amount that for all intents and purposes is equivalent to 8. The circle is squared!

In addition, from the apex of the outer circle's vertical diameter, lines have been drawn (here shown in red) to the left and right extents of the original circle's horizontal diameter. The relative vertical height of the resulting triangle is, as we've seen, 1.27202. The base of this triangle is 2 units, and so its hypotenuse turns out to be the by now familiar 1.618034 units. These are exactly the same proportions that are in the Great Pyramid, and so we have successfully derived the "squaring the circle" synthesis diagram presented initially - and perhaps much more.

The derivation just detailed is one that is fairly well known in certain circles (forgive the pun). What follows, is not as well known.

As suggested earlier, the Great Pyramid was also designed to allow for a second, and quite different, means by which to 'square' the circle. The previous derivation squares the circle in terms of perimeter and circumference. The following diagram squares the circle in terms of area - with but one minor catch.

In Diagram 7, the 1.618034 lines are shown in blue (yes, these have the same length as the sides of the just derived pyramid). Vertical lines (in red) have been drawn in which connect the ends of the 1.618034 lines, and these new lines (in red) have been extended above and below the horizontal diameter to the same distance that they each stand from the original circle's vertical diameter (the diagram's mid-line). This distance, when computed, equals .7236 the length of the original circle's radius. Horizontal lines (also in red) are then drawn connecting the (red) vertical lines at their tops and bottoms to create a new, inner square. Circles are drawn centered on the points where the left and right sides of this new square each cross the horizontal diameter (at a distance of .7236 from the diagram's midpoint), with each circle using this distance from the midpoint (.7236) as their radius.

The newly created square has each side equal to 2 x .7236 = 1.4472. The area of this square is therefore 1.4472 multiplied by itself, which is 2.09439, and this amount is exactly 2/3rds of 3.14159. The minor catch is that the circle isn't actually squared, but instead is "2/3rds" squared. However, the diagram can be continued to create a rectangle that does contain the precise area of the circle.

All that has been done in Diagram 8 is to extend the inner square vertically, both above and below the horizontal midline, by an additional amount equal to .3618 (that is, 1/2 of .7236). The resulting rectangle is therefore half again larger than the inner square, and hence has an area exactly equal to Pi.

Many have tried to find a geometric construction that directly squares the circle in regard to the areas involved, but no satisfactory means has yet been found. Egyptian mathematics did resolve this problem for computational needs by "squaring" 8/9ths the length of a circle's diameter to get a circle's area. (See footnote 3 in the "Circle and Square" section.) In 1882, the German mathematician Ferdinand von Lindemann proved that a diagrammatic route to finding such a square is not possible, and so the above appears to be as elegant a solution to this problem as there is. The architects of the Great Pyramid apparently came to the same conclusion, admiring the fact that, in any event, this method falls right in with the "perimeter to circumference" squaring of the circle - and is also not without a certain charm of its own. They evidently had sufficient regard for this overall treatment that they determined to guide the placement of the pyramid's interior passageways and chambers in line with the particulars of this derivation's component parts.

The upper half of the inner square is here outlined in red, and a diagonal is drawn from the lower right corner to the upper left, of this rectangle. This is the "ideal" line of the pyramid's Ascending Passage. The ceiling of this passage increases dramatically in height at the beginning of the upper half of the passage, this half known as the "Grand Gallery". As shown by the diagram, the increase in ceiling height begins at the point where the right-hand side 1.618034 line (also outlined in red) passes through. The height of the Grand Gallery is simply the distance between - a) the point where the half-square diagonal (i.e., the Ascending Passage) intersects the vertical mid-line and b) the base of the lower 1.618034 triangle (this line shown here in purple - note: this triangle is formed by the two 1.618034 lines which emanate from the nadir of the vertical diameter). This height is then extended down along the Ascending Passage to determine the actual point at which the Grand Gallery starts.

The passage to the Queen's Chamber leads off from this same starting point, and the chamber itself is nestled against the left-hand side .5 radius circle.

The passage to the King's Chamber begins at the upper end of the Grand Gallery, and the chamber is nestled against the edge of the left-side .618034 circle. (The center of this circle is the red hash-mark on the left-hand original circle radius. If not already familiar with how to determine the .618034 division of a line, refer to the Pentagon section for this procedure. Note also that to simplify things a bit, I am leaving out the ante-chamber, which here would sit just to the north of the King's Chamber. To aid visibility, the relative heights of the chambers themselves are given larger than scale.)

The Descending Passage is determined by two points. The first locator is the spot at which the the left-side 1.618034 line (shown in green), in rising from the bottom of the vertical diameter, intersects the left-side .5 radius circle. (This intersection is highlighted by a small red circle.) The other locator point is the .618034 intersection of the right-hand radius. The Descending Passage - as theoretically extended - passes through these two positions.

The passage to the Subterranean Chamber begins at the outer edge of the right-side .5 radius circle. The chamber itself nestles against the left-side .5 radius circle.

All of the locations mentioned so far have been determined from the diagram itself, and not from merely superimposing a scaled outline of the pyramid's interior onto the diagram. It must be admitted that there is some room for play in that a choice must be made as to which of the inner circles is best suited to match the actual horizontal passage and chamber distances within the pyramid. More on this will be discussed below.

Before we launch into seeing how well things match up with the actual surveyed positions within the pyramid, there is yet one more diagram to consider. This last one deals with the placement of the pyramid's so-called "air shafts".

I will deal with the King's Chamber "air-shafts" first. As we look east to west through the structure, the King's Chamber is offset somewhat from the pyramid's center-line. Both of the shafts exit this chamber from a point about 39 inches above the floor. The northern shaft travels horizontally for 12 1/2 feet before angling upwards, while the southern shaft angles up after 5 1/2 feet. In the diagram, the southern channel (on the left) heads for the point where the Pi rectangle intersects the top side of the outer square. The channel to the right has its end point at the upper-right corner of this same outer square. The paths of both shafts lie tangent to their respective .7236 radius circles.

The Queen's Chamber sits (seemingly) symmetrical to the pyramid's center-line, and its "air shafts" exit the chamber from points about 5 feet above the floor, and extend horizontally about 6 1/2 feet before they angle upwards. They head for the respective upper corners of the 2/3rds Pi square.

A remarkable twist regarding the Queen's Chamber shafts is that unlike the King's Chamber shafts, those of the Queen's Chamber does not actually exit to the outside of the pyramid. Instead, they dead-end about 50 feet short of reaching daylight. The diagram does not necessarily explain why these shafts were designed to fall short. The diagram, however, may provide a reason for why they stop where they do. More on this in the next section.

I end the derivation diagram here, but do not claim that it is at the same point at which the pyramid's architect may have ended it. I see Diagram 10 as being the basic plan onto which more may well have been added for various refinement purposes.

All of the comparative lengths from the diagram given below have been computed using basic geometric and trigonometric techniques. (Anyone with questions on specific methods should feel free to email me.) Comparisons of lengths listed here are in relation to W.M. Flinders Petrie's extensive surveys of the Great Pyramid (as published in his work, The Pyramids and Temples of Gizeh, see references below), and with measurements now available from Rudolph Gantenbrink's published results (also see below).

In the derivation diagram the radius of the original circle was, for simplicity's sake, given the arbitrary but convenient length of 1 unit. As it turned out, 1 unit then became the relative length of 1/2 of the side of the derived pyramid (the derived pyramid being directly proportional in size to the Great Pyramid). Petrie calculated one half the length of the pyramid's side at its base to be 377.8 feet. I say calculated, because he was inferring the length of the pyramid's sides as the builders had built them, not as he found them. Most of the pyramid's original exterior "casing" stones had been removed to build the city of Cairo in the 1300's A.D.

The derivation diagram proves to be extremely accurate in all but one particular. The relative elevations above what Petrie calls the "pavement" level are all lower in the diagram than in Petrie's findings - and this by the fairly constant amount of between 3 to 4 feet. When both diagram and survey of the pyramid are plotted out to the same scale, one finds that all interior positions match up correctly. The innards of the pyramid, as designed by this diagram, were evidently raised some 3 to 4 feet in height. In other words, the true 'grade level' for the structure is near the top of the first course of stone blocks. My guess is that this was done either as an aid in establishing a level foundation, to raise the Subterranean level above the seasonal water table extreme, or for the reason given at the end of this essay. There will be more on this shift in height as we go along

1) The entrance to the pyramid (and hence the entrance to the Descending Passage) is on the pyramid's north side and is listed by Petrie as being 55.66 feet (668 inches) above the "pavement" level. He also says that this entrance point lies in a vertical plane that is 43.66 feet (524 inches) south of the pyramid's original northern base edge. 668 divided by 524 equals 1.275, which is indeed very close to 1.272 - this being the tangent of 51°50' - the "ideal", if not intended, base angle for this pyramid.

In comparison, the derivation diagram shows the entrance to be at a relative height of .137095 units, and located .107776 units south of the north edge of the pyramid. Since Petrie calculated half the pyramid's length to be 377.8 feet, and we have set this length in the diagram equal to 1 unit, the diagram's entrance height figure of .137095 units therefore calculates to be 51.8 feet (621.5 inches). The distance south of the north edge of .107776 units then equals 40.7 feet (488.6 inches).

As I forewarned, in the diagram the entrance height is about 4 feet lower than Petrie's finding. The entrance position relative to the north edge in the diagram is less as well - about 3 feet less than Petrie's finding. To understand why, picture the diagram as an overlay placed over the Petrie survey (with both using the same scale). When the interior passageways are lined up to match (which they will), it will be seen that there is then a gap at the ground level (here shown in green and exaggerated somewhat for effect). The bottom of the derivation diagram stops short of Petrie's "pavement" level by roughly 4 feet. As a result, the actual pyramid has a 4 foot layer that is not accounted for in the diagram, and hence the actual pyramid also protrudes northward at the base by about 3 feet more than the layer above. We will see that all of the interior features of the actual pyramid have been raised 4 feet, and moved south by 3 feet, relative to the diagram as a result of this shift.

2) The diagram has the Descending Passage inclined at the angle of 26°33.8' (it is the diagonal of a 1 x 2 rectangle, and as such lies at an angle whose tangent - opposite/adjacent - equals .5). Petrie measured this passage to be at 26°31.3'. This is close enough to the diagram's prediction to cause very little discrepancy in comparative elevation findings. In the diagram, the Descending Passage intersects the "pyramid's" northern 1/2 base length at the "Golden Ratio" .618034 division point. However, in the actual pyramid and because of the shift, the Descending Passage cuts through the pyramid's base at about the .5888 division point.

3) Where the Ascending Passage meets the Descending Passage Petrie says is 1,111 inches down the passage from the entrance point. The diagram computes this length to be 1,122 inches. (I should note here that Petrie bases most of his measurements as computed to be along a passage floor, and is therefore taking into account in his overall plan the various passageway heights. The diagram is not able to be so specific, and so it would have been up to the pyramid's builder to decide whether the diagram should be interpreted as pinpointing a passage floor, ceiling, mid-point, etc.)

4) The diagram has the Ascending Passage inclined at the angle of 26°33.8'. Petrie measured the mean angle of the entire Ascending Passage to be 26°12'50" - noting that it begins at about 26°5' and ends inclined at 26°16'40". Using Petrie's figures for height and horizontal lengths, I calculate that he understood the average angle of the entire Ascending Passage to be 26°16'. (I can only speculate why the angle of this passage is not closer to that of the Descending Passage, or why its slope is somewhat variable. An interesting explanation is that the intended use of the Grand Gallery was as an observation post from which to map the heavens prior to the pyramid's completion. Refer to Richard Proctor's work at the Bibliography link below for more on this theory.)

5) The point at which the Grand Gallery begins is listed by Petrie as being 1,547 inches from where the Ascending Passageway, if it were to be theoretically extended, would touch the Descending Passage floor. The diagram has this length at 1,532 inches. It is at the beginning of the Grand Gallery that the passage to the Queen's Chamber intersects into the floor of the Ascending Passage and heads horizontally south.

6) Petrie calculates the height above the 'pavement' at this Queen's Chamber passage point to be 852 inches. The diagram has it at 804.8 inches. Again the diagram is about 4 feet (48 inches) lower than Petrie. The Queen's Chamber and passage extend south 1,730 inches by Petrie's measure, ending at the interior south wall of the Queen's Chamber. (This length includes the Chamber's interior north to south measurement of 206 inches.) The diagram predicts this total Queen's passage and chamber length to be 1,818 inches (in the diagram, the space allotted for the Queen's Chamber and the passage to it from the Ascending Passage extends to the leading edge of the left-side .5 radius circle). The 88 inch discrepancy between the diagram and Petrie's figures may be due to the fact that the architects were accounting for the thickness of the chamber's south wall, which Gantenbrink finds to be 79 inches. Note that this means that in the diagram the Queen's Chamber is not bisected by the diagram's mid-line - (as it is in the actual pyramid). Instead, the center-line of the diagram lies about 35 inches south of the center of the Queen's Chamber. More on this below.

Petrie also notes that the Queen's Passage has a drop-step of 17.6 inches which occurs at a point 1,307 inches from this passageway's point of beginning at the Ascending Passage. This means that everything south of this step is now at 834.4 inches above the pavement level, not the original 852 inches above. This drop in elevation brings Petrie's accrual's into even closer theoretical alignment with the diagram. I will propose two possible reasons for the Queen's Passage Step below.

7) Returning now to the Ascending Passage, as stated before, the Grand Gallery begins at Petrie's 1,547 inch mark. Here things get interesting, and revealing. The Ascending Passage continues upwards through the Grand Gallery, ostensibly ending after 1,815.5 inches (according to Petrie) at the mysterious Great Step (shown as triangle ABC in the depiction below)..

As Petrie notes, the Ascending Passage comes to an abrupt end at the base of the Great Step (Point A), but that it actually would have ended at point C (at 1,883.6 inches from the Gallery's starting point) had the step not been in the way. He details that the walls of the Grand Gallery actually continue on past the face of the step as well, ending at point C - again, where the passage would have ended had there been no step. He gives 61.7 inches for the top of the step (BC) and 68.1 inches for the distance CA. He correctly points out that the step's face (BA) is directly in line with the pyramid's vertical mid-line. Curiously, he then continues to measure southward to the King's Chamber not from this mid-line (point B), but from point C - where the Ascending Passage by all rights should have ended. What is going on here?

Petrie obviously was faced with a bit of a conundrum which he simply accepted as odd and continued on. Why this Great Step? Why not end the passage on the pyramid's mid-line? Why extend the Grand Gallery walls to point C? (Interestingly, Piazzi Smyth believed that the mid-line of the pyramid did run through point C - see Plate IX in his book referenced below.) As discussed along with Diagram 11, the answer to all this has to do with the way in which the constructed pyramid was adapted from its derivation diagram.

Diagram 12 below, is more correctly the precursor to Diagram 11 and yes, you are seeing double in it. Here we have the derivation diagram (in red) placed at the grade level before being raised the vertical four feet amount (here in green). The blue lines represent the post-shift arrangement, showing the position of the diagrammatic layout after having been conceptually slid upwards along the pyramid's north face and into place. Note that -

a) Although the red vertical mid-line will lie directly over the blue vertical line when the shift has been made, the beginning position of the red vertical line WILL STILL REMAIN as the final pyramid's center line. Raising the pyramid's innards upwards by sliding it along its exterior side will not change the location of the pyramid's exterior baseline dimensions, and hence will not change the actual mid-line axis. As said above, this is because that with the shift described, a pedestal of sorts is created that still continues northward some 3 feet. Making an overlay to use as a visual aid may be found helpful in fully grasping this.

b) The vertical blue line, then, is the line which in the actual pyramid demarcates the virtual end point of the Ascending Passage at the southern end of the Great Step (Point C in the previous drawing), and it also demarcates the true end of the Grand Gallery (i.e., where the Gallery's walls and roof end). It is the mid-line of the derivation diagram, not of the actual pyramid. Also note that the shift does not increase the height of the pyramid. The south face meets the apex of the red line (it is the pyramid's center-line, not the blue line)

c) The top of the Great Step, as measured by Petrie is about 5 feet in length. The shift accounts for about 3 feet of this. The 2 foot difference is due to the fact that the Ascending Passage was, for reasons still uncertain, built less steep than the diagram dictates. Although less steep, the builders decided to extend the length of the Ascending Passage so that it would still reach the original elevation point of the King's Chamber Passage (1,692 inches). Rather than push the true mid-line further south (and thereby increase the length of the entire pyramid), it was decided to allow for this difference by extending the location of the "vestigial" mid-line - and hence the length of the Great Step - southward.

This all then leads to an explanation for the existence of the Great Step. It is the architect's acknowledgement of the duality between the mid-line of the pyramid as it sat derived on the drawing board and how this same line existed fully realized in the pyramid. Because of the way in which the diagrammatic pyramid was slid into place from the base (i.e., at a 51°50' angle), these meridians are not the same, and the pyramid contains, in effect, two mid-lines. The Great Step is a bridge between these two.

The height above "pavement" for the passageway to the King's Chamber is listed by Petrie as 1,692 inches. The diagram gives 1,640 inches, again about 4 feet less. Petrie lists the angle of the Ascending Passage at this point to be about 26°17', and so one might expect that some of this 4 foot difference in height would have been reduced relative to the diagram by this lower angle. It does not do so because the length of the actual Ascending Passage was made precisely that amount longer than that of the diagram to maintain the same relative height of this passageway.

8) For the length of the distance from Point C to the south interior wall of the King's Chamber Petrie gives 475 inches. In the diagram, if a line is extended from the Ascending Passage endpoint (Point C) southward to intersect the edge of the .618034 radius circle, a distance of 530 inches is provided. Gantenbrink lists the thickness of the King Chamber south wall as being about 67 inches. This plus the 475 inches measured by Petrie makes for a very close fit.

9) Petrie lists the Descending Passage as having a total length of 4,144 inches as calculated from the pyramid's entrance. By extending the diagram's Descending Passage line to the lower edge of the right-side .5 radius circle, a length of 4,122 inches is provided. By going horizontally from this point to the left-side .5 radius circle, a distance of 716 inches is available. Petrie measures this length to be 672 inches, ending at the south interior wall of the so-called Large Subterranean Chamber. The horizontal difference between these two allotments is 4 feet, and as this chamber is entirely dug out of bedrock one can not necessarily account for this as being due to the thickness of the chamber's wall - as was offered in the King's and Queen's Chamber discussions. It is possible, of course, that a need was felt to allow for a symbolic amount of space equal to what the chamber wall would have required had it been built above ground.

10) Diagram 10 also reveals the final destination points for the "air shafts" which emanate from both the King and Queen Chambers. As may be evident, comparing the diagram to the actual pyramid can be a complicated affair. In the instances of these shafts, they exit each chamber north and south at differing heights from the floor, traverse horizontally for differing distances, and then angle upwards on their way. The northern shaft to the King's Chamber even curves around parts of the Grand Gallery before settling down. The diagram has none of this, though in making my calculations I have incorporated Gatenbrink's findings for height above chamber floor and length of horizontal run for each shaft. For the Queen's Chamber shafts I have also factored in the decrease in height due to the Queen's Passage step. Other than these adjustments, the diagram angles are based on diagram heights. I refer interested readers to Petrie's information and Rudolph Gantenbrink's website for the data needed to do these computations.

- For the Queen's Chamber South Shaft - Gantenbrink finds that it angles upwards at an average angle of 39°36.5'. The diagram yields an angle of 37°48'.
- For the Queen's Chamber North Shaft - Gantenbrink finds that it angles upwards at an average angle of 39°7.5' . The diagram yields 38°24'. (Remember that in the derivation diagram the Queen's Chamber is not centered on the pyramid's mid-line.)
- For the King's Chamber South Shaft - Gantenbrink finds that it angles upwards at an average angle of 45° exactly. The diagram has it at 46°1.5'
- For the King's Chamber North Shaft - Gantenbrink finds it angles upwards at an average angle of 32°36'. The diagram has it at 31°35.5'.

These findings are quite close to Gantenbrink's, and assuming that his are correct, I think it fair to say that the derivation diagram does a creditable job of predicting the destination points of these "air shafts". The fact that they do not line up perfectly is, I believe, likely due to the numerous variables and unknowns previously discussed, and not to a failure of this theory. The largest angular discrepancy derived for the shafts is about 2 degrees, and so we are talking about a relatively small difference from the actual position of origination.

A word needs to be said about the architect's apparent choice of destination points for these shafts. For the Queen's Chamber shafts, there is clear intent to connect, or even "suspend", the chamber from the upper corners of the "2/3rds" Pi square. With the King's Chamber, the connection is to both the Pi rectangle and to the outer square. The need felt by the architect to establish these added connections between these chambers and these specific points in the diagram, to my mind, opens up a completely new and challenging direction for inquiry.

11) The Blocked Air Shafts - Gantenbrink has discovered that the southern Queen's Chamber shaft does not exit the pyramid, but actually ends abruptly some 50 feet before reaching the exterior surface. (It has also recently been found that the northern Queen's shaft is also plugged, and apparently at the same distance from the Queen's Chamber.) We can only speculate as to why these shafts were designed in the first place, and therefore we are left having to compound such speculation when considering why those of the Queen's Chamber were built intentionally obstructed. The diagram doesn't provide definitive answers, but it does offer some interesting food for thought.

The blocked end of the southern shaft of the Queen's Chamber appears to coincide with the point in the diagram at which the shaft exits from its transit within the left-side .5 radius circle. Gantenbrink says that the shaft ends after a run of about 57.5 meters (188.6 feet) from that point just outside of the chamber at which it begins angling upward. Using a diagram derived placement, I calculate that the shaft intersects the .5 circle at about 3 feet past this 57.5 meter mark. If, as therefore seems possible, there is an intended linkage here, then what is its significance?

12) The Queen's Passage Step is yet another of the pyramid's design mysteries. Two explanations for it come to mind: a) It is there to lower the height of the Queen's Chamber so that it will fit more precisely between the diagram's left and right inner circles; and b) It is there to provide a marker of an important pyramid meridian, as was the case with the Great Step.

a) The Queen's Passage step occurs (according to Petrie) 1307 inches from the start of the Queen's Chamber passageway, and has a drop of 17.6 inches. This lowering does seem to improve the Chamber's fit between the diagram's two inner .5 circles by 5 inches. As given previously, the diagram counts the Queen's Passage and Chamber as 1,818 inches while Petrie has 1,730. Gantenbrink measured the south wall of the Queen's Chamber to be 79 inches thick, and so adding 1,730" to 79" means that the space needed for chamber, chamber wall and passage is 1,809 inches in the actual pyramid. The diagram provides 9 inches too much. Lowering the passage by 17.6 inches means there is theoretically now only 4 inches extra space.

b) Whatever the reason that it was decided to place a step in this passageway, there is also the related question of why it was placed where it was. The Queen's Passage Step has a close vertical alignment with the point at which the Descending Passage meets the horizontal Subterranean Passage and, interestingly, a similarly close alignment with the point at which the Ascending Passage exits its .7236 radius circle. (See Diagram 10.) In the derivation diagram, the Queen's Passage Step occurs 364 inches from the center of the pyramid, while the Descending Passage ends at a point 356 inches from the center line and the Ascending Passage exits the .7236 circle 355 inches from this same center line. I believe it likely that the intent here was to have these all in perfect alignment. I therefore have to wonder if the 8 inch difference in the passage step's location might not be due in part to the choice of point of beginning from which Petrie began his measurement for the passage length.

I will end with a final observation that I have been pondering. As noted earlier, in the derivation diagram the center line of the pyramid passes through about 35 inches to the south of the center of the Queen's Chamber. Following the shift, however, the center line of the pyramid winds up being directly in line with the center of the Queen's Chamber. Can it be, then, that the entire purpose and magnitude of the shift was simply to impose this specific alignment? If so, it was clearly done at the expense of the Grand Gallery whose upper end was shifted away from the center line and to a very curious ending at the south side of the Great Step. Was it that the architect simply felt that at least one of the chambers absolutely had to be at the center of the pyramid - even at the expense of other alignments - and that the Queen's Chamber was closest? Or, was there some other critical reason to have the Queen's Chamber thusly aligned, and if this other reason was so critical, why was it not accounted for in the pre-shift placement? These kinds of details can certainly get a body chasing his tail in a hurry. Interesting, though.

We have seen that a specific geometric diagram which "squares" the circle both in regard to its circumference and to its area appears to hold promise as being the original design template for the placement of the interior passageways and chambers of the Khufu Pyramid at Giza.

The application of this template to the pyramid offers explanations for: 1) the angle of incline of the pyramid's sides; 2) the pyramid's length to height ratio; 3) the relative location of the pyramid's entrance; 4) the angle and length of the pyramid's Descending Passage; 5) the placement, angle, and length of the Ascending Passage; 6) the position of the beginning and ending points of the Grand Gallery; 7) the height of the Grand Gallery; 8) the elevation above grade of both the King's and Queen's Chambers - and the lengths of their respective passageways; 9) the depth below grade of the Large Subterranean Chamber, and the length of its passageway; 10) the reason for the Great Step design detail; 11) the placements of the so called "air shafts"; and 12) a possible rationale for the presence and placement of the intentional block in the Queen's Chamber southern "air shaft". Also given is a possible reason for there being a step in the Queen's Passageway.

All of these correlations flow from a single diagrammatic construction, a construction which provides an entirely new vantage from which to consider the capabilities and concerns of the builders of this pyramid. Many new questions have been raised, and many new lines of inquiry have been identified from which to pursue answers to these questions.

 References

Gantenbrink, Rudolph. Gantenbrink's findings referred to in this paper are all available at his web site: (http://www.cheops.org.). See especially his CAD drawing of the pyramid.
Lawlor, Robert. Sacred Geometry. Crossroad. New York, 1982.
Petrie, W.M.Flinders. The Pyramids and Temples of Gizeh. London, 1883. Link here for the online version of this work.
Smyth, Piazzi. The Great Pyramid: Its Secrets Revealed. Crown, 1978 (A reprint of the 1880 original.).

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