The Bent Pyramid

Leon Cooper

The Bent Pyramid: Part I

The Basic Derivation Algorithm

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

In each of these four pyramids, a
two-fold 'squaring of the circle' - a squaring in terms of the*
circumference* of the circle __and__ in terms of* *the*
area *of the circle* *- was apparently used to create a
geometric diagram which then provides the basis for:

1) the height above base level for each entrance passage;

2) the descending angle of each entrance passage;

3) the length along the floor of each descending passage; and

4) the lateral location of each entrance passage - and of the interior chambers themselves.

The methods to be described below lead to results which are almost all within inches of, and in a few cases exactly match, published survey findings. The diagrams presented not only explain why the passageways and chambers were built where they are in the pyramids discussed, but these diagrams also give very strong clues as to where as yet undiscovered features are likely to be located.

I will repeat as we go along, but let me say at the outset as clearly as I can, that all of the mathematics and geometry to follow are achievable through very basic and simple empirical means. Although much of the discussion here will be in terms of modern understandings (for example, decimal unit notation is used and angles are often spoken of in terms of degree measurement), it will be shown that all of these undertakings are empirically achievable within the context of the 'royal cubit' measuring rod, and within the context of the mathematical - and angle measurement - systems which current scholarship acknowledges to have been in use as early as the Old Kingdom period. For those wishing to see a "modern" computational analysis for the lengths derived within each diagram, these can be found by linking to the Computations page.

We begin by
noting that the Egyptians had indeed developed an empirical method by
which they could 'square the circle' in terms of the circle's area.
This means that they could - for any given circle - determine the
size of the specific square which would contain the same area as that
circle. Their method was to measure the diameter of a circle, find
8/9ths of this diameter length, and then square the 8/9ths
amount.^{1}
This simple algorithm is at once easy to remember, easy to
implement, and remarkably accurate. The diagram below shows such an
8/9ths square centered on the circle's diameter.

Since each side of the square is
equal to **8/9ths** of the circle's *diameter*, centering the
square around the circle's midpoint clarifies the finding that
*half* of the square's side (S/2 = OR) will be equal to
**8/9ths** of the circle's *radius* (OB). The relevance of
this distinction will become apparent shortly.

Seeing that the Egyptians had
determined a means to represent the *area* of a circle in terms
of a square, it is then reasonable to wonder if they were curious to
learn whether this same square (or some other square) might not have
(for all intents and purposes) the same total perimeter length as the *circumference* of the
circle. There is no known written proof that the Egyptians had
devised a "formulaic" method for determining the circumference of a
circle, but it is credible that they would have sought a means to do
so. As it turns out, they did have directly at hand a measurement
system which would have easily and accurately allowed them to find
the exact square having the same perimeter length as any given
circle's circumference.

The foremost standard unit of
measurement used by the Egyptians was the "royal cubit". This length,
equal to about **.**524 meters, was marked out into 28
subdivisions which were called "fingers". If the diameter of a circle
is made equal to a royal cubit in length, then this diameter will as
a result be 28 "fingers" long. If a length of cord or twine is marked
off into these same "finger length" units, it can easily be found
that the circumference of this circle will then measure to be almost
exactly 88 of these finger length units. If one then draws a square
having a total perimeter length equal to these 88 units, this square
will by necessity have a side that is 22 units long (from 88 divided
by 4).

We see this "circumference" square centered on its circle in this next diagram. Note that since the diameter of the circle equals 28 fingers, the circle's radius (i.e., OB in Figure 2) will be 14 fingers in length.

By centering the square in this way
we can see that half the side (i.e., OT) of the "circumference"
square will be 11 fingers long, and hence will be **11/14ths** of
the length of the circle's radius (OB). It is therefore clear that by
"squaring the circle" in terms of circumference we have derived a
square that is __not__ the same size as the square seen above for
the circle's area correlation (which was a square whose half-side was
**8/9ths** the length of the radius). It is also clear that it
would have been child's play for the Egyptians to have arrived at the
above circumference result. A number of Egyptologists have indeed
acknowledged this possibility - if not probability.^{2} (The reader may wish to try for himself the method described above in order to confirm how easily and accurately it can be done.)

The pathway to the derivation of pyramid interior design now lies in the direction of merely combining the above two 'squaring the circle' findings into a single diagram.

The above diagram reflects this
combination, and so contains the square having the same area, and the
square having the same perimeter, as the circle whose radius is
length OB. Therefore, half the side of the outer square (i.e., S/2 =
OR) will be **8/9ths** the length of radius OB, and half the side
of the inner square (S/2 = OT) will be **11/14ths** the length of
radius OB. This diagram is the derivation beginning point.

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

The present pyramid exterior rises
from grade level with a general angle of about 54°28'. At a
height of about 47**.**25 m this angle abruptly changes to near
43°22'.^{4}
Leaving aside why an initial angle of 54°28' was used, or why it
was then reduced during construction, we will look at how the
'squaring the circle' design protocol provides rationales for:

1) the elevation at which the the bend occurs;

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

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

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

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

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

7) the length along the floor of this passage.

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

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

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

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

Again, the pyramid's half-base length
(OS in this instance) is made equal to 1 unit. A line from the center
of the diagram to the intersection at Point B therefore creates a
right triangle having a base of 1**.**1 and a hypotenuse of
1**.**4. If OS, half the pyramid's base, is now made equal to the
actual pyramid's half-base length of 94**.**8 m, then the
corresponding elevation of Point N will be 47**.**78 m, a very
close match with the elevation at which the pyramid's bend has been
surveyed to occur.^{6}
Knowing this measurement and the angles involved, the relative height
of OA (the height of the constructed pyramid) can be computed to be
1**.**1084 units, here rounded off to 1**.**11. Therefore, the
square which has the same area as a circle of radius 1**.**11 (the
inner square of Figure 4) can be found to have a half- side length of
1**.**11 x **.**8888 = **.**9866.

Next, we have that a line from the
center of the circle to the intersection at Point E will create a
right triangle with a base of 1**.**1 and a hypotenuse of
1**.**11. With OS equal to the 'as built' length of 94**.**8 m,
Point K will compute to have an elevation of 11**.**7 m, which is
almost exactly the surveyed elevation (11.9m) for the entrance to the
Bent Pyramid's lower descending passage on the pyramid's northern
side.^{7}

The intended slope angle for the
lower passage is now supplied by the angle of line OD, and the actual
(as surveyed) length of this passage is supplied by the part of line
OD that extends from Point O to where OD intersects the side of the
pyramid at Point H. The angle that OD makes with the horizontal is
27°16', and the relative length of line segment OH is 78 m. Line
OH is then next moved to Point K to become the lower descending
passage (Line KZ), whose length Maragioglio and Rinaldi found to be
78**.**6 m.^{8}

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

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

The Upper Descending Passage

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

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

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

The 'circle and square' intersection
at Point Y in Figure 5 allows for the creation of a right triangle
with a base of 1 and a hypotenuse of 1**.**1. The resulting angle
that the line OY therefore makes with the pyramid's base is
24°38'. Point W - the point at which line OY crosses the
pyramid's side - has a relative elevation above the pyramid's base
line of 32**.**74 m. This elevation for Point W is in exact
agreement with Dorner's survey results for the elevation of the Bent
Pyramid's upper passage entrance.^{10}

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

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

In Figure 5, Line OG computes to have
a relative length of 77**.**47 m. Maragioglio and Rinaldi report
the full length of the actual western descending passage to be
67**.**66 m.^{12}
The difference here may be explained by looking at the finding, as
alluded to earlier, that the originally planned pyramid for this site
was substantially smaller, having been measured to have had a half
base length of about 78**.**6 m, and an exterior slope angle of
between 57 and 60 degrees.^{13}
Point X in Figure 5 identifies the point at which the line OJ crosses
the exterior side of this vestigial pyramid, and computation will
show that line OX is of a length (= 68m) well suited to have been the
diagrammatic rationale for the length of the as-built descending
passage.

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

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

The Bent Pyramid: Part 2

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

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

The Lower Chamber

From the Part 1 discussion we saw
that, in the derivation diagram, the relative height of the entrance
to the Lower Descending Passage is 11**.**68 meters above the
pyramid's base. (1 unit in the diagram is made equal to the actual
pyramid's 94**.**8 meter half the base length). The lateral
location of this entrance is directly on the east to west midline of
the pyramid's north side.^{15}
If a perpendicular is dropped from the point at which the radial line
to Intersection Point # 13 in Figure 6 crosses the pyramid's side, it
will meet the pyramid's base exactly on the midpoint of the
half-side. When the half-side midpoint is superimposed over the
pyramid's full north side midpoint, the derivation rationale for the
central axis location of the Lower Passage is
accomplished.

The diagram's length for the Lower
Descending Passage has earlier been shown to be 77**.**95 m,
meaning that it ends below the base level of the pyramid. (See
Figure
4 in Part 1, and the analysis
in the Computation
section). Figure 7 (below) shows the particulars of this
situation.

As can be seen in Figure 7, the
derivation diagram's Descending Passage ends 24**.**03 m below the
base level of the pyramid at a point that is 17**.**17 m north of
the pyramid's center. From this juncture, the passageway turns to the
horizontal and, according to Maragioglio and Rinaldi's scaled
drawings, it proceeds southward about 5**.**7 m until it ends at
the south wall of an antechamber.^{16}
(See Figure 8 below.) The entrance to the main Lower Chamber is
located in this same south wall, at a point that is 7**.**25 m
above the antechamber floor.^{17}
Adding this fact to the derivation diagram's findings will place the
Lower Chamber floor at a point that is (24**.**03m - 7**.**25 m
=) 16**.**78 m below the pyramid's base level.

As a result of these placements, the
north wall of the Lower Chamber (which is also the upper south wall
of the antechamber) would then appear to be 17**.**17m -
5**.**7m = 11**.**44 m north of the pyramid's center. If in
Figure 6 a perpendicular is dropped to the pyramid's base from the
point at which the radial line to Point # 18 crosses the pyramid's
side, it will meet the pyramid's base at a point that is 10**.**97
m to the left of the half-side's midpoint. If this midpoint is then
superimposed over the midpoint of the pyramid's full western side, it
will provide the apparent intended location of the Lower Chamber's
north wall. Again, note that this radial line crosses the pyramid's
side above the bend. (Refer to the Computations
section to see how these intersections above the bend can be
calculated.)

That this is the correct method for
determining the intended north wall location is supported by the fact
that the same radial line is involved in successfully providing the
elevation below grade at which this chamber sits. In Figure 6, if the
perpendicular dropped from the point at which the radial line to
Point #18 crosses the pyramid's side is shifted down to the point at
which the radial line to Point #11 crosses the pyramid's side, this
perpendicular will then extend 16**.**7 m below the pyramid's
base. (Note that this is the same basic procedure as was seen in
Figure
13 of the Red Pyramid essay).
This 16**.**7 m depth is in nearly exact agreement with the
diagram's earlier prediction which - in concordance with Maragioglio
and Rinaldi's measurements - found that the floor of the 1st Chamber
lies 7**.**25 m above the elevation of the end of the Descending
Passage, and hence is (24**.**03 m - 7**.**25 m =) 16**.**78
m below the pyramid's base.

Remembering that the North / South
axis of the pyramid is also the axis of the Lower Entrance Passage,
we can learn from Maragioglio and Rinaldi's drawings that the east
wall of the Lower Chamber is about **.**75 m east of the north /
south axis.^{18}
If a perpendicular is dropped from the point at which the radial line
to Intersection Point # 16 in Figure 6 crosses the the side of the
"tangent = 1**.**4" pyramid (the pyramid created by extending
upwards the slope angle of the lower section - see the dotted line
pyramid side in Figure 6), it will meet the pyramid's base at a point
that is 1**.**09 m to the right of the half-side's midline.
Super-positioning this half-side midpoint over the pyramid's full
south side midpoint will then provide the likely derivation for the
positioning of the Lower Chamber's east wall location.

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

The Upper Chamber

The diagrammatic derivation for the
elevation of the entrance to, and the length of, the Upper Descending
Passage was given in Part 1 of this essay. Not covered was the
derivation of the lateral placement of this entrance along the
pyramid's western side, and also the particulars for the Upper
Chamber. Maragioglio and Rinaldi state that the entrance is
13**.**7 m south of the the pyramid's western face midline, and
that the passage itself is about 1 m wide.^{20}
Given that the 13.7 m distance is from the midline of the western
side to the midline (axis) of the passage, it will then be
(13**.**7 m - **.**5 m =) 13.2 m from the western side's
midline to the north wall of the passage. If a perpendicular is
dropped to the pyramid's base from the point at which the radial line
to Intersection Point # 8 crosses the pyramid's side, it will meet
the pyramid's base at a point that is 13**.**3 m to the right of
(i.e., in this case south of) the half-side's midpoint. Superimposing
this scenario over the midpoint of the pyramid's full western side
will then provide the apparent rationale for the north to south
location of the Upper Entrance and Passage. (Note that this is the
same radial line that was used to determine the location of the bend
in this pyramid. See Part 1 of this essay.)

As it turns out, the locations for
all of the Upper Chamber's four walls can be determined via the
lateral location protocol. To begin with, survey findings show that
the east wall of the Upper Chamber is 13**.**3 m east of the
pyramid's north to south axis, and therefore this location can also
be determined by employing the same perpendicular associated with the
Point # 8 line used above, but this time by aligning the half-side
midpoint with the midpoint of the pyramid's full __southern__
side.^{21}^{
}(See Figure 9 below).

The east to west measure of this
chamber is 5**.**26 m, and so the chamber's west wall sits
(13**.**3 m - 5**.**26 m =) 8**.**04 m east of the pyramid's
north to south axis. If a perpendicular is dropped from the point at
which the radial line to Intersection Point # 11 crosses the side of
the 'tangent = 1**.**4' pyramid (the pyramid created by extending
upwards the slope angle of the lower section - as shown by the dotted
line pyramid side in Figure 6 above), it will meet the pyramid's base
at a point that is 7**.**94 m to the right of the half-side's
midline. If the midline of this scenario is superimposed over the
midline of the pyramid's full southern side, the west wall location
will be identified.

As seen in Figure 9, the south wall
of this chamber is located about **.**2 m further south than the
south wall of the horizontal passage leading into it. Since this
passage is a continuation of the Upper Descending Passage, its axis
is 13**.**7 m south of the east / west axis of the pyramid. And
since the passage is 1m wide at this point, the chamber's south wall
will then be (13**.**7m + **.**5m + **.**2m =) 14**.**4 m
south of the pyramid's east / west axis. A perpendicular dropped from
the point at which the radial line to Intersection Point # 7 crosses
the pyramid's side (this is below the bend, and so the tangent of the
pyramid's side equals 1**.**4), will meet the pyramid's base at a
point that is 14**.**39 m to the right of the half-side's midline.
If one superimposes this scenario over the midline of the pyramid's
full western side, the south wall location will be
derived.

With the north to south length of the
chamber having been measured to be 7**.**97 m, the chamber's north
wall is then (14**.**4 m - 7**.**97 m =) 6**.**43 m south of
the pyramid's east / west axis. If a perpendicular is dropped from
the point at which the radial line to Intersection Point # 12 crosses
the pyramid's side (we're back now above the bend, and back to a
'pyramid side' tangent of **.**9444), it will meet the pyramid's
base at a point that is 6**.**58 m to the right of the half-side's
midline. If one superimposes this scenario over the midline of the
pyramid's full western side, the intended north wall location will be
derived.

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

Unfortunately, there is apparent
conflicting survey information regarding the elevation of the Upper
Chamber. In their diagrams, Maragioglio and Rinaldi seem to imply
that the elevation of the floor of both the Upper Horizontal Passage
and the Upper Chamber varies from being directly at the pyramid's
base level to being 3**.**2 m above it.^{22}
I propose that the protocol which provides the Upper Chamber
elevation is similar to that which has been shown to have been used
for the derivation of the 3rd Chamber of the Red Pyramid. Referring
again to Figure 6, if a perpendicular is dropped from the point at
which the radial line to Point # 21 crosses the pyramid's side, and
if this perpendicular is then shifted up to the point at which the
radial line to Point # 22 crosses the pyramid's side, it will end
2**.**27 m above the pyramid's base level. This is the same
procedure as was shown in Figure
13 in regard to the Third
Chamber of the Red Pyramid, and in both cases the determination is
made by the lines which lead to the final two intersection points. It
of course remains possible that the original intent here was to have
this chamber's elevation be at the pyramid's base level, although
this would be a departure from what is seen in the analyses for the
interior-most chambers in the Red, Khafre, and Khufu pyramids.

Further Observations

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

The interior layout derivations shown
here for the Bent Pyramid strongly suggest that the design rationale
was fully dependent upon the final, and as-built, exterior shape and
dimensions of the structure. This finding would appear to be in
contradiction with the evidence that suggests that the original
intent at this site was to build a pyramid with a half-base length of
about 78**.**6 m, and having an exterior whose sides were inclined
at an angle of about 60°.^{24}
It is perhaps not a coincidence that a fully realized pyramid of this
exterior slope angle and 78**.**6m half-base length would have had
a height almost exactly equal to that of the actual as-constructed
Bent Pyramid. This convergence opens the door to the possibility - if
not likelihood - that the "precursor" pyramid was part of the plans
for the Bent Pyramid from the beginning.** **A forthcoming
essay by the present author will further explore the design
relationship of the precursor pyramid with the finished
structure.

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

Concluding Thoughts

The algorithm presented in this essay does not profess to explain all of the aspects of Egyptian pyramid design, but rather is focused on describing the foremost imperatives driving the design of interior passageways and chamber locations, at least as can be seen in the three pyramids discussed here. An accompanying paper details how these same principles are in evidence in the interior layout of the Khufu Pyramid. A remaining question is then whether the concerns and methods of the proposed algorithm were indeed those of the architects of these pyramids. I submit that the ability of the protocols to simultaneously account for the angle, length, and elevation of each passage, and to do this for not only one but for a series of structures, argues against the accuracy of the results being due to some form of amplified coincidence.

An important factor acknowledged to have been left unresolved thus far is the issue of why there was a shift, or transfer, of a prominent line from one intersection point to another such point in the diagram for each pyramid. The answer, I believe, likely lies in one, or two, not necessarily contradictory directions. Either these shifts were done in an attempt to have the pyramid embody as much of the 'circle and square' convergence information as possible, and/or they were done in an attempt to preserve the secrecy of what was at the time deemed to be privileged knowledge.

If an intentional discontinuity had been placed by the architects into the original sectional plans prior to these plans being given to a pyramid builder, it would have meant that any builder - or any unauthorized person - who gained access to the plans for more than one section of the pyramid would have found it nearly impossible to work the geometry backwards to the causative knowledge. The relatively minor, or secondary, shifts (such as were touched upon in the analysis of the location of the Khafre Pyramid's lower passageway and main chamber) were likely also to have been intentionally worked into the builder's plans for the very same reason.

As mentioned earlier in footnote 1,
there is surviving written evidence which shows that the 8/9ths
correlation was being used by scribes during the Middle Kingdom
period to compute a circle's area in 'square' units. It is assumed,
therefore, that during the even earlier Old Kingdom period the 'area'
squaring of the circle knowledge had already been discovered and was
not of a privileged nature. However, these same surviving texts do
not appear to touch upon the perimeter to circumference squaring of
the circle correlation.^{25}
This latter 'squaring' may in fact have been the knowledge that was
to be kept restricted, if indeed it was secrecy that was the major
concern.

**The Royal Cubit Rod**

A few words need to be said about the
means by which the scribes could have determined the relative lengths
of the various lines, and the slopes of the various angles, that are
seen in the diagrams. A knowledge of trigonometry is helpful nowadays
in computing these factors, but it is likely that the ancient
Egyptians made their determinations through direct measurement. If
the proposed diagrams are drawn carefully and of workable size, then
surprisingly accurate relative lengths can be determined
empirically.^{26}

We know that the royal cubit was
divided into seven 'palms' with each palm containing four 'fingers',
and that each of these resulting twenty-eight 'fingers' could then be
further subdivided into from two to sixteen finer
subdivisions.^{27}
Using a cubit rod that is ruled in this way, the relative base to
height measure of an angle can be taken directly from the diagram.
For instance, in regard to the 27°16' angles seen in the
diagrams, if the royal cubit's 28 'fingers' are each divided into 9
sub-units, then for each horizontal run of 1 full cubit (containing
28 x 9 = 252 sub-units) a vertical rise of 130 of these sub-units can
be found by direct measurement. Alternately, for each vertical rise
of 252 sub-units (= 1 royal cubit), a horizontal run of 489 units
will be found.^{28}
The builders may have been given their construction directives in
terms of such a 'rise and run' format, although in some circumstances
they may have simply been given a premarked template of appropriate
size with which to guide the laying out of a particular
angle.^{29}
The angle given to the builder may also have been a version of the
angle from the diagram that was slightly rounded off to facilitate
construction. In cases where such a rounding off may have been used,
the constructed angle can not then be expected to exactly match the
design angle as it is derived in the parent diagram. Whatever the
methods that were used, the Egyptians were clearly capable of doing
exacting work, and were inventive enough to find ways to implement
that which their designs suggested.

The theory that has been proposed in this essay develops a rationale which successfully determines the interior layout designs of four of the most important pyramids of the Old Kingdom period, showing how these layouts are an integral part of, and in fact derive from, each pyramid's height to base design ratio. Given the consistent level of accuracy of the theory's predictions, and given its ability to explain much of what has previously been notably obscure, it is hoped that a broadened consideration will soon be given to the theory's further predictive capabilities and implications.

1. Problem 50 of the Rhind
Mathematical Papyrus (RMP) finds the area of a circle by squaring
8/9ths of the circle's diameter. Problem 48 of the RMP is generally
interpreted as showing that this 8/9ths understanding was arrived at
diagrammatically. See R. J. Gillings, *Mathematics In the Time of
the Pharaohs* (New York, 1982), 139 - 46. It is worth noting that
Gillings specifically refers to the scribe who wrote the RMP as being
"the first authentic circle-squarer in recorded history" (p. 145). An
interesting relationship exists between the proposed 'S/2' value of
8/9ths of the radius and the proposed 11/14ths value, in that by
squaring 8/9ths one gets a near numerical equivalent to 11/14ths.
This connection between 8/9ths and 11/14ths may very well be the
relationship that Archimedes is referring to in Proposition 2 of his
treatise on the measurement of a circle. See T. L. Heath, *The
Works of Archimedes*, (Cambridge, 1897), 93.

2. See Legon, "The 14:11 Proportion at Meydum", p. 22, and also, Robins and Shute, "Irrational Numbers and Pyramids", p. 44. A "formula" for finding the area of a circle was a relationship much needed by scribes in computing amounts of grain stored in cylindrical containers. The extant papyri do not reveal immediate evidence that the scribes knew how to compute the circumference of a circle. However, there has been suggestion that perhaps Problem 10 of the Moscow Mathematical Papyrus may provide an instance of this capability. See T. E. Peet, "A Problem in Egyptian Geometry", J.E.A. 17, pp. 104-107.

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

4. For the angles quoted here see H.
Mustapha, 'The Surveying of the Bent Pyramid at Dahshur', ASAE 52
(1952), 597. There has been ongoing discussion regarding the slight
variations in some of these angles as detected by survey, and what
these variations might signify. For an inroad into this issue see J.
A. R. Legon, 'The Problem of the Bent Pyramid', GM 130 (1992), 49-56.
As with the Red Pyramid, there is the distinct possibility that the
design of the interior features of the Bent Pyramid was based on one
specific pyramid height to base ratio, while the actual height to
base ratio used in construction was allowed to vary ever so slightly
from the ideal in order to incorporate other desired numeric or
geometric relationships for the pyramid's exterior. This latter point
is made by Legon in his article referenced above (p. 55), and is also
implicit in some of Rossi's presentation in* Architecture and
Mathematics in Ancient Egypt* (see especially pp. 210-221). A
forthcoming essay by the present author will fuirther explore the
details of this issue. Note that Mustapha's finding for the elevation
at which the bend occurs is about 2 m. higher than the 47 m. given by
J. Dorner in 'Form und Ausmasse der Knickpyramide', MDAIK 42 (1986),
56. In addition, Petrie, *A Season in Egypt*, p. 30, reported
this elevation to be at 47**.**25 m.

5. With OS being made equal to 1
unit, OP will be 1**.**4 units long. 11/14ths x 1**.**4 yields
an S/2 of 1**.**1 units for the square having a perimeter equal to
the circumference
of the circle on OP. The actual pyramid height (OA in Figure 5 =
1**.**11 units) multiplied by 8/9ths yields an S/2 of **.**9866
for the square having the same area as the circle on OA. As shown in
the text, once the height of NT in Figure 5 is found, the length of
OT is easily determined and thence so is the height AM. OA follows as
NT+ AM.

6. As previously
stated, Petrie in* A Season in Egypt*, 30, gives 47**.**25 m.
for the elevation of the bend. Here, he also gives his finding for
the length of half of the pyramid's base to be 94**.**8 m. The
diagrammatic derivation of the elevation of the bend can be made
solely via the 1**.**4 pyramid's partculars, and so is fully
independent of any exterior angle that might have been chosen for the
upper part of the pyramid.

7. Petrie, *A Season in Egypt*,
30, gives 11.96 m. for the elevation of the entrance. Mustapha, ASAE
52, 599 gives 11**.**8 m., while Dorner in MDAIK 42, gives
11**.**9 m.

8. Maragioglio and Rinaldi,
*L'Architettura Parte III*, 60. Mustapha, ASAE 52, 599,
interpreted this lower passage to be 79**.**53 m.

9. Petrie, *A Season in Egypt*,
30.

10. Dorner, MDAIK 42, 56, gives 32.76
m; Mustapha, ASAE 52, 599, gives a height of 33**.**3
meters.

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

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

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

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

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

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

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

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

19. Maragioglio
and Rinaldi, *L'Architettura Parte III*, Plate 10, Figure 2. The
dimensions of the Lower Chamber are listed as being 4**.**96 m
east to west and 6**.**3 m north to south.

20. Maragioglio and Rinaldi,
*L'Architettura Parte III*, p. 66. Maragioglio and Rinaldi do
not make explicitly clear, either in the text or in their diagrams,
if it is the axis of the Upper Descending Passage that is in fact
13**.**7 m south of the midline of the side. The published results
of Mustapha (p. 599) and Fakhry (pp. 87 - 94) are also not specific
on this point. I have assumed that it is the axis that is meant, and
the Upper Chamber's analysis correlations appear to bear this out.

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

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

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

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

25. Although I am not aware of
anything in the written record that directly relates
to either
the perimeter mode of 'squaring the circle', or to its associated
11/7ths and 11/14ths relationships, Problem 38 of the Rhind
Mathematical Papyrus is not without interest. This problem
essentially asks the scribe to divide the number one by 22/7. No
mention is made of circles or squares, yet the situation involves
numbers which recall those seen in a perimeter 'squaring' instance.
See Gillings, *Mathematics In the Time of the Pharaohs*,
205.

26. Anyone wishing to prove this fact for themselves should have little difficulty in achieving the required accuracy if the initial radius of the circle in the diagram is made, at minimum, a royal cubit in length.

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

28. The Egyptians referred to angular measure in terms of 'sekeds', with a seked being the number of cubits and/or 'palms' and 'fingers' of horizontal run required by an angle for each vertical rise of one royal cubit. The 27°16' angle, with the stated horizontal run of 489 of the 1/9th units for each rise of 1 royal cubit (= 252 of the 1/9th units), therefore has a seked of 1 royal cubit, 6 palms, 2 1/3rd fingers (which is the same as 1 royal cubit, 6 and 7/12th palms). In a similar manner, the seked for the design angle of the Red Pyramid, with its height to half-base ratio of 17/18, has a seked of 1 royal cubit and 5/12th palms. The design of the royal cubit rod permits its user to choose in each instance a 'finger' subdivision which best suits ease of measurement and numerical manipulation. Interestingly, when each finger is divided into 9 sub-units, thereby giving the royal cubit a total of 252 of these sub-divisions, we find that 8/9ths of 252 is exactly 224, and 11/14ths of 252 is exactly 198. It is possible that this is the subdivision that was the one most often used by the architects both when setting out the initial diagram, and in the subsequent determination of the various seked relationships. See the Computation section for a further discussion regarding the use of sekeds.

29. An issue of ongoing debate is
whether, during construction, slopes were measured in terms of ratios
(such as has been presented in this essay), in terms of sekeds, or as
sekeds that have been 'rounded off' to the nearest finger or palm.
For a discussion of this issue see Roger Herz-Fischler, *The Shape
of the Great Pyramid* (Waterloo, 2000), 34-45; and also see
Corinna Rossi, *Architecture and Mathematics In Ancient Egypt*,
(Cambridge, 2004), 203-214. Rossi includes mention of the use of
wooden triangular templates to guide slope construction. Legon also
discusses this issue in his article "On Pyramid Dimensions and
Proportions".

References

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

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

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

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

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

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

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

**Legon, J. A. R.,** 1989. "The
Design of the Pyramid of Khaefre". GM 110, 27 - 34.

**-----------** 1990. "The 14 to
11 Proportion at Meydum". DE 17, 15 - 22.

**-----------** 1991. "On Pyramid
Dimensions and Proportions". DE 20, 25 - 34. This paper is available
on line at http://www.legon.demon.co.uk/pyrprop/propde.htm

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

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

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

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

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

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

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

**-----------** 1883. *The
Pyramids and Temples of Gizeh*. London. The text of Petrie's
original publication is currently available on line at
http://www.ronaldbirdsall.com/gizeh/index.htm.

**Robins, G. & Shute, C. C.
D.** 1990. *The Rhind Mathematical Papyrus*. New York:
Dover.

**-----------** 1985.
"Mathematical Bases of Ancient Egyptian Architecture and Graphic
Art".

Hist. Math. 12, 107 - 122.

**-----------** 1990. "The 14 to
11 Proportion in Egyptian Architecture". DE 16, 75 - 80.

**-----------** 1990. "Irrational
Numbers and Pyramids". DE 18, 43 - 53. This paper is available on
line at
http://www.hallofmaat.com/modules.php?name=Articles&file=article&sid=39

**Rossi, Corinna.** 2004.
*Architecture and Mathematics In Ancient Egypt.*
Cambridge.

**Scott, N. E.** 1942. 'Egyptian
Cubit Rods', Bulletin of the Metropolitan Museum of Art 1

**Vyse, H. **1842. *Operations
Carried On at the Pyramids of Gizeh in 1837, V. 3*.
London

**Abbreviations:**

MDAIK = Mitteilungen des Deutschen Archaologischen Instituts, Kairo

KMT = KMT: A Modern Journal of Ancient Egypt

GM = Göttinger Miszellen

ASAE = Annales du Service des Antiquités de l'Égypte

Hist. Math. = Historia Mathematica

DE = Discussions in Egyptology

Copyright ©2004 L. Cooper (rc@atara.net) All Rights Reserved.