Copyright Ernest P. Moyer, author
March, 2001
Revised February, 2003
Thoroughly revised September, 2004
Introductory
All four great pyramids have entrance passages on their north faces. In addition, the Bent has a second entrance on the west side high up on the structure, while Giza 2 has a second entrance cut into bedrock outside the base on the north side. The passages are very small; they are roughly one meter square. They cannot be negotiated except by crouching on hands and knees. The small entrances are in sharp contrast to the huge size of the Pyramids.
(Please note that I do not examine other passage properties in this paper. Here I concentrate on the slopes only.)
Except for the bent pyramid west passage all slope downward linearly from the entrance until they reach the level on which they terminate, leading to, or at, a chamber near the center of the respective pyramids. The entrance on the west face of the bent pyramid changes angle of decline part way down.
Refer to the cross section diagrams of the respective pyramids. These diagrams are from the book by I. E. S. Edwards, The Pyramids of Egypt, The Viking Press, New York, 1972. This edition is a reprint in full color of the 1961 edition by Penguin Books. Please note that I do not make the King assignments shown by Edwards.
The angles of the slopes generally vary from approximately 24° to 30°. The angle of the passage cut into bedrock outside the base of Giza 2 differs from this generalization; it slopes at 2l° 40'.
My sources are:
1. W. M. Flinders Petrie, The Pyramids and Temples of Gizeh, Field and Tuer, London, 1883, First Edition.
2. H. Vyse and J. S. Perring, Operations Carried out at the Pyramids of Egypt in 1837, 3 vols, J. Fraser, London, 18401842.
3. A. Fakhry, The Monuments of Sneferu at Dashur, in two Volumes, Government Printing Office, Cairo, 1959. Another version was published as The Pyramids of Sneferu at Dahshur, University of Chicago, 1961.
4. V. Maragioglio and C. Rinaldi, L'Architettura delle Piramidi Menfite, in seven volumes, Rapallo, 1964.
This table summarizes the angles of
the passage slopes of the several entrances. Where the measurements are
given with greater precision I round off to the nearest minute of arc. Since
some of the measurements are less precise the analysis is limited by that
restriction. I shall explain the Ideal angles in following discussion.
Measured Values 
Ideal 


Pyramid 
Passage Angle 
Exterior Angle 
Mathematical Form 
Exterior Angle 
Onehalf Angle 
Giza 1 North 
26° 31' 
51^{o} 51' 
14/11 
51° 50.6' 
25° 55' 
Giza 2 North 
25° 55' 
53° 10' 
4/3 
53° 7.8' 
26° 34' 
Bent West 1 
30° 09' 
54° 31'  7/5 
54° 27.7' 
27° 14' + 2° 57' 
Bent West 2 
24° 17' 
27° 14'  2° 57' 

Bent North 
25° 24' 
43° 21' 
16/17 
43° 15.8' 
26° 34'  1° 26' 
Flat North 
27° 56' 
43° 38' 
20/21 
43° 36.2' 
26° 34' + 1° 26' 
Giza 2 Bed Rock 
21° 40' 
(43° 21') 
21° 38' 
In the discussion below I refer to the first three exterior measurements as steep angles, and the last three as shallow angles. The double angle of the Giza 2 Bed Rock passage is a calculated value, which I show with parenthesis.
No comprehensive explanation for the various slopes has ever been offered, beyond the simple speculation of seked (seqt) ratios. One is led into this by the most famous of the passages, that of Giza1. It has a seked ratio of 1/2. Unfortunately, the other passages are not simple. One must invoke complex seked ratios to explain them. Some idea of this complexity can be seen from the following chart, where I list some ratios near the measured values while others are shown for comparison.
Seked Ratio 
39/100 
2/5 
9/20 
47/100 
12/25 
1/2 
53/100 
29/50 

Geometric Ratio 
.39 
.40 
.45 
.47 
.48 
.50 
.53 
.58 
Degrees 
21° 18.3' 
21° 48.1' 
24° 13.7' 
25° 10.4' 
25° 38.5' 
26° 33.9' 
27° 55.4' 
30° 6.8' 
Over the many centuries a mythology developed that the passages point to the North Star. Because the pyramids are oriented so close to true north, it would seem that the passages would point to the north polar position. This idea is incorrect for two reasons. First, the angles are off as much as six degrees from the true north polar position on the celestial sphere. The angle necessary to point at the north polar position is the same as the latitude at which the pyramids are located, approximately 30° N. Only the one bent passage is near this angle, and it is on the west face.
Second, the North Star is not fixed with time. Due to the precess of the equinoxes, the stars drift past and around the north polar position at a rate of about one degree in seventy years. This means that any star in the North Pole position in 2700 BC has long since drifted to another location, and that Polaris, the present pole star, will drift away also, if the earth continues in its present motion. Since 2700 BC, the earth has moved around the great circle almost 70 degrees. This covers a considerable area of the northern sky because the earth is tilted at an angle of 23° to the plane of its orbit.
Clearly, the architect of the Great Pyramids did not intend the entrances to point to any star. However, location on the north faces pointing near the north polar position draws attention to them. The architect, to attract the attention of later students, probably chose this method.
Examination of the passages shows that they were designed as part of the overall scheme of the geometric properties of the pyramids.
Petrie measured the entrance passage angle of Giza 1 at 26° 31'. This angle is very nearly the halfangle of the 345 exterior design slope of Giza 2 of 53° 8'. The ideal halfangle of the 345 is 26° 34'; the error of the measured angle from the ideal is 3'. If calculated according to the 5400 number of arc minutes in a quarter circle, this error is a mere 0.06%.
Note that the halfangle of a 345 triangle has a slope of 1/2 (26° 33.9'), as I show above.
Within the 3' error:
The angle of the Giza 1 entrance passage slope is onehalf the angle of the exterior slope of Giza 2.
Furthermore, the measured entrance passage angle of Giza 2 is 25° 55'; the ideal halfangle of the Giza 1 exterior slope determined by Petrie at is 25° 55'. Therefore, within measurement error:
The angle of the Giza 2 entrance passage is exactly onehalf the angle of the exterior face slope of Giza I.
Note that the halfangle of the assumed 14/11 (51° 50.6') slope of the Giza I exterior is 25° 55'.
The two pyramids complement one another in their exterior face angles and their entrance passage angles.
This fact is surprising and has never been discussed in any of the literature on the pyramids. No student of the pyramids has ever reported this fact, as far as I am aware.
But this fact has dramatic implications:
This design seems hardly possible as the result of accident.
If intentional the two great Giza pyramids were designed as one project.
I conclude that the two pyramids could not have been designed and built independently of one another.
Both designs had to be known before either was built.
If the four pyramids are tied together by exterior dimensions, and the two Giza structures are tied to one another by their entrance passages, do the entrance passages of the Bent and the Flat also demonstrate mathematical connection with one another, or with the two Giza pyramids?
The entrance on the west side of the Bent begins at an angle of 30° 09'; part way down it changes to 24° 17'. These cannot be half angles of any exterior, since double of the first is 60° 18', and double the second is 48° 34'. The first is more than 5° beyond the greatest exterior steep slope of 7/5. The last is about 3° less than the Giza I exterior steep slope and 5° higher than the Flat exterior shallow slope, falling between the steep angles of three structures and the shallow angles of two structures. This is well beyond the error range of any great pyramid measured face slope.
After brief examination, I found the solution to this query quite easily. The sum of the angles of the two passage slopes at 54°26' is the same as the slope of the exterior of the Bent frustum ideal of 7/5, within less than 2' of angle.
The sum of the two angles of the Bent west passage slopes is the same as the exterior angle of the Bent frustum at 7/5, within measurement error.
Thus I am able to explain the Bent west passage angles. However, the entrance on the north face of the Bent presented another puzzle. It has a measured angle of 25° 24'. Double this angle is 50° 48', at least 1° less than the steep exterior slope of Giza I. Thus it is lower than the lowest steep angle.
In attempt to understand I next examined the entrance of the Flat pyramid slope at 27° 56'. Double this is 55° 52'. This last value is about 1° 24' greater than the steep slope of the Bent frustum. Thus it is higher than the highest steep angle.
To repeat, double the Bent north entrance passage slope is well below the exterior steep slopes of the three pyramids that range between 51.8 degrees and 54.5 degrees, while double the Flat north entrance passage is well above that range.
I next considered the possibility that the north passage angles on both the Flat and the Bent summed to another exterior slope angle: 25° 24' + 27° 56' = 53° 20'. This is 10' greater than the measured exterior slope of Giza 2, and 12' greater than the ideal Giza 2. This is somewhat above the error range of all other proposed solutions to the passage angles described above, but a plausible relationship. (12' on 53° is 12 parts out of 3200, or an angular error less than 0.4%.) Within this error it would appear that:
The sum of the angles of the two north entrances passages at Dahshur is equal to the angle of the exterior slope of Giza 2.
If this is true, we find another profound implication. If the sequence of construction had the Dahshur pyramids built prior to the Giza pyramids:
The builders of the two Dahshur pyramids had to know of the future construction design of Giza 2.
Further:
The two Dahshur pyramids were built with knowledge of the design of one another,
regardless of construction sequence. Otherwise, their complementary passage slope angles could not sum to the Giza 2 exterior slope angle.
The design of all four pyramids had to be known before any one was built.
The exterior designs of the four great pyramids do not demand that they be related to one another. They are just curiously related by their geometric ratios. However, through this intricate design of the passages the architect showed that they were absolutely designed as a common project.
The Giza 2 Bedrock Passage
The only passage of the four great pyramids I have thus far not considered is cut into bedrock outside the base of Giza 2. It could have been added as an afterthought to the design of the four pyramids. The location does not force any conclusions about the manner in which the four are related to one another.
It has a measured angle of 21° 40'. Double this angle is 43° 20'. This value is 4' greater than the slope of the shallow Bent top. It is about 16' less than the exterior shallow slope of the Flat. Given this difference we are led to infer that it is related to the Bent top and not to the Flat. Within error:
The angle of the slope of the Giza 2 outside passage is the halfangle of the exterior slope of the Bent top.
Until now no one could propose a satisfactory explanation for this extra passage dug into the bedrock outside Giza 2. Now we can understand. The regular entrance slope of Giza 2 reflected Giza1; it could not serve to reflect the Dahshur structures. Therefore:
The outside Giza 2 passage was an extra cut to show a relationship to the Dahshur construction project, and specifically to the Bent pyramid.
In this manner the architect tied together the two pyramids at Dahshur with Giza. The sum of the passage angle slopes of the two Dahshur north passages reflect the exterior slope of Giza 2, while the Giza 2 extra cut reflects the Dashur Bent. Hence, Giza 2 serves as the focal point to tie the Giza project to the Dahshur structures, based strictly on their passage slope angles.
The four structures are tied to one another according to the angles of the slopes of the exteriors and the angles of the slopes of the entrance passages, where the latter represents the half angles of the former.
I represent the relationships in the Figure below. The solid lines show an exact half angle. The tail of the arrow originates on the block representing the pyramid containing the passage; the head of the arrow points to the pyramid that it represents in halfangle. The dashed lines represent those passages that sum to an exterior slope, but are not simple half angles.
Note that the entrance passage slopes at Giza directly represent halfangles of one another and of the Dahshur Bent top slope. Stated otherwise:
The angles of the passage slopes at Giza are directly onehalf the angles of the respective companion exterior slopes.
In contrast the angles of the entrance passage slopes at Dahshur are not directly the half angles of exterior slopes but sum to exterior slope 1/2 angles. The angles of the Bent west passage sum to the Bent top angle, while the angles of the two Dahshur north passages sum to the exterior angle of Giza 2.
The angles of the passage slopes at Dahshur are split half angles which sum to the angles of the respective companion exterior slopes.
In this manner the architect intentionally made the passage slopes a unique feature of the respective construction sites: Giza different from Dashur.
Here I have followed the order in which I made the discoveries: Giza 1 and 2, the split angles representing the Bent 7/5 exterior slope, the sum of the two Dashur pyramid north passages, and the angle of the passage outside Giza 2. It would seem that I was being led along a path of discovery. After I recognized the two Giza complimentary structures, it was natural to examine the Dashur pyramids. The sum of the angles representing the Bent frustum on the west side of the structure then led me to examine the two Dashur north passages.
The placement on the west side of the pyramid of the Bent passage representing the Bent frustum seemed rational; it was different from the north passages because the representation was different, a passage reflecting on its own structure. The architect was directing our attention to this difference.
But what was the significance of the split angles at Dashur?
If the two Dashur north passages had been equal to one another, onehalf angles without a split, representing Giza 2, all three passages would have been the same as Giza 1, 26° 31', or thereabout. They would not have been unique. And we might be led to believe that the passages were merely a 1/2 slope, not reflecting exteriors, and not connecting the pyramids to one another through design. Later students would then have seen nothing peculiar to the passages. The architect directed our attention to his design, and thereby made them uniquely different by using split half angles.
Following through on this logic I next came to the question of how he chose his values.
I found the answer by accident. I was using my engineering slide rule to compute the tangents of the several angles, and to determine the departure of the Dahshur north entrance passage angle from the ideal half angles. The difference between the two angles in the Bent west passage is 5° 52' (30° 09'  24° 17'). One half of this angle is 2° 56'. I set the cursor on my slide rule to find the tangent of 27.234°, the Bent frustum theoretical half angle. (I was looking for the ideal halfangle of the frustum slope.) This gave me a tangent reading of 0.512. I then looked for the tangent reading of 2.933°, half the difference of the two angles.
To my surprise I did not need to move the cursor on the slide rule. (The slide rule is laid out with different scales that are multiples of ten to one another.) The value of the tangent of 2° 56' is 0.0512, onetenth of the tangent of the frustum half angle.
In other words, the tangent of the difference between the two slope angles divided by two is 0.1 of the tangent of the frustum half angle. But the frustum half angle is the sum of the two slope angles divided by two.
tan (30° 09'  24° 17')/2 = 0.1 X tan (30° 09' + 24° 17')/2
tan (5° 52')/2 = 0.1 X tan (54° 26')/2
tan (2° 56') = 0.1 X tan (27° 13')
Within error I found that 0.0512 = 0.1 X 0.514, or thereabouts, not easily distinguishable on a slide rule.
This procedure may be stated as a mathematical expression:
tan ([A  B]/2) = K tan ([A + B]/2)
where A and B represent the two angles, and K is the ratio.
Unfortunately, this cannot be a general formula; it can only describe specific cases.
To illustrate, consider that we maintain the difference the same at (5° 52')/2. Then make the angles considerably smaller, for example 20° 52' – 15° = 5° 52'. The sum of these two angles is 35° 52'. Divided by 2 this becomes 17° 56'. The tangent of 17° 56' is 0.324. Clearly this last value is not easily divisible by the tangent of the difference at 0.0512.
The question then becomes how the architect knew about a relationship that would produce a ratio of 0.1.
We can rewrite the above formula to solve for K.
I opened my Plane Trigonometry textbook to find this relationship and was dumbfounded to discover that this formula is based on a modern mathematical rule called the Law of Tangents. A and B are the angles of an oblique triangle. The only difference is that K is defined by
where a and b are the sides of the oblique triangle.
I could now calculate the relationship of a to b by making K = 0.1
This reduces to a = (1.1/0.9) b. (1.1/0.9 = 1.22222 . . .)
In other words, the K value of 0.1 is good for all triangles where a and b obey this relationship. Since a and b can be an infinite number, there are an infinite number of a and b to make K values of 0.1.
But we have no triangle to show sides a and b. I could suggest that the a and b dimensions are simulated by the length of the respective portions of the passage, from the exterior to the bend in the passage, and from the bend to where it terminates at the inner chamber. However, these measurements are unknown to me, and I question whether anyone has ever attempted to determine them. The one would be about 20% longer than the other.
However, this lack does not void the relationship described above.
I felt that the value of K = 0.1 was not a coincidence, that it was chosen by the architect to show his finesse with tangents.
I next examined the relationship of the two north passages at Dashur. I tabulate the results below.
A (Bent) 
B (Flat) 
Sum 
Difference 
Tan ½ Sum 
Tan ½ Difference 
Ratio 


Measured 
25° 24' 
27° 56' 
53° 20' 
2° 32' 
0.502 
0.0221 
0.044 
Ideal 
24° 48' 
27° 40' 
52° 28' 
2° 52' 
0.493 
0.0250 
0.051 
The ideal values show that the ratio of the two tangents is now 0.051, or, within error, just 1/2 of the value of K obtained for the Bent west passage.
Now I was certain that the K values were not accidental.
Tabulating different split angles and K values shows how the angles vary. I offer this in order for the reader to grasp the practical constraints that faced the architect.

K 

0.05 
0.1 
0.2 
0.5 

(A + B) 
2 (A + B) 
tan 
Split Angle (A  B) 

20° 
10° 
0.1763 
1.01° 
2.02° 
4.04° 
10.07° 
30° 
15° 
0.2679 
1.53° 
3.07° 
6.13° 
15.26° 
40° 
20° 
0.3640 
2.09° 
4.17° 
8.33° 
20.67° 
50° 
25° 
0.4663 
2.67° 
5.34° 
10.65° 
26.25° 
53° 8' 
26° 34' 
0.5000 
2.88° 
5.75° 
11.47° 
28.19° 
54° 28' 
27° 14' 
0.5147 
2.95° 
5.89° 
11.74° 
28.84° 
60° 
30° 
0.5774 
3.29° 
6.61° 
13.17° 
32.21° 
The architect had to choose angles that would be realistic. He would not choose a K value of 0.5 because the split angles become too great. We can see that even for a K value of 0.2 the split angles of 11.47° and 11.74° at the design slopes of the pyramids are too great. This would require passage slopes on the order of 30° and 20°. These values would interfere with our recognition of the simple half angles. Therefore, he went to lower K values.
He chose a simple K value of 0.1 because it would become apparent to someone attempting to investigate the reasons for the slopes. This is exactly what I did. If he had chosen a K value of some unusual number, such as 0.18, the value would not have been obvious. I get the uncanny feeling that he knew I would be using a slide rule with multiple decimal scales. I would not have discovered the relationship with a trigonometric table. Or, stated more realistically, he knew that the discovery of his design would depend upon someone familiar with mathematical tools similar to slide rules.
Then, in order to separate the 53° 10' of the Giza 2 exterior slope from the 54° 31' of the Bent top slope he used 0.1 K value for the first, and 0.05 K value for the second. This gave him 5.75° and 2.95° for the differences in the split half angles. If he had used 0.1 K value for both they would have been too close to one another. Thus he was constrained by what was practical, and how a future investigator would discern his design intent.
But there was another limitation. The more shallow he designed the angles the more difficult it became to discern them because they were placed in Trigonometric Tables that showed little increment between angles. For example, the tangent increment between 0.049 and 0.051 runs from 2° 48' to 2° 56'.
Now I knew the reason for the split halfangles. He was telling us about his knowledge of the Law of Tangents, or his equivalent.
None of the foregoing analysis makes rationale sense unless we understand the architect as using angular measure, and not seked ratios. He just was not that simple minded. He knew and understood angular measure.
Curiously, this knowledge has not come down to us in the Egyptian mathematical treatises of later centuries.
Following is a chart showing how the various angles of the four Great Pyramids are related to one another. The distances on this chart are to scale. This chart will permit the reader to grasp how distant the halfangles are to the exterior angles, and how they could be developed by the architect to portray the halfangle relationship  all in massive buildings of stone.
The error in the various exterior and passage slopes is indicated on the second part of the graph, where I have shown the absolute departure, not according to direction. It can be seen that the passages appear to follow a normal distribution of error up to 6 minutes of arc, except for the Bent north passage. Given this small amount of error we can be confident that the halfangles have no other solutions.
More exactly, the error in the Bent north passage may be due to error in the measure of the slope, the construction, or, more likely, settling of the structure. As I stated in a companion paper, the Bent has suffered settling sometime in the past. This is evident on both the north and west passages where glaring fractures can be seen. The genius of the construction can be discerned when we note that the Best west passage has suffered almost no significant alteration to the angles, but settling has caused the north passage to increase in slope about 24 cubits from the entrance to a reported angle of 28° 38' . Perring reported a shift in passage angle to 26+°. Edwards reported 25° 24'. This is where the large discrepancy exists.
But we have the puzzle of why the architect chose to display half angles in the first place. What was his motive?
This depends on the peculiar relationship between half angles and Pythagorean triangles.
Before proceeding with that relationship I here show graphically how the half angles are related to the angles of the pyramids.
I shall then return to develop those relationships in another paper.