5. THE QUEEN CHAMBER'S

GEOMETRIC DESIGN

If these concepts I expressed here are true, and I certainly believe they are, these constitutes evidence that the ancient Egyptian engineers knew more about geometry than they are credited by tradition and found evidence. The design of the Queen Chamber is a perfect example that they knew how to create the Golden Section, although they could have use another name to describe it. We have to remember that this design corresponds to the Old Kingdom time, the pyramid’s era. If they designed this small chamber using this process, in the same way, they could have designed any other structure, like the Great Pyramid.

I used the same concepts and principles to design this Queen Chamber than to design the Great Pyramid’s structure. Finally, I found equal positive results.

Queen Chamber's Original plans

The following is my idea about the procedure to trace the geometric configuration of the Queen's Chamber.

As I sated, the Queen Chamber's geometry also supports the Phi concept design used for the Great Pyramid. It follows a geometrical pattern using the figure of the circle and following the Golden section, as used for the Great Pyramid structure.

Therefore, using the following procedure, the general geometrical configuration of the eastern wall part of the Queen Chamber can be easily established. As in the Great Pyramid design, the design drawing can be set without setting the measurements involved. After the designed drawing if finished, the value of the radius of the circle is set, and then, all measurements can be calculated, exactly as was done for the Great Pyramid. As said, the design can be done without numbers, calculations, angles, etc. At the end of the design, just plug in the radius, or diameter of the circle, and calculate all the angles and dimensions from the design drawing.

If the height of the Queen Chamber is set as 12 Egyptian cubits, assuming and Egyptian cubit is equal to so many inches, its height will vary accordingly to the equivalency used for the cubits. What I meant is, if the height is 12 cubits and somebody measures it as 245.1 inches (as Petrie did), the cubit equivalency will be = 245.1 / 12 = 20.425 inches. However, if another investigator measures 247.367 inches, the cubit will be = 20.6139 inches. As a matter of fact, this equivalency, shows and interesting relation, if the cubit is set as 20.61387436 inches, which is a more exact number, in relation to the 20.6139 inches equivalency, then you have:

(20.61387436 inches / 12) / 3.2808 = 0.523598776 meters.

This means that 1 cubit = 20.61387436 inches = 0.523598776 meters

If 1 cubit = 0.523598776 meters, see what happens with 6 cubits:

6 cubits = 6 (0.523598776) = exactly 3.141592654 meters = (π) meters

Surprised! 6 cubits = (π) meters

Therefore: 1 Egyptian cubit = (π / 6) meters

There is more... 12 cubits is equivalent to = 2 π meters = 6.2831853 meters = 20.613874 ft = 247.366492 in.

I want to make clear; that I am not saying that the Egyptians knew about meters or feet as unit of measurement. This could have happened by a coincidence! However, it is interesting, No?

To change from cubits to meters, it would be easier to multiply the number of cubits by (π / 6). For example, the sides of the GP = 440 cubits, then, 440 (Pi / 6) = 230.38346 m.

The pyramid’s height is = 280 cu, it follows that 280 (π / 6) = 146.6076 m.

However, my geometric design, which follows, establishes what I called a generic design. You can use as the Queen Chamber’s height any number that you prefer, in any unit of measurement. For example, set the radius, or diameter of the circle, to read the Queen Chamber’s height of 12 cu. The radius would be 6 cu.  Since the drawing is completely proportional to the radius, or diameter, you can calculate any other dimensions, in the appropriate unit of measurements you set for the radius. For example, in the design you don’t have to set the width of the chamber; the drawing sets it for you. You don’t have to set the wall’s height, or the angles. The drawing set them for you. This is the convenience of a circle design following a geometric process. You don’t need to input any other measurement than the radius, or diameter. This is the same procedure I used for the Antechamber, the granite leaf stone, and the "boss" designs.

I already pointed out that 6 cubits (the radius) could be equivalent to π meters. Therefore, the radius which composes the Queen Chamber’s design, would be exactly equivalent to 3.141592654 meters = 10.306937 ft. The diameter = (2) (10.306937) = 20.613874 ft. = 247.366492 in.  This is highly interesting, if the result of a coincidence, surely it is a good one!

Well, after all of these things have been cleared, we can continue the examination of my method for the Queen Chamber.

Queen Chamber’s Design

My geometric procedure to design the Queen Chamber is related to the geometry involved in the eastern wall and its niche. No religious concepts, or other considerations concerning their builder’s behavior are implied. I consider that this geometric solution could stand as a complete explanation for the design geometry applied to the Queen Chamber’s east wall and its niche location, where the φ function and its knowledge is clearly shown by the Egyptians. As a reference, the following figure is a drawing plan of the actual Queen Chamber.

 

 

 My research firmly suggests that the designs of the eastern wall and the niche were done at the same time. They were not separated one from the other, nor the dimension were randomly chosen. If they were taken at random, they will not fit into my design. As you will see, I applied a logic geometric process-involving Phi, to set the chamber’s dimensions and proportions. The location of the niche clearly responds to these measurements, and at the same time shows us the knowledge the designer have about the Golden Section (φ). It’s cross section show the dimensions set by the designers to establish the area for a statue, or altar, called a niche. Nonetheless, the vertical and horizontal measurements for each rectangular section that compose the niche’s front section, follows a mathematical arrangement. The Egyptian engineers set up here a scale of their cubit unit of measurement, for all of us to see.

Procedure to design the Queen Chamber

Although I showed the distances in my design, the drawing is dimensionless until finished and set the Queen Chamber’s height = 12 cu = 2 (Pi) meters = 20.61387436 feet = 247.3664923 inches.

1. Set a circle with diameter (AH), as shown in the figure representing the Queen’s Chamber vertical distance from the floor to the center of the ceiling (12 cu). Draw its vertical and horizontal diameters.

 


 


2. With center at A, trace another circle, of equal radius (6 cu). Point A represents the top of the chamber’s ceiling).

3. Draw a horizontal line to join points B and C, which intersect the circumference of both circles. The line BC represents the width of the Chamber.

4. From the intersection points B and C, draw incline lines to point A (center of the ceiling).

 5. From B and C, set vertical lines to intersect the floor-line showed as points G, and I. The vertical distances CG and BI represent the height projection of the north and south walls of the Chamber.

6. Draw a line from D to F to join the intersection points between the lower circumference and the projected chamber’s wall lines. This horizontal line will become the shafts floors. The top section elevation of the lower niche apparently represents the ceiling of the shafts.

7. The line DF shows the projection of the lower section of the ventilation-shafts, and its vertical distance from the floor-line. The elevation of Line DF also corresponds to the height of the entrance to the chamber.

The geometric configuration shown in the drawing is proportional in its parts. Therefore, you can use any unit of measurement as the diameter of the circle, and determine the other dimensions.

The formulas I developed to determine the dimensions of the Queen’s Chamber, in accordance with the drawing geometry are as follows:

Chamber’s height = D = 12 cu

Radius = D / 2 = 6 cu

Height of the walls =(H) = (3/4)(D) = 9 cu

Width of the Chamber = (W) = (Ö3 / 2)(D)

= (Ö3)(R) = 10.39230 cu

The tangent of ceiling’s angle = 3 / (10.39230 / 2) = (.05773502).

The corresponding angle of the inclined ceiling = 30°.

 





As already calculated, the width of the chamber is 10.392 cu. Here comes an interesting part. Apparently, the ancient engineers, using a very intelligent procedure, decided to use exactly 10 cubits as the actual distance between the north and south walls. Therefore, they provided 0.20 cu to each side of the north and south walls, to use a 10 cu distance between the walls. This can be verified with the other measurements we will obtain.

 By doing this type of design, the upper section of the walls and the ends of the inclined ceiling is slightly changed from its position forming a triangle with a measurement of 0.20 cu horizontal, and 0.113 cu vertical. These displacements are also seen at the same locations in the Queen’s Chamber, as shown in the original sketch plan.

Dimensions of the designed plan

After setting the dimensions of the Queen’s Chamber, we can set up the location of the niche. The niche was designed following the cubit measure as its basis. Apparently, the ancient Egyptian engineers left a cubit scale mark here to be read from future colleagues. In the top rectangle of the niche, its horizontal measurement is equivalent to one cubit. The horizontal increments for the rectangle’s widths are 1, 1.5, 2, 2.5 and 3 cubits. Therefore, each next length lower section, the width increases by 1/2 cubit. The length of all the corbels was set as 1/4 of a cubit.


 

 

 

  

 

 

 

 

 

 

 

Vertical section:

The height of the first 3 vertical section of the niche adds up to 4 cubits. Each section would be equivalent to (4/3) of a cubit. The 4th section represents (5/3) of a cubit, and the 5th section that touches the chamber’s floor is equivalent to 3.333 cubits.

Dimensions of the Niche:

Horizontal sections:


1st width =     (1.0) cubit =       0.5236 m =
π / 6 m =     20.6138"


2nd width =    (1.5) cubits =     0.7854 m =
π / 4 m =     30.9207"

3rd width =     (2.0) cubits =     1.0472 m = π / 3 m =     41.2276"

4th width =     (2.5) cubits =     1.3090 m =
π Pi / 12 m=  51.5345"

5th width =     (3.0) cubits =     1.5708 m =
π / 2 m =     61.8414"

Vertical Sections:

1st height     =      (4 / 3) cubits =   0.6981 =     2 π / 9 =      27.4852"

2nd height    =     (4 / 3) cubits =   0.6981 =     2 π / 9 =      27.4852"

3rd height     =      (4 / 3) cubits =   0.6981 =     2 π / 9 =      27.4852"

4th height     =      (5 / 3) cubits =   0.8727 =     5 π /18 =     34.3565"

5th height     =      3.333 cubits =    1.7452 =     5 π / 9 =      68.7060"

 

 

THE NICHE AT THE QUEEN'S CHAMBER


Note: All the corbels sections have a horizontal length of (1/4) cubits (= 5").


This section is important and apparently denotes the knowledge of the ancient Egyptian engineers about what we called the golden proportion, Golden Section, Phi, or the function of 1.618. This seems like if the ancient engineers desired to expose to us their knowledge of this function. They located the niche in a position that clearly will identify the use of the Phi function. Many scholars and Egyptologists are in disagreement with the use of this function of Phi, although it is extremely easy to work with it. It comes more easily from the triangle with side’s 1: 2, than the usual (phi) rectangle. The sides of the right triangle are in the ratio 1: 2, and the diagonal represents the square root of 5. By definition, the function of Phi is equivalent to (
Ö5 +1) / 2. If you draw a right triangle with side’s 1: 2, we get:

The hypotenuse (Ö5), plus the length of the short side (1) divided by the length of the long side (2), is equal to the Golden Number (= φ).

The hypotenuse (Ö5), less the length of the short side (1), divided by the length of the long side (2), is equal to the inverse of the Golden Number (= 1 / φ).

The hypotenuse (Ö5), plus the length of the long side (2), divided by the short side (1), is equal to the Golden Number cubed (= φ³). Phi cubed is a value very important in the design of the GP, as well in other pyramids. It might even come from the mastabas time, since it generates the 76 degrees angle that was used for the mastabas sides.

The hypotenuse (Ö5), less the length of the long side (2), divided by the length of the short side (1), is equal to the inverse of the Golden Number cubed (1 / φ³).

The sum of the three sides of the triangle (
Ö5 +1 + 2) is equivalent to the Golden number cubed, plus one (= φ³ + 1).

As stated, and as a computer, this triangle will provide us the Phi function formula. Wherever you find this triangle, you will find the value of Phi embedded, like in the Great Pyramid’s dimensions. For example: if you see the f³ symbol, don’t panic! It is the same as Ö5, plus 2. Why the scholars had considered this function “off-limits” from the Egyptian works? Really, I don’t know, there are no reasons to dismiss the f function from the Egyptian history. They say that there is no evidence of its use? What about the numerous rectangles and triangles with sides in the ratio 1:2, used in their monuments? These triangles and rectangles are f generators.

After this short explanation of the f proportion, no matter what name the Egyptians used to call it, we will continue with the niche’s design. The ancient engineer wanted to locate the center of the niche exactly at a φ distance from the northern wall. This is something that has preoccupied the Egyptologists forever. Why the center of the niche was not centered in the eastern wall? I gave a reasonable answer. Because they knew about the golden section and possibly think it has some hidden power, not disclosed. Anyway, it was decided to locate this φ point, measuring from the northern wall. Since the entrance to the Queen’s Chamber is located at the northern wall, probably it was better to set up the niche away from the entrance to the chamber.

 This means that the length of the east wall had to be divided at the point where it creates a ratio equal to φ (= 1.618). This could be done by calculations or by geometry. By calculations, the result is (10.00 cubits / 1.618) = 6.18 cu from the northern wall. With this distance measured from the north wall to the south wall, the builders establish the mark of the center of the niche.  At exactly this point, they raised a vertical line to establish the niche’s center. From this center, they traced the complete niche’s configuration in the wall, and then, excavated the niche. This simple solution answers the design method of the eastern wall.

See in the figure the corresponding distances following this geometric design. As you will find out, the dimensions agree with those that had been measured in the Queen’s Chamber.

How to set, geometrically, the φ function

in the east wall as described.

The geometric method is also simple. At floor elevation, set a nail (1) at point A, close to the northern wall and another (2) at point C, at the southern wall following a level chord. The distance would be 10 cu. From the southern side nail measure vertically 5 cu, toward the ceiling, and place another nail (3) at point B. You placed 3 nails (A,C,B) to establish a triangle with sides 10 cu, 5 cu and the diagonal from A to B. The triangle has the side proportions of (1:2), that are 5 cu to 10 cu. Using a chord set at nail 3 to nail 2 distance (B-C), trace and arc to intersect the diagonal chord and mark point F at the location. Now, set the chord using the distance from nail 1 to the previous diagonal mark (A-F), trace and arc to intercept the base line (A-C), and mark point D. Point D set the location of φ from the northern wall. As seen, three nails and a chord, is all the necessary equipment to set the φ point at the eastern wall.

 

 

 

 



The figure shows the final measurements

 Calculated for the Queen Chamber’s design.

Dimensions:

Height of the chamber: = 12.00 cu

Width of the chamber: = 10.00 cu

Center of the niche from the north wall: = 6.180 cu

Elevation of the shaft’s bottom surface

from the chamber’s floor: = 3.00 cu

Height of the entrance passage: = 3.00 cu

Distance between the pyramid’s axis and the

Niche’s axis = 1.187 m = 24.32"






All shown dimensions were calculated from the designed plan. As it is known, all construction work will have acceptable tolerances. It is impossible that the builders construct exactly each measurement in the plans. However, the dimensions should be as closed as possible to the plans, with a minimum tolerance. To prescribe tolerances for the dimensions is wise and to accept that humans built the construction.

 

THE GEOMETRY OF THE QUEEN'S CHAMBER

 If the value of D is set to any number with a unit of measurement, the other sections of the chamber and the niche will be automatically set at the same time, due to its geometrical configuration.

          After setting the dimensions of the Queen’s Chamber, we can set up   the location of the niche for the statue. The niche was design following the cubit measure as its basis. The top of the niche is one cubit length in its width. The length of all the corbels was set as 1/4 of a cubit. Therefore, each next length section for the width increases by 1/2 cubit. The ratios for each length of the squares formed are 1, 1.5, 2, 2.5 and 3 cubits.

           The height of the first 3 vertical section of the niche adds up to 4 cubits. Each section would be equivalent to (4/3) of a cubit. The 4th vertical section represents (5/3) of a cubit, and the 5th section that touches the chamber’s floor is equivalent to 3.333 cubits. Most probable, it was set as 3 cubits plus 0.333 cubits for the foundation of the base of the statue.

           To locate the niche in the east wall, its center was placed using the golden section, no matter what name the Egyptians used to call it. It was possible to locate this point measuring from the south wall to the North, or from the north to the south wall. Since the entrance to the Queen’s Chamber is located at the north wall, it would be better to set it up starting from the north wall, to avoid people’s traffic.

           This means that the length of the east wall have to be divided at the point that creates a ratio equal to φ (=1.618) between the two divided sections, measured from the north to the south wall. Therefore, the midpoint of the top-width of the niche would be located in the intersection between the vertical alignment of this (φ) point location, and the horizontal line which intersect the two circles (the wall’s height). 

     Characteristics of the Niche

 Setting the center of the niche at φ location.

 

      Corners of the ceiling and walls, at the north and south wall.

 Few people know about its existence.

This is very important. The cubit measurements in the niche, which I showed, are the result of an extensive analysis I did of these measurements, as measured by the investigators. Probably nobody noted before the mathematical relation that exists between them, and this is the first time you notice it.


The niche’s measurement measured by Pochan, shows the vertical

measurements’ of each niche’s section as:


1. 0.71 m 2. 0.71 m

3. 0.71 m

4. 0.84 m

5. 1.70 m


If you analyze this numbers in meters, considering a cubit = 0.5236 meters

1. = 0.71 m = 4/3 cu = 1.33 (0.5236) = 0.70 m
2. = 0.71 m = 4/3 cu = 1.33 (0.5236) = 0.70 m
3. = 0.71 m = 4/3 cu = 1.33 (0.5236) = 0.70 m
4. = 0.84 m = 5/3 cu = 1.66 (0.5236) = 0.87 m
5. = 1.70 m = 10/3 cu  = (3.33cu)(0.5236) = 1.74 m

This shows how the builder could use thirds of a cubit for the vertical alignment of the niche.

The measurements are there; I just figure them out from the documents, and expressed them in parts of a cubit.

I did the same thing with the horizontal sections. Pochan showed the vertical section, Petrie the horizontal.

Petrie says ” The general form of the niche was a recess 41 inches (2 cubits) deep back, 62 inches (3 cubit) wide at the base, and diminishing its width by four successive overlapping of the sides (at each wall course) each of 1/4 cubit wide, until at 156 inches high, it was only 20 (1 cu) thus of the 3 cubit width of the base one cubit was absorb on each side by the overlapping, leaving one cubit width at the top. This cubit is the regular cubit of 20.6 inches.

Section at the base = 3.00 cu = 3.00 cu =(3.00) cu

Section 4 = 3.00 cu less (0.25 +0.25)= 2.50 cu = (3/2) cu

Section 3 = 2.50 less (0.25 +0.25) = 2.00 cu = (2.00) cu

Section 2 = 2.00 less (0.25 +0.25) = 1.50 cu = (3/2) cu


Section 1 = 1.50 - (0.25 +0.25) = 1.00 cu = (1.00) cu

Here you have the results of my complete analysis of what the eastern wall and the niche’s show. The measurements could be compared with other measured by other investigators. The results will be the same.

I understand that all these information is contrary to what it had been learn and teach about Egyptian designs and their theological aspects. I know people are not going to accept it initially, but this could explain what religions don’t. I am satisfied with my answers, but as I state before, any one can do the process by them.

I used here the same principles, I used for the Great Pyramid’s design, and other pyramids, with equal positive results. All designs are based in the figure of a circle, using determined geometric configuration, as in this case.

I hope not all doors I am knocking at, result as the Egyptian “false doors”.

 

 

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