Criteria for Pyramid Designs
The pyramids were structures of much importance and meaning for the ancient Egyptians. The pyramid designers must have developed a criteria and geometrical methods to produce their designs and constructions. However, if these geometrical procedures and methods ever existed, they have been forgotten.
Nevertheless, the important dimensions to build a regular pyramid with square base are the height (h), the measurements of the sides of the square base (b), and the angle of elevation or slope of the faces (q). The ratio between the pyramid’s height and half the measurement of the length of the base (b / 2), establishes the value of the tangent (tan q) of the angle of elevation of the faces. The tangent of the slope angle of the faces can be defined as:
tan q = h / (b / 2)
It can be noticed in the formula, that the numerical value of the height and the length of the sides with their unit of measurement must be known, in order to determine the numeric value corresponding to the function of the tangent. Having only the slope angle available will not permit the building of a defined pyramid. Additionally, it is needed either the height, or the length of the sides; with the tangent of the angle, the other unit will be determined.
I do not believe that this general procedure to design the pyramids offers many alternatives to the Egyptians, to express their possible standards of value for the pyramid’s construction. In addition, this procedure is not very helpful in the construction phase.
I will present my idea about the geometrical design process and different standards of value, used by the Egyptian designers to create their pyramid designs. My idea includes two phases. The first is to select the type of geometrical configuration to be used for the construction from an infinite number of possible configurations.
This phase does not specify quantities or measurement units in the design. It is related only to the type of geometric configuration. For example, two of them would be: when the square is inscribed in a circle, and when the circle is inscribed in a square. Others could be the configuration of the triangle with sides in proportion 3:4:5, or a configuration having a special angle. It can be also the ratio of the area of the circle to the area of the square, the configuration created by using the Golden Number, and many others.
In the second phase, the dimensions of the pyramid will be determined and set. This phase will be related to numbers and units of measurements set by the designers to define and build the pyramid. For example, the height of the pyramid could represent a number of significant values for the pharaoh, like the number of years from an important event, the number of conquered countries, or maybe a number with a personal significance. The units of measurements will correspond to those used by the designers at those times.
Returning to the first phase, if the formula tanq = h / (b / 2) is evaluated, it can be noted that if the height (h) represents the radius (R) of a circle, the value of 2 (h) will represent its diameter (D). In this case, the formula can be simplified as tan q = (2 h) / b = tan q = (D / b). Therefore, when using the circumference of a circle for the pyramid’s design, the formula tan q = (D / b) represents the slope of the faces of the pyramid, where the radius represents the pyramid’s height and (b) the base length, as shown in figure 143.
The horizontal diameter (AB) will represent the surface of the terrain where the pyramid will be built, and the vertical diameter (QK), the vertical axis of the pyramid. These standards will apply to the geometrical design of any pyramid.
The lines traced from point Q (top of the circle), to any point P in the circumference of the circle from point A to point K, will cut the horizontal radius AO, at any point, which will be defined as point H. In addition, point H will identify the location of the corner of the base of the pyramid.
The distance from point H to the center O of the circle represents half the base’s length. Since HO represents half the length of the sides, and this distance is equivalent to the distance from point O to point T, the distance from point H to point T, represents the length of the base of the pyramid.
To project the square base of the pyramid to the vertical sectional view of the pyramid, from the center O of the circle, trace another circumference with radius equivalent to HO, or OT. Designate point S and U at the intersection points of the circumference of this circle with the vertical axis. The lines that define the four corners of the base will pass through the four already defined points (H, T, S and U).
To define the corners of the square, from point H, and HO as radius, trace an arc as shown in figure 143. With an equal radius, repeat the procedure at points S, T, and U. The intersection points between the arcs, identified as I, II, III and IV in the figure, define the four corners of the square. Join the points with straight lines to trace the projection of the horizontal square of the base, as seen in the vertical sectional view of the pyramid.
The triangle defined between the points Q, H, and T, represents the vertical sectional view of the pyramid. The square formed between points I, II, III, and IV defines the projection of the base of the pyramid. Note that an unlimited number of lines can be traced from point Q, crossing the radius AO and the extension of its line, to intercept the circumference of the circle. These permit to define an infinite number of pyramids.
Figure 144 shows four examples of these lines traced from point Q, crossing the radius AO, and intercepting the circumference of the circle. These lines are QA, QP, QP’ y QP". The lines intersect or cross the radius AO at the points marked A, H, H’ y H" in the circle’s circumference.
Figure 145 shows the four cross sectional view of the pyramid’s configurations created with the lines as traced in figure 144. The triangles QAB, QHT, QH’ T ‘ and QH"T" identify the four configurations. It can be seen that the lines originated in point Q, when move from point A to point K in the circumference, they reduce the size of the base of the pyramid until it vanishes when reach K.
Nevertheless, the line originating in point Q could also go beyond the limits of the horizontal radius. The line can be traced to intercept the horizontal extension of the radius, as shown in figure 146 with the line QM. Note that when the radius AO is equivalent to half the length of the sides of the base, that is, R = b / 2, the circle is inscribed in the square of the base. The slope of the faces would be (D / b) = (2 / 2) = 1. This tangent corresponds to a 45º angle.
It can be observed that when the value of half of the side’s length of the base of the pyramid is larger than the radius, (MO is larger than AO), the square of the base (shown between points 1, 2, 3, and 4) is located outside the circumference of the circle. In these cases, the slope angle of the faces is less than 45º. This type of configuration, is I found in the north pyramid at Dahshur, also known as the Red Pyramid. In addition, it is also found in the upper section of the Bent Pyramid, also located in Dahshur (see Chapter 12).
Another interesting example occurs when the square of the base is inscribed by the circle, as shown in figure 147. This appears to be the configuration used to design the lower section of the Bent Pyramid.
In figure 147, if the Pythagorean theorem is applied to the right triangle between points O, II and T, the radius (R = hypotenuse) is equivalent to the square root of (b / 2)² plus (b / 2)². Therefore, R = (b / Ö2). Since the slope of the faces is equivalent to tan q = (D / b), the tangent of the slope angle is: tan q = 2R / b = 2 (b / Ö2) / b = 2 / Ö2 = Ö2. The value of Ö2 is equal to 1.414213562, and the angle having this tangent function is 54° 44' 8.2" (54.73561°). After the geometric design is finished, two simple crossed and perpendicular lines, could represent the pyramid configuration, one vertical and one horizontal, as shown in figure 148. The vertical line represents the diameter of the circle, while the horizontal line represents the sides of the base. The intersection of the lines represents the center of the circle.
If another circle, or ellipse as shown in figure 149, is drawn in the upper section, the configuration will look very similar to the famous Egyptian Cross. It could be possible that this configuration originated such an important religious symbol. The complete cross would represent the geometrical configuration of the pyramid, and its dimensions, the proportional measurements of the structure.
This pyramid identification system, although speculative, could be further simplified and be represented by only two perpendicular lines. This is shown by the configuration of a vertical line crossing a horizontal line at point O, as shown in figures 150 and 151. The vertical line QO represents the circle’s radius and the horizontal (HO) half of the side length.
To build the pyramid, as stated, is necessary to establish a numerical value and a unit of measurement for its height and the side lengths. These values will establish the slope of the faces. The geometrical configuration does not show the units of measurement for the pyramid. It is the designer’s job, in the second phase, to determine the dimensions he or the pharaoh desires for the final plans and construction of the pyramid.
Imagine that a pharaoh wants to build a pyramid, based on the configuration of the right triangle, having the proportion of 3, 4, and 5, in its sides. This is a very well known triangle and is generally used to demonstrate the Pythagorean theorem. Figure 152 shows a sample of this triangle. Its height (QO) has four units, and its base (HO), 3 units.
Since OQ represents the height, and HO, half the base length, the tangent of the angle would be R / (b / 2) = 4 / 3 = 1.33333. The angle whose tangent function is 1.33333 corresponds to 53° 7' 48.4'’ (53.13010°). Figure 153 shows the geometrical configuration created with this triangle.
In the second phase, a value and a unit of measurement will be selected and set for the height of the pyramid. The designers must determine both. In my theory about the Great Pyramid configuration, it appears that its height was a measurement equivalent to the product of 153 multiplied byp, expressed in feet units. In this occasion, just as a guess, let assumed a height equivalent to the product of 150 multiplied by p, that is, 471.24 also expressed in feet units. Now, the height of the pyramid divided by the tangent of the slope of the faces is equal to half the length of the base (b / 2). The tangent of the slope is (D / b) = 8 / 6 = 1.33333.
Therefore, half the base of the pyramid would be equal to 471.24 / 1.3333 = 353.43', consequently, the length of the base (b) = 2 (353.43) = 706.86 feet. As a summary, the height of the pyramid (R) is 471.24', the base length (b) is 706.86', and the slope angle (q) of the faces is 53° 7' 48.4'’.
The dimensions and angles of this example represent, very closely, those given in the references for Chephren’s Pyramid, the second of the three great pyramids [Ref. #43, Petrie, Flinders W. M., page 32, height = 472' (more or less 13 inches). The slope of the face is 53° 10', more or less 4']. Due to the approximation between the measurements of Chephren Pyramid and the model created with the configuration of the triangle with sides in proportion 3:4:5, many people attribute the configuration to Chephren Pyramid design. In Chapter 12 is presented a detailed analysis of another geometrical configuration that satisfies the measurements and characteristics of Chephren Pyramid. Although the results of the configuration seem to be very similar to those presented here, the configuration is not the same and gives better results for the pyramid.
The last example of the different configurations will be that shown in figure 154, corresponding to the Great Pyramid, in accordance with my theory. Among all geometric configurations available to design a pyramid, there is no doubt that this is the best proportioned, the most harmonic, the most beautiful, and the one having so many unique characteristics. This geometrical configuration is formed by the length equivalent to(Ö5 - 1). If I were a pharaoh, I would also select this configuration, as Pharaoh Khufu did. The slope of the faces is 51.827292° (51° 49' 38.23'’). The characteristics of this configuration are indeed, interesting (see Chapter 11). It is worth to question, if the designers of Khufu’s Pyramid knew about these characteristics shown by this type of configuration. I believe the answer is positive. It seems improbable that the Egyptian designers selected a slope angle for the faces of the Great Pyramid equal to the slope angle of this configuration, just by accident.
Since the radius of the circle is assumed to be one, the diameter is equivalent to two. The slope’s tangent is equivalent to (D / b). However, it has been shown and proven that b = D / Öf = 2 / Öf (see Chapter 11). Therefore, the tangent of the slope’s angle is automatically set by the configuration, and is: tan q = (D)/(D / Öf) = Öf = 1.27201965. The angle with a tangent’s function equivalent to 1.27201965 is 51.8273°, that is, 51° 49' 38.25".
The perimeter of the square base (P) is equivalent to 4(b). Since b = D/ Öf, then P = 4 b = 4 (D / Öf ) = (4 / Öf (D). There is an interesting situation in this matter. The value of (4 / Öf) is equivalent to 3.144605511, which is known as an approximation of the value of Pi (p = 3.141592654). This makes the formula for the perimeter equal to P = 3.14460551(D), very similar to the formula to determine the circumference of the circle, which is C = p D = (3.141592654)(D. Since the value of D is the same in both formulas, the results will be very similar, although not equal. The difference in the two formulas would be = (p - (4 / Öf) = 0.003012857, which represents a difference of less than 3 inches in the length’s sides.
Figure 154It is possible that the similarity between the results of these two formulas make many people believe that the circumference of the circle with the pyramid’s height as its radius is equal to the perimeter of the base. I understand that this mathematical relation has also led many people to believe that the Great Pyramid was designed according to the function of p.
The Circle, the Triangle and the Square
The circumference of a circle shows a curve line turning at an equal distance around a fix point. The square figure presents its four equal straight lines, each one forming a 90° angle with the adjacent, forming a closed area. The triangle, in this case, is an isosceles triangle, composed of two congruent sides and its two equal base angles.
How can we compare and relate these three different geometrical figures? Is it possible to combined the mathematical and geometrical characteristics of these figures to make each one dependable from the other? What would be their relation? Can we use and apply this relation to the designs of Egyptian pyramids?
Is it possible that the ancient Egyptians designers used this standards and configurations?
I believe it can be done, and there is a great possibility that the Egyptians used it, I will demonstrate how. The three geometrical figures will be interlaced mathematically using the following methods and procedures. Note that this process, although using another approach, is similar to the one presented in the previous analysis and will reinforce the theory of the previous analysis. This process refers to the initial phase in the pyramid designs, that of selecting and using the geometrical configuration. As stated before, the second phase would be to determine and select the measurements and units to completely define the pyramid.
The formula C = (p) D defines the circumference (C) of a circle (see figure 155, while the perimeter of the square (P) (see figure 156), stands for the total length of its four sides. As shown in figure 157, the tangent of the angle between the hypotenuse (H’Q’) and its base leg (H’O’) corresponds to the right triangle between points H’, O’, and Q’, which is given by the ratio between its height Q’O’ (= h) and half the base length H’O’ (= b / 2).
To continue the analysis, let’s examine the figure of the circle. Its figure shows the horizontal diameter (A-B) and its vertical diameter (Q-K). As it is known, the diameter of a circle is equivalent to twice its radius, or D = 2 (R). In reference to the square, its perimeter (P) is equivalent to the sum of its four sides, which ultimately, can be defined with the formula P = 4 (b), where (b) represents the length of each side.
If the ratio between the circumference of a circle and the perimeter of a square (C / P) were established, the proportion would be equivalent to (C / P) = (p D) / (4 b). This formula can be regroup as (C / P) = (p / 4) (D / b), where (p / 4) represents a constant value, and the ratio (D / b) is variable and by itself, implies the tangent of a slope angle. The ratio of (D / b) is equivalent to the ratio that defines the function of the tangent of the base angles, of an isosceles triangle.
Notice what it had happened, by combining the formulas of the length of the circumference of the circle and the length (perimeter) of the square, we obtained their mathematical and geometrical relation, by means of a third geometrical figure, that of a right triangle; in this case, the figure of an isosceles triangle. By this means, the three figures: the circle, the square and the right triangle are connected mathematically and geometrically. Interesting enough!
Let’s refer to figure 158 to explain the use of the meaning of the ratio (D / b). If the altitude from the vertex angle of the isosceles triangle shown in figure 157, that is, (Q’O’), is placed to coincide with the radius (QO) of the circle, from point Q to point O, the vertical diameter of the circle would also bisect the base of the triangle, since the triangle altitude bisects the base. The base length (b) of the triangle will be displayed along the horizontal diameter (A-B), and will change or vary, in accordance with the slope angle of the sides.
In the isosceles triangle formed by points (QTH), the tangent of the base angle formed by the side HQ, is defined by the expression (R) / (b / 2). Since R = (D / 2), the formula can be changed to (D / b). Therefore, (D / b) represents the tangent of the slope angle of the line HQ. This slope angle is equivalent to the slope angle of line YQ, which is an extension of the line HQ. In addition, it can be seen that D = QK, divided by b = YK, which stands for the tangent of the hypotenuse YQ. From another point, since HO = OT, the base length (b) of the triangle is equivalent to the distance (HT), therefore a square could be easily figured out and traced with these measurements, as illustrated between points I, II, III, and IV.
The Pyramid design concepts
The geometrical design of the pyramids in Egypt seems to fit completely in this combined configuration pattern of the circle, the square and the triangle. Therefore, the principles presented here about the mathematical and geometrical relations of the three combined figures, could be also applied to the design of the Egyptian pyramids. The fact that my calculations seem to agree with the studied pyramids, support this theory, specially the analysis of the three pyramids in the Giza complex, and the Red and Bent Pyramid in Dahshur. Nevertheless, as in any investigational work, concepts and rules has to be established in order to have uniformity in the system.
The isosceles triangle (QTH), with its variables base length (b) and base angle (slope), stands for the vertical cross-sectional view of a pyramid, as seen when the vertical plane cut through the center of its faces. The base angle represents the pyramid slope, and its tangent, would be equivalent to the ratio (D / b). The rules I established to set the many pyramid geometrical configurations are as follows:
1. The design of the pyramids is based on the circumference of a circle.
2. The radius (R) of the circle represents the pyramid’s height.
3. The horizontal diameter of the circle represents the base line, or ground level where the pyramid will be built. The vertical diameter indicates the vertical axis.
4. The pyramids design involves a geometrical process to delineate the figure of a triangle, which represents the pyramid’s cross-sectional view through the center of its faces, and a square, that symbolizes the projection of the base of the pyramid, as seen in the vertical plane.
5. In the design process, the inclined lines of the isosceles triangle (QTH), which represent the sides of the faces of the pyramid, should be continued to cross the horizontal diameter, or base line, to intersect the circle’s circumference. The line joining these two intersection points (horizontal) is traced to establish the location, of what appears to me, is one of the control points used for the design and construction of the Egyptian pyramids. As stated before, I called this location point X. The location of point X (see figure 159) is defined as the intersection of the line joining the two intersection points of the extended face lines with the circle’s circumference, and the vertical axis. This location seems to be very important to trace the internal and underground work to be done in the pyramid. The location of point X in the pyramid configuration changes in accordance with the type of configuration used for the design. Apparently, the descending passages of the different pyramids seem to be aligned with the location of this point.
6. The space comprising the area between the pyramid’s base line, the extended inclined lines of the faces, and the horizontal line joining their intersections with the circle’s circumference, establishes the limits for the alignments of corridors, descending passages, and the location of chambers to be built under the pyramid’s base.
7. As stated, the design of each pyramid is based on different geometrical configurations. Examples of these configurations would be: an inscribed square, a circle inscribed in a square, an inscribed equilateral triangle, the triangle with sides in proportion 3:4:5, the circle with the Pi (p) configuration (p = 3.14159265... the circle with the Phi (f) configuration (f) = 1.618033989... which represents the Golden Number and Golden Section, a circle inscribed in a regular polygon, etc. The ancient engineers, or the pharaohs, could select their favorite configuration to build his pyramid.
Note from figure 159, that if the end point Q, of the slope line (QH), is fixed and the straight line is freely rotated from this fixed point, its angle could vary from almost vertically (90°) when it approaches the vertical axis, to almost 0° when it extends to a point at an infinite distance in the extension of the horizontal diameter line.
The distance (HT) represents the length (b) of the base of the pyramid. If a square is drawn using the distance HT as its sides, the figure will show a vertical view of the projection of the base of the pyramid, as seen in the pyramid vertical plane.
In summary, I had found, stated and evidenced, that (D / b) stands for the tangent of the slope of the pyramid, and that the formula (C / P) = (p / 4) (D / b) can also be rewritten as (C / P) = (p / 4) (multiplied by the tangent of the slope of the pyramid).
However, we know that besides the Great Pyramid, the ancient Egyptians built nearly 100 other pyramids around Egypt, and each one shows different heights, slope angles, and base lengths. Therefore, my geometrical method should apply to all.
Nevertheless, as general information, let’s examine the following tabulation I prepared to show the relation between the ratio (C / P) and the pyramid’s base length for the design of an unlimited number of possible pyramid models. The length of the base of the pyramid can be calculated from the formula (b) = (P / C) (p / 2), and when the ratio (P / C) is given.
(C/P) (P/C) = 1 /(C/P) (b) = (P/C)(p/2) Slope Angle (°)
.1 10 15.70796 7.256084°
.2 5 7.85398 14.28661°
.3 3.33333 5.23599 20.90544°
.4 2.50000 3.92270 27.01489°
.5 2.00000 3.141593 (= p ) 32.48164°
.6 1.66667 2.61799 (~= p²) 37.37783°
.7 1.42857 2.24399 41.70965°
.8 1.25000 1.96350 45.52762°
.9 1.11111 1.74533 48.88993°
1.0 1.00000 1.570796 (= p / 2) 51.85397°
1.1 0.90909 1.428000 54.47316°
1.2 0.83333 1.309000 56.79525°
1.3 0.76923 1.208831 58.85058°
1.4 0.71429 1.122000 60.70757°
1.5 0.66667 1.047200 ( =p / 3) 62.36345°
1.6 0.62500 0.981750 63.85473°
1.7 0.58824 0.924000 65.20306°
1.8 0.55556 0.872665 66.42679°
1.9 0.52632 0.826735 67.54139°
2.0 0.50000 0.785398 ( =p / 4) 68.56011°
2.1 0.47619 0.747998 69.49425°
2.2 0.45455 0.713998 70.35349°
2.3 0.43478 0.682954 71.14614°
2.4 0.41667 0.654498 71.87937°
2.5 0.40000 0.628319 ( =p / 5) 72.55939°
2.6 0.38462 0.604152 73.19169°
2.7 0.37037 0.581780 73.78081°
2.8 0.35714 0.560999 74.33122°
2.9 0.34483 0.541654 74.84627°
3.0 0.33333 0.523594 ( =p / 6) 75.32939°
4.0 0.25000 0.392700 78.89129°
5.0 0.20000 0.314159 (= p / 10) 81.07295°
6.0 0.16667 0.261799 ( = p /12)(~= f² / 10) 82.54241°
7.0 0.14286 0.224400 ( = p /14) 83.59819°
8.0 0.12500 0.196350 ( = p /16) 84.39297°
9.0 0.11111 0.174533 ( = p /18) 85.01263°
10 0.10000 0.157079 ( = p /20) 85.50922°
From the tabulation, it can be noticed that as the ratio (C / P) increases, the length of the base of the pyramid diminishes and the slope angle of the pyramid increases. As it is known, most of the Egyptian pyramid slopes vary from about 80° to 40°.
Observe that when (C / P) = 0.6, the base length (b) = 2.61799, value that closely approximates the value off² (= 2.6180339). This data provides for the calculation of an approximate value of (p). Since (C / P) = pD / 4 (b) = 0.6, then pD = 2.4 f², and p = 1.2 f², this approximation between the constant (p) and f is known.
The length of the base (b) in the tabulation must be multiplied by the pyramid’s height (= circle’s radius) using the desired unit of measurement, since R was originally considered equal to one (R = 1), for the tabulation.
Therefore, it appears that the ancient Egyptian designers, used these mentioned principles and rules, applied to determined geometrical configurations, to develop the geometrical design of their pyramids.
Next, I will show the wide range of geometrical configurations and their relation with the design of the geometry of the pyramids.
Geometrical Configurations Used for the Design of Pyramids
The designs of the Egyptian pyramids, apparently, follow a previously selected geometrical configuration. Each pyramid had its own and unique configuration. I divided these types of geometric configurations in 6 categories.
In all categories, the radius of the circle (R) represents the height of the pyramid, the length (b) indicates the sides of the square, and the square represents the pyramid’s base. The ratio of the circle’s diameter and the length of the base, that is, (D / b), defines the tangent of the slope angle of the pyramids. To completely define a pyramid, its height and the base length must be known, or, otherwise, the tangent (D / b) of the slope angle and the height, or the tangent and the base length. When a standard geometric configuration is used, the slope angle of the faces is automatically set by the configuration. This means that by setting only the pyramid’s height, or the base length, the pyramid will be defined.
The formulas I developed to find the basic dimensions for the pyramid’s configuration are illustrated in each category. Each one of the categories could be used to create an infinite number of four sides, right pyramids. Note that in each next category, the size of the square (pyramid’s base) increases from the inside of the circumference, until it is completely outside. These categories are:
1. When the square is inside the circle.
2. When the circle circumscribes the square.
3. When the corners of the square are outside the circle, but its sides cut its circumference.
4. When the square circumscribes the circle.
5. When the circle is inside the square.
6. When the circle is inside the square (Special Case).
In this special case, the ratio between the circumference of the circle and the perimeter of the square (C/ P) is equal to(p / f³). Note from this formula that the perimeter of the square is equal to the diameter of the circle multiplied by f³, or P = D f³, and also that the radius R = 2 (b) / f³.
Category 1 When the square is inside the circle.
As shown in Figure 160, the square (base of the pyramid) is inside the circle. The tangent of the slope angle is (D / b) and depends on the values of the measurements used. The base length would be (b) = D / divided by the tangent of slope. The perimeter of the square P = 4 (b) = 4 (D) divided by the tangent of slope).
In this category, the pyramid’s slopes vary from a minimum angle of 54.735611° (54° 44' 8.2"), when the square is inscribed in the circle, to a maximum of less than 90°, when the base’s length is minimal. The pyramid designs using this category are more likely to be small in size, as those built in Egypt and in Modern Sudan, or Nuvia. The slope angles of this category seem to be too steep for the construction of larger size pyramids, considering the building materials and construction techniques presumed to be available to the ancient Egyptians engineers.
Category 2 When the circle circumscribes the square.
Figure 161 shows the configuration when the circle circumscribes the square (base of the pyramid). I believe that the ancient engineers considered this configuration to be very important. This category represents the turning point between the previous category (when the square is inside the circle), and the next category (when the corners of the square are outside the circle, but its sides cut its circumference). Note that the slope angle in this configuration is automatically set to 54.735611° (54° 44' 8.2". As already explained, the categories that have a defined slope angle will create a completely defined pyramid when its height is established.
It looks obvious to me that the ancient Egyptian engineers did combine the geometry of this important configuration having a slope angle of 54° 44' 8.2", with the special configuration presented in category 6 (when the circle is inside the square (Special Case), to create a composite design to build the pyramid, known today as the Bent Pyramid. This will be explained later.
The following formulas applied to this configuration:
R = b/Ö2, D= Ö2 (b), b = Ö2 (R), C / P = p D / 4 (b) = (p / 4) (D / b)
Where the tangent of the slope angle = (D / b) = Ö2 (b)/ Ö2 (R) = (Ö2)(Ö2 (R) / Ö2 (R ) = 1.414213562 = Ö2. The angle corresponding to this function of the tangent is 54° 44' 8.2", and the ratio C / P = is equivalent to (p / 4) (Ö2).
Category 3 When the corners of the square are outside the circle, but its sides cut its circumference.
When the slope angle is less than 54.735611° (54° 44' 8.2") and the length of the base increases; the corners of the pyramid’s base will be located outside the circumference (see figure 162). Nevertheless, the pyramid’s sides will cut the circumference. The slope angle in this category varies from to 54° 44' 8.2" to 45°.
This is the category that satisfies the geometrical design of the three great Pyramids in Giza. The ratio (C / P) =p D / 4 (b) = (p / 4) (D /b), where b = D / (tangent of slope). Substituting the value in the formula, (C / P) = p D / 4 (D / tangent of slope) = (p / 4) (tangent of slope) = (C / P) = (p / 4) (tangent of slope).
As an example, using the Great Pyramid’s configuration, if the tangent of the slope is made equivalent to Öf =1.27201965, the slope angle would be 51.82729237° (51° 49' 38.25"), that is, the slope produced for the (Phi = f) configuration. The ratio between the circumference and the perimeter would be (C / P) = (p / 4) (1.27201965) = 0.999041897. Note that C / P is less than one (1), the number required to make the circumference of the circle equal to the perimeter of the base (C = P).
If the ratio of (C / P) is made equal to 1, that is, when (p / 4) multiplied by (the tangent of slope) = 1, the tangent of the slope would be equivalent to (4 / p) = 1.273239545. The angle corresponding to this tangent is 51.85397401° (51° 51' 14.3). This angle would correspond to the value of the (p) configuration. It seems to be a contradiction from many scholars, since they accept this slope angle as used in the Great Pyramid, but they do not accept the circle’s design, as a possible design method.
Category 4 When the square circumscribes the circle.
This is a very simple category. The base of the pyramid is equal to the circle’s diameter (b = D). Therefore, the slope angle is automatically set to 45° (see figure 163). The function of the tangent of this angle is D / b = D / D = 1. The angle having this function is 45°. Note that P = 4 (b) = 4 D. If the length of the base of the pyramid increases more than the diameter dimension, the base will be located completely outside the circle’s circumference, as will be shown in the next case.
Category 5 When the circle is inside the square (General Case).
This configuration shows that the pyramid’s base is completely outside the circle, therefore, the slope angle is less than 45°. The perimeter of the square is P = 4 (b). Since (D / b) = (tangent of slope), b = D / (tangent of slope). The ratio (C / P) =p D / 4 (b), consequently, with b = D / tangent of slope, (C / P) = p D / (4)(D / tangent of slope) = (p / 4)(tangent of slope).
Category 6When the circle is inside the square (special case).
In this special case, the ratio between the circumference of the circle and the perimeter of the square (C/ P) is equivalent to(p / f³). Note that the perimeter of the square is equal to the diameter of the circle multiplied by f³, or P = D f³, and also that the radius R = 2 (b) / f³.
This category is similar to the previous, but in this special case, the ratio between the circumference of the circle and the perimeter of the square, (C / P), is equal to (p / f³).
That is, the ratio (C / P) = p D / 4 (b) = p D / P = p D / D f³, which finally shows that (C / P) = (p / f³). Having (C / P) = (p / f³), it can be established that the perimeter (P) of the square base for this pyramid would be equal to the diameter (D) of the circle, multiplied by f³, that is, P = D f³, and that R = pyramid’s height = 2 (b) / f³.
The slope angle of this configuration is fixed by it configuration and is equal to 43.358198° (43° 21' 29.51"). This special configuration shows the existence of a unique geometrical and mathematical relationship between the circle’s circumference and the perimeter of the base, or, between the pyramid’s height and its base length. The height of the pyramid would be equivalent to the product of twice its base length and the inverse value of Phi cubed.
As it have been shown, these mathematical relations presents us the rare case of a pyramid where the diameter of the circle that generates the design multiplied by p represents its circumference (C = D p ), while the same diameter multiplied by f³, represents the perimeter of its square base (P = D f³). Additionally, from the formula D / b = 4 / f³, it can be calculated that the length of the base would be (b) = (f³/ 2) (R). The amazing characteristics shown by this configuration, surpass only by the configuration used to build the Great Pyramid, could be a reason why the ancient Egyptians seems to use it to build the Red Pyramid, and combined its configuration with another important one (When the circle circumscribes the square), as the probably method to design the Bent Pyramid. In Chapter 12 it is further discussed the geometrical configuration of the Bent Pyramid.
This analysis about the mathematical and geometrical relations between the figures of the circle, the triangle and the square (as projections of the solid figures of the sphere, the pyramid and the cube), could indicate that the ancient Egyptians had to know very well the science of Geometry and that the circumference of the circle was used for their pyramid designs using different geometrical configurations, as presented here.
There is no doubt for me that the Egyptian designers make use of their best known geometrical configurations, using the circle, the triangle and the square, to build their pyramids as their after-life resting place. These results support and strengthen my theory about the Great Pyramid’s geometrical design.
Egyptologists, Archeologists, Topographers, Engineers, Technicians, Specialists, and Authors, can be sure that they will have work in the matters of pyramids for many years to come. The modern human beings, with their advanced technology, computers, space travels, will not rest until they discover the well-kept secrets of the pyramid’s design and construction.
They will continue excavating deep in the earth, until they find the way in which the Egyptians accomplished such a huge works using the limited instrumentation and technology available five centuries ago. When this happens, at that time, the theories developed through the ages to explain their construction and design, could be evaluated, judged, and the corresponding merits awarded.