The "Perfect Symbol", is how I named the following geometric configuration. The symbol shows the figures of a circle, a triangle and a square, grouped in what appears to be, perfect harmony and arrangement. It shows a pattern of beauty and elegance, ideal for the design and proportioning of figures and images.




For many years I had in mind the existence of this symbol. Nevertheless, although I looked up in all math-books, and books containing symbols I could find, but I found no positive results. Well, since I could not find the searched symbol, I decided to find myself a geometrical method to trace or design it. The specifications I used to develop the symbol were that it should be traced using only a straightedge to set the lines, and a compass to trace the circle. Besides, no numerical values, or math calculations should be used to generate the symbol.


After several years working in my research, I finally found the geometrical process to trace the symbol using my specifications. I was more than delighted with the symbol I created? Discovered? or rediscovered? After I studied it carefully, I noticed that the symbol has unique geometrical characteristics. Its configuration shows that it is a representation of the Golden Number and the Golden Section.


The configuration I developed for this Perfect Symbol is drawn using only a straightedge and a compass, following a geometrical method. It looks as shown in the illustration.


Surprisingly, for me it is the "key" for the design of all the pyramids and their internal chambers. It easily transformed into what is called the "boss", carved at the side of the frontal granite leaf at the entrance to the antechamber of the Great Pyramid. I also used the symbol to design the antechamber. This is the reason why you will see the symbol all around this book.


Sometime ago, many people, including myself, associate a similar symbol, to the cross-sectional area of the Great Pyramid in Egypt. These symbols: the circle, the triangle and the square were drawn just as a model of the Great Pyramid, having no dimensions or set face angles. Every drawing was different from the other. However, we were obviously wrong since the existence of this Perfect Symbol in particular, is completely independent of the pyramid's design. This symbol belongs to the Science of Geometry, since it existed and could have been drawn before the construction of the pyramids, or drawn without the knowledge of the existence of the pyramids.



            The Great Pyramid




The Great Pyramid designed using the Perfect Symbol


The process to trace the Perfect Symbol is very simple, and as it will be seen, its geometric characteristics are incredible and surprisingly arranged in the symbol. The following is the geometrical process to draw the symbol.


1. With the straightedge, trace a straight line, as shown in Figure 1, from any point A to a point B.




2. Select a point (O) in the line and using the compass, trace a circle (any convenient size, adequate to generate the drawing). To define the horizontal diameter, mark in the circumference the points A and B in the AB' line.




3. To draw the vertical diameter of the circle, from point A, using a longer measurement than the radius, trace an arc. Do the same, from point B, with an equal radius.  Mark the two points of intersection between the two arcs (points

a and b). The line joining the two points will be perpendicular to the horizontal diameter and will pass through the center of the circle. Now, you can set point Q and K in the circumference, to define the vertical diameter.




4. Using the compass and point B as center, trace a semicircle (OL) using radius BO, to establish point L in the line A’B’. Using the compass, from center L and radius LB trace an arc tangent to point B in the circle. With the straightedge, set a line from point L to point K and mark point M at its intersection with the previous traced arc.




5. With the compass. with point K as center and radius KM, trace an arc to intersect the circumference of the circle at both sides. Mark point N and P at these two points of intersection. Join points P, Q, and N, to create the triangle PQN. Mark points H and T, at the intersection of lines PQ and QN with the horizontal diameter AB. Mark point X, at the intersection of line PN with the vertical diameter QK.



6. The configuration shows two triangles: (1) triangle HQT, and (2) triangle PQN. Both triangles are very important in this configuration, as it will be shown.




7. Point O, the center of the circle is located at the center of the base (HT) of the triangle QHT. From the center O, and using the length OH, or OT, set the distances OS and OU in the vertical diameter. Construct a square having length HT as its sides, which will cross through points H, S, T and U.




8. With points H, S, T and U already defined, the corners of the square can be set. From point H, using HO as radius, trace an arc as shown. With an equal radius and using S as center, trace another arc. Do the same at points T and U. The four intersection points between the arcs, identified as I, II, III, and IV, identify the four corners of the square.


9. Join points I, II, III and IV to create the square. This complete the design of the "Perfect Symbol", the same figure shown in the article.




As I stated, this symbol has nothing to do with the Great Pyramid design. As shown, I traced the drawing without any reference to the Great Pyramid. I used no math calculations, no numbers, or formulas. However, when  a unitary radius value is set for the circle, all the dimensions for the drawing can be calculated.

From another point, the Perfect Symbol can be defined mathematically in terms of Phi, which is the Golden Number. Leonardo da Vinci's famous drawing for the human proportion is based on this symbol. Therefore it is important to denote that the Perfect Symbol contain information regarding the Human Proportions, as set by our Creator.





Since the Perfect Symbol shows the figure of a circle, a triangle and a square, it also seems to fit the Great Pyramid design, where the circle is the base of the design, the triangle, the Great Pyramid's cross-sectional view, and the square, the base of the pyramid. Anyone can draw a circle, a triangle, and a square with a similar look as the perfect symbol, but they are not equal. The Perfect Symbol figure defines its proper angles, and proportions. That is not the case of drawing a circle, a triangle and a square with random dimensions. Now, if we examine and calculate the Perfect Symbol dimensions based on the Great Pyramid's height as the radius of the circle, we can arrived to the apparent conclusion that the Egyptian Engineer used the Perfect Symbol design, for the design of the Great Pyramid. But, there is more evidence to prove that the Perfect Symbol was used to design the Great Pyramid.




We can say that the Perfect Symbol did not come from the pyramid's architectural design, but it is possible that the pyramid design came from this simple and small symbol. There is enough evidence, as it will be shown in this book, that the Great Pyramid's geometrical design could have its origin on this Perfect Symbol. It is feasible that because its unique characteristics, it could have been known by the Egyptians in their geometrical work, and based on its extraordinary characteristics, they decided to used its configuration for the Great Pyramid's design.





I noticed in the Great Pyramid structure, that the descending passage, starting from its entrance is inclined and aligned toward the position of point X in the symbol. When the descending corridor alignment approaches the location of point X, the corridor direction was changed to horizontal. Why the Egyptian architect set up the descending corridor to go underground the pyramid? Why the alignment of this descending passage coincides in the direction to reach point X. and is realigned to go horizontally to the underground chamber? I decided to investigate these situations. An Anyway, for this, we need to set some rules. I set the following rules to work out this geometrical comparison for both designs.









 Mathematical solution for the Perfect Symbol




Leonardo da Vinci's Circle and Square

proportional for his drawing of the Human Proportions 


In order to be consistent with my idea, I adopted the following rules and symbolisms to work out this exercise.


Let us assume that the Great Pyramid's, design was traced using the principles of the symbol's geometry.


1. The radius of the circle is equivalent to the pyramid's height, and equal to one (1). The radius of the Perfect Symbol is also equivalent to one (1).

Nevertheless, any dimension for the pyramid's height can be used, and then calculate the whole configuration.


2. The horizontal diameter line (AB) represents the natural level of the terrain (where the pyramid was built.)


3. The vertical diameter (QK) represents the pyramid's vertical axis.


4. The triangle QHT represents the Pyramid's cross-sectional view (the Great Pyramid's sectional view, in a plane cut through the center of its sides).


5. The triangle QPN, generated by the symbol represents a virtual pyramid, where its section PHTN is located below the ground level of the Great Pyramid and was used to design the underground work done below the pyramid.


6. The square (between points I, II, III, and IV) represents the square base of the pyramid as seen in a vertical plane. The distance HT = b, represents the length of the sides of the square.


7. The distance OX represents the vertical distance from the center O, at the base of the pyramid, to the location of point X, below the center of the pyramid. Point X represents the point in the vertical diameter of the circle, where the diameter is divided in two sections, whose ratio will be equivalent to 1.618, the Golden Number. In other words, (QX / XK) = 1.618.


8. From another point of view, the distance OX' represents the vertical distance from the center O at the base of the pyramid to point X' shown above the center the base of the pyramid. In this special Geometric Configuration, the Phi point could also be located over the center of the base of the pyramid. In this case, it divides the diameter in two other sections, where its ratio is also equivalent to Phi (1.618). This would be represented by the ratio (X'K / QX') = 1.618 = Phi.


9. The inclined lines QH and QT represent the sides of the pyramid to be built.


10. To find the location of the Phi point, divide the length of the diameter of the circle by 1.618. The resulting length is the vertical distance from point Q at the top of the circle to the Phi point. This means that if you divide this distance by the remaining section of the diameter, your result would be equal to 1.618.


11. The distance from point X' over the center O of the Pyramid would be the same distance as OX below the center O. Since the location of point X, below the base of the pyramid and that of point X' over the base are located at points equal to Phi, make us think that the pharaoh's mortuary chamber could be located at either of those location.


12. The dimensions of these geometrical configurations are proportional to the radius of the circle, and the angles are fixed and will not change, no matter the dimensions set for the radius.


13. As a curiosity, the length of the lines PQ and QN, from the triangle PQN, is equal to the length of the base HT of the pyramid. Therefore, it can be stated that the squares constructed over the lines PQ and QN, are exactly equivalent to the base of the pyramid (the square between points I, II, III and IV).


From the other part, if the Egyptians knew about the existence of this peculiar symbol, and its unique and extraordinary characteristics, they could have certainly used it for the design of the Great Pyramid.


The Golden proportions, as set by the Perfect Symbol, due to its esthetics, harmonious, figure generator properties, and other characteristics, has many uses in architecture, painting, and designs. Although the value of Phi is well known, there are many of its properties that are still unknown.

It is being said that the golden proportion was used in the design and construction of the United Nations Building in New York. Anyway, it is known that was used in the building of Medieval cathedrals to proportion the architectural elements. It seems to me that the Egyptian engineers used Phi in proportioning the vertical sections of the pyramids. From another point, pyramids built by the Mayas, Aztecs in Mexico, and adjacent countries, seem also to reveal that they used Phi to proportion the horizontal level of the pyramid design. In other words, apparently in Egypt, the Phi constant was used to proportioning the vertical cross-section, while in Mexico they used Phi in proportioning the top view plans of their pyramids.

One of the curious properties of Phi, on the Great Pyramid's design is that it facilitates the location of the most important places in the Great Pyramid. For example: take the pyramid’s height and divide it by Phi, and continue the same operation with the obtained results. Assume the pyramid's height as 480.66 ft.

480.66’ / Phi = 297.04’ = Radius of the internal circle within the pyramid, center at its base, and tangent to the sides of the Pyramid.

297.04’ / Phi = 183.60’ = Height of the Queen Chamber over the location of point X (the phi point in the vertical diameter).

183.60’ / Phi = 113.46’ = the vertical distance from the center of the GP to the location of point X.

113.46’ / Phi = 70.12’ Elevation of the Queen Chamber over the pyramid’s base line. If you do a similar process with the diameter of the circle, that is, 961.32 ft,

961.32’ / Phi = 594.13’ location of point X from the apex of the Pyramid.

594.13’ / Phi = 367.19’ elevation of point X over the lower point of the vertical axis (diameter). Ad to the previous result, you get the diameter of the circle,

367.19’ / Phi = 226.94’ = elevation of the pyramid’s base over point X, if the pyramid drawing is inverted.

226.94’ / Phi = 140.25’ = elevation of the King Chamber’s floor over the base line of the pyramid.

Was this process used by the Egyptians to set the different locations in the structure? Anyway, these are the marvelous expositions of this perfect symbol's configuration. We do not have to establish the location and dimensions of these places to get a beautiful structure, the geometric configuration gives then to us.
All these numbers are not invented, or crafted; it is something that can be confirmed by calculations.

How far can we go with Phi? I don't know, but see in the picture the official map of Stonehenge, where I drew my pyramid’s perfect symbol's configuration. It seems to fit exactly in the plans.

I know that the figure of a circle and the Great Pyramid had been used for many theories, such as to make predictions, forecast, prophesies, etc. That could have affected the scholars from looking seriously to this method. However, in this occasion it could be the right solution, if rightly explained and documented. Egyptologists require from me as evidence, a papyrus showing my Great Pyramid’s configuration, and properly identified as such. If this happens in a near future, I will change the way of viewing of the Pyramid’s designs. If this papyrus is found, but not properly identified, it will go into one of the great Egyptian Museums, maybe forever. Who knows, if this papyrus is already unidentified in some papyrus collections. The problem is that you have to know what you are looking for. In this case you have to look for the Perfect Symbol, to get the required evidence.