"The Golden Number"

Phi = φ = 1.618033989...

Besides being present in the proportion of the human body and in its habitat, even in music, is a standard in mathís and geometry. This famous proportion, known by many, and fear by others, represents a parameter for building beautiful structures and paintings. That is why it is used in Art and architecture. It is called the Golden Number (φ), the Golden Section and many other names and is a mystery for many people.


However in Leonardo da Vinci's times, it was commonly known and used by artists and architects. The famous drawing of Leonardo, showing the figures of a man superimposed inside a circle and a square, clearly exposed this number and its geometrical configuration.

This proportion and number, which many do not want to talk about (probably because they do not know about its properties. is very easy to obtain, mathematically, or geometrically. I have been studying this number for many years. I found in my calculations that this number emerges very easily from the figure of the triangle with sides in the proportion or ratio 1: 2. As it is known, the ratio of the radius of a circle perpendicular to its diameter and forming two sides of a triangle is equal to 1: 2. Therefore,
φ is also present in that configuration.

Letís examined why I stated that
φ and its variables emerges from the right triangle with sides in the proportion 1: 2.


Draw a right triangle with sides in the ratio 1: 2.

I discovered and established the following formulas.



Triangle with sides in ratio 1:2   -  Phi Generator





Radius/ Diameter  =  1 / 2





Letís call the sides as: the longest (hypotenuse), the middle (equal to 2) and the shortest (equal to 1). Letís use simple arithmetic, only addition, subtraction, division and multiplication to see how to obtain the φ value and its corresponding relations.


Try these calculations by yourself:

1. The longest side, plus the length of the shorter side), divided by the length of the middle side, represents the Golden Number

= (
÷5 +1) / 2 = φ = 1.618033...

2. The longest side, less the shortest side, divided by the length of the middle side, is equal to the inverse of the Golden Number.

= (
÷5 - 1) / 2 = the inverse of φ

3. The longest side, plus the middle side, is equal to the Golden Number cubed.

= (÷5 + 2) = φ 3

4. The longest side, less the middle side, is equal to the inverse of the Golden Number cubed.

= (
÷5 - 2) / 1 = 1 / φ 3

5. The sum of the three sides of the triangle, the longest, the middle and the shortest, represents (
φ 3+ 1 ).

= (
÷5 + 1 + 2) = φ 3 + 1

There are many other configurations of the value of Phi, which can be determined using this triangle and this method. It can also be determined geometrically, the square root of Phi, and the inverse of the square root of Phi.
Now, letís see how Phi proportion goes into the Great Pyramid's structural design. Most of the geometrical configuration in the Great Pyramid is based on the triangle with sides in the ratio 1: 2. Letís see and example: the King Chamber length and width: Its length is 20 cubits and its width is 10 cubits. Add the diagonal plus the width, and divide by the length, the result is Phi = 1.618033, that is, (
÷5)(10) +10 / 20) = 1.6180339. Check also that the chamberís height is equivalent to Phi less the ratio of the width over the length multiplied, by the width. In mathematical form (1.618033 - 10 / 20) (10) = 11.18 cubits. Using feet units, (1.618033 - 17.17 / 34.35) (17.17) = 19.195 = 19 feet and 2.3 inches, which is the chamber's height.

Now, as I noticed, most of the Great Pyramid's geometry is based on the triangle with sides in the ratio 1 : 2. That is the reason why the corridors gradients in the pyramids are 26.5650į; the slope is 1: 2. Surely, there must be a variation with the actual measurements; the construction shows small differences in measurements with the original plans. Therefore, the calculations and proportions of these distances will show the inherent value of
φ. That is not something to fear. It is there, as a function of this humble triangle.

In reference to the Great Pyramid, you can check the following geometrical configuration: If from the intersection of the ascending and descending passage, you continue the alignment of the descending passage to intercept the pyramidís axis, and also do the same with the ascending passage, you have created an isosceles triangle. The ascending passage line will end at the base of the great step in the vertical axis. If the intersection point of the two passages is extended horizontally to intercept the pyramidís vertical axis, then, you have defined two triangles with sides in the ratio 1: 2.


The length of the horizontal line from the intersection of the passages to the pyramidís vertical axis is equal to the vertical measurement from the intersection of the descending passage and the vertical axis, to the base of the great step. This creates the two right angles with sides in the ratio 1: 2. As stated, all the geometrical and mathematical relations of these measurements are caused by the presence of
φ in these triangles.

Now, since the triangle having the ratio 1: 2 in its sides can be interchange with the radius and the vertical diameter of a circle, the figure of a circle could be used to define the Great Pyramidís geometry. This means that to fit the pyramidís geometry, we need to design the geometrical configuration of
φ in a circle. It is important to note, that this is not the same as determining it in a rectangle or a square.

This is what I did 30 years ago before I published my book in Spanish (already sold the same year and no copies available), to define the external and internal geometry of the Great Pyramid. Therefore, If we forget about the existence of the Great Pyramid, and using the figure of a circle, setting the radius to symbolize the pyramidís height and equal to unity, you will be able to trace the internal geometry of a model pyramid, with no calculations, no crunching numbers, no formulas. Your drawing will be proportional to the radius (pyramidís height). You just have to set the pyramidís height and calculate all distances in the drawing. You will be surprised! The only number, which fits as the modelís height to make all the measurements equal to those in the Great Pyramid, is 480.66 37feet. This, curiously, represents the product of 153 multiplied the value of π (= 3.14159Ö). Please note I am not implying that my fellow ancient engineers use the feet or inches units of measurements. Just that they used a distance, which in their units of measurements is equivalent to our 480.6637 feet.


The dimensions of the following drawing are calculated from my designed pyramid's model. They are not the dimensions measured at the Great Pyramid. All the dimensions shown are the results of only one measurement, which I assigned to the radius of the circle (the radius equal to the pyramid's height). Therefore, all dimensions were calculated base on the pyramid's height. It is incredible, but these measurements correspond with those dimensions measured at the Great Pyramid. Why the Egyptologists are not interested in looking to this? I do not know. Their dogmas do not permit to evaluate math proves?  I have shown and proved with calculations that both dimensions are equivalent and correspond to each other. I am sure that someday, many Egyptologists and other scholars will be surprised with these findings, after having them for many years in their hands.

People talk about the Phi configuration in the Great Pyramid. Many are in favor, others denies it. But what are we talking about? I can tell you, I have never seen in any reference book, from the hundreds I had examined on this matters on pyramids, architecture, the golden number, etc., that shows this amazing configuration of the Perfect Symbol. Even in books addressed only to the Golden number. It took me several years to fully define, or developed it. This occurred 30 years ago. After all this period of time, I have never seen the Symbol in any place, only in my works.


Is it a coincidence that the geometrical design of a model pyramid, in all its sections, passages, angles, characteristics, designed as a unitary circle, when the radius is set as the pyramidís height, the measurements of each section agreed with those measurements, angles and characteristics as measure in the Great Pyramidís structure? No way. There will be too many coincidences.


My design model indicates that if the inclined line of the faces is extended to intercept the circumference of the circle, its length from the top of the pyramid to the intersection point is equal to the base length of the pyramid. This requirement is basic to satisfy the Perfect Symbol configuration. I measured this distance with my computer in my model and it has 440 cubits. I decided to ask this question to an expert in Egyptology, working in a special job in Egypt, but have an Internet web site. He claimed that he have the most exact computer drawing of the pyramid. He accepted my question and tried the experiment. To me, if my condition happens, there will be no doubt that my theory is completely correct, and prove that:


1.          The pyramid was designed using the Perfect Symbol.


2.          That the lines extending the faces establish, at their intersection with the circumference, the area worked with chambers and corridors construction under the pyramid.


3.          The Perfect Symbol will help to set and define the pyramid's architecture, outside and inside as well.


I waited for his experiment. He answered, ďNo, the measure is 440.34 cubits, which is not equal to the 440 cubits of the pyramidĒ. For me the answer was just exact, perfect. The difference was only about 7 inches in 9,068.9 inches! Less than the differences measured between the sides of the base of the Great Pyramid, and they are considered equal. As you will understand, my prediction of its length is from a design work, and the expert measurement is from actual dimensions measured at the structure. I was more than happy with the result.

The Phi studies should be of particular interest to those concerned with ancient Egyptian structures. To know it, and how it works, allow us to a better understanding of the Egyptian structures and their geometrical relations. This is clearly the reason for the repetition of certain measurements inside the Pyramidís structure and their mathematical relation found by many people. If these principles are applied to other pyramids, you will find equal results. It is not crunching numbers, is applied geometry and mathematics.

In our period of time, the Golden Section
φ is almost forgotten and itís teaching in Math's, Geometry, Arts, and other sciences are very limited. Other science curriculums never have heard about it. I would recommend that it should be included in the appropriate curriculums. The Golden Section represents a parameter in Arts, harmony, and beauty. Our Creator included it in the formation of the cosmos, in the physical universe, and even in the proportions of the human figure. Why to refuse to accept and use it?  The Perfect Symbol gives us its merits...



Finally, we have to conclude that ... there is nothing wrong with Phi.