4.
WHAT IS WRONG WITH PHI?
"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 math’s 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.
Let’s 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
Let’s call the sides as: the longest (hypotenuse), the middle (equal to 2) and
the shortest (equal to 1). Let’s 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
φ.
= (Ö5
+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.
= (Ö5
 1) / 2 = the inverse of
φ
3. The longest side, plus the middle side, is equal to the Golden Number cubed.
= (Ö5 + 2) =
φ ^{3}
4. The longest side, less the middle side, is equal to the inverse of the Golden
Number cubed.
= (Ö5
 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 ).
= (Ö5
+ 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, let’s 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. Let’s 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, (Ö5)(10) +10 / 20) = 1.6180339.
Check also that the chamber’s 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 26.5650°; 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 pyramid’s
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 pyramid’s 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
pyramid’s 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 Pyramid’s geometry. This means that to fit the
pyramid’s 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 pyramid’s 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 (pyramid’s
height). You just have to set the pyramid’s height and calculate all distances
in the drawing. You will be surprised! The only number, which fits as the
model’s 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 (= 3.14159…). 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 pyramid’s height, the measurements of each section agreed with
those measurements, angles and characteristics as measure in the Great Pyramid’s
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, “No, the measure is 440.34 cubits, which is not
equal to the 440 cubits of the pyramid”. 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
Pyramid’s 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 it’s 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.
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