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KAVALA-GEMATRIA-NUMEROLOGIA BIBLICA-ETC: El Número de Oro; Phi; la Divina Proporción
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Reply  Message 1 of 162 on the subject 
From: BARILOCHENSE6999  (Original message) Sent: 08/06/2011 15:58



Phi in the Bible


Although perhaps not immediately obvious, phi and the golden section also appear in the Bible.  Also see the Theology page.

The Ark of the Covenant is a Golden Rectangle


Ark of the Covenant

In Exodus 25:10, God commands Moses to build the Ark of the Covenant, in which to hold His Covenant with the Israelites, the Ten Commandments, saying,

"Have them make a chest of acacia wood-
two and a half cubits long,
a cubit and a half wide,
and a cubit and a half high."

The ratio of 2.5 to 1.5 is 1.666..., which is as close to phi (1.618 ...) as you can come with such simple numbers and is certainly not visibly different to the eye.  The Ark of the Covenant is thus constructed using the Golden Section, or Divine Proportion.  This ratio is also the same as 5 to 3, numbers from the Fibonacci series.

In Exodus 27:1-2, we find that the altar God commands Moses to build is based on a variation of the same 5 by 3 theme:

"Build an altar of acacia wood, three cubits high; it is to be square, five cubits long and five cubits wide."


Note:  A cubit is the measure of the forearm below the elbow.

Altar of Exodus 27

Noah's Ark uses a Golden Rectangle


In Genesis 6:15, God commands Noah to build an ark saying,

"And this is the fashion which thou shalt make it of: The length of the ark shall be three hundred cubits, the breadth of it fifty cubits, and the height of it thirty cubits."

Thus the end of the ark, at 50 by 30 cubits, is also in the ratio of 5 to 3, or 1.666..., again a close approximation of phi not visibly different to the naked eye.  Noah's ark was built in the same proportion as ten arks of the covenant placed side by side.

Noah's Ark

The Number 666 is related to Phi


The relationship of phi and 666

Revelation 13:18 says the following:

"This calls for wisdom. If anyone has insight, let him calculate the number of the beast, for it is a man's number. His number is 666."

This beast, regarded by some as the Anti-Christ described by John, is thus related to the number 666, one of the greatest mysteries of the Bible.

Curiously enough, if you take the sine of 666º, you get -0.80901699, which is one-half of negative phi, or perhaps what one might call the "anti-phi."  You can also get -0.80901699 by taking the cosine of 216º, and 216 is 6 x 6 x 6.

The trigonometric relationship of sine 666º to phi is based on an isosceles triangle with a base of phi and sides of 1.  When this triangle is enclosed in a circle with a radius of 1, we see that the lower line, which has an angle of 306º on the first rotation and 666º on the second rotation, has a sine equal to one-half negative phi.

In this we see the unity of phi divided into positive and negative, analogous perhaps to light and darkness or good and evil.  Could this "sine" be a "sign" as well?

In addition, 666 degrees is 54 degrees short of the complete second circle and when dividing the 360 degrees of a circle by 54 degrees you get 6.66... The other side of a 54 degree angle in a right angle is 36 degrees and 36 divided by 54 is .666.

Phi appears throughout creation, and in every physical proportion of the human body.  In that sense it is the number of mankind, as the mysterious passage of Revelation perhaps reveals.

Also see the Theology page.

The colors of the Tabernacle are based on a phi relationship

The PhiBar program produces the colors that the Bible says God gave to Moses for the construction of the Tabernacle.

As it says in Exodus 26:1, "Make the tabernacle with ten curtains of finely twisted linen and blue, purple and scarlet yarn, with cherubim worked into them by a skilled craftsman."

Set the primary color of the PhiBar program to blue, the secondary color of the PhiBar to purple and it reveals the Phi color to be scarlet.

This reference to the combination blue, purple and scarlet in the construction of the tabernacle appears 24 times in Exodus 25 through 39, describing the colors to be used in the curtains, waistbands, breastpieces, sashes and garments.

See the Color page for additional information.

Insights on the Ark of the Covenant and 666 contributed by Robert Bartlett.

Insights on the Altar in Exodus 27 contributed by Sir Hemlock.

Insights on the Tabernacle colors contributed by J.D. Ahmanson.



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Reply  Message 148 of 162 on the subject 
From: BARILOCHENSE6999 Sent: 21/04/2014 23:49

The Great Pyramid at the Giza Plateau, Egypt, is mysterious, mystical and strategically placed on the face of the Earth. It is aligned with the four cardinal points more accurately than any other  structure, even the Greenwich observatory. The builders displayed enlightened understanding of engineering, mathematics, physics and astronomy. They also had a profound knowledge of the Earth's dimensions. Many scientist and scholars now think it was built earlier than the reign of Cheops (Khufu) at 2600 B.C. They disagree with the traditional view of Egyptologist and attribute it to an advanced civilization before the rule of the Pharaohs. I am convinced they are right. For more details follow the "Egypt/Center of Earth" link.
The Pyramid lies in the center of gravity of the continents. It also lies in the exact center of all the land area of the world, dividing the earth's land mass into approximately equal quarters. It lies in the middle of Egypt and in the middle of Lower and Upper Egypt.
The Plate XX from an original 1877 copy of  Piazzi Smyth's "Our Inheritance in the Great Pyramid". Charles Piazzi Smyth (1819-1900) was Astronomer Royal for Scotland and a respected Scientist. 

The north-south axis (31 degrees east of Greenwich) is the longest land meridian, and the east-west axis (30 degrees north) is the longest land parallel on the globe. There is obviously only one place that these longest land-lines of the terrestrial earth can cross, and it is at the Great Pyramid! This is incredible, one of the scores of features of this mighty structure which begs for a better explanation.

Pyramid Statistics  © 2000 by Larry Orcutt,A total of over 2,300,000  blocks of limestone and granite were used in its construction with the average block weighing 2.5 tons and none weighing less than 2 tons. The large blocks used in the ceiling of the King's Chamber weigh as much as 9 tons. 

Original entrance of the Great Pyramid. Massive blocks of limestone form a relieving arch over the entrance.

  • The base of the pyramid covers 13 acres, 568,500 square feet and the length of each side was originally 754 feet, but is now 745 feet. 
  • The original height was 481 feet tall, but is now only 449 feet.

The majority of the outer casing, which was polished limestone, was removed about 600 years ago to help build cities and mosques which created a rough, worn, and step-like appearance. The base measurements of the Great Pyramid are: north - 755.43 ft; south - 756.08 ft; east - 755.88 ft; west - 755.77 ft. These dimensions show no two sides are identical; however, the distance between the longest and shortest side is only 7.8 inches. Each side is oriented almost exactly with the four Cardinal points. The following being the estimated errors: north side 2'28" south of west; south side 1'57" south of west; east side 5'30" west of north; and west side 2'30" west of north. The four corners were almost perfect right angles: north-east 90degrees 3' 2"; north-west 89 degrees 59'58"; south-east 89 deg 56'27"; and south-west
90 deg 0'33". When completed, it rose to a height of 481.4 ft., the top 31 feet of which
are now missing. It's four sides incline at an angle of about 51deg. 51 min. with the ground. At its base, it covers an area of about 13.1 acres. It was built in 201 stepped tiers, which are visible because the casing stones have been removed. It rises to the height of a modern 40-story building.


The pyramid is built partly upon a solid, large, bedrock core and a platform of limestone blocks which can be seen at the northern and eastern sides. The builder of this pyramid was very wise to choose this site because most of the stones, with the exception of the  casing stones, some granite and basalt stones, could be cut right on the spot and in the nearby quarry. This practical choice made it possible to reduce considerably the time and back-breaking labor needed to drag the stones from distant quarries across the Nile.

The many surveys done on the pyramid proved that the Egyptians located the sides of the pyramid along the four Cardinal Points with extreme accuracy. Whether they used the stars, and/or the rising and setting sun, cannot be determined. One this is certain, that whatever method they used was direct and very simple.

Once the sand, gravel and loose rocks had been removed, down to the solid bedrock of the plateau, the whole pyramid site was open-cast quarried into blocks, leaving a square core for the center of the pyramid (the core is approximately 412.7 ft square, and rises approx. 46.25 feet high). These blocks were then stored outside a low wall; made of mortared stone that surrounds the core (the outside dimensions of the wall are approx. 887.3 feet square). Today there still remains the foundation of this wall on the north, south and west sides of the pyramid, at an average distance of 65 feet from the outer edge of the base casing stone.

This core gives the pyramid stability from the downward and horizontal forces that will develop from the superimposed loads of blocks of stones that are piled up, as the pyramid rises. Also, from the prevailing north-west winds that exert enormous pressures on the huge areas of the pyramid's faces, thus increasing these forces further.

Leveling of the entire pyramid site was accomplished by flooding the area inside the wall with water, leaving just the high spots. These them were cut down to the level of the surface of the water. Next, some of the water was released and the high spots again were cut down to the water's surface. This
process was repeated until the entire pyramid site, between the core and the four walls, was leveled down to the base of the pyramid's platform.


A few of the fine limestone casing blocks remain at the base of the northern side and show how accurately the stones were dressed and fitted together. The core masonry, behind the casing stones, consists of large blocks of local limestone, quarried right on the spot, built around and over the
bedrock core. The size of this core cannot be determined, since it is completely covered by the pyramid.

The casing stones were of highly polished white limestone, which must have
been a dazzling sight. Unlike marble, which tends to become eroded with time
and weather, limestone becomes harder and more polished.


The size of the blocks are based on a chance discovery in 1837 by Howard Vyse. He found two of the original side casing blocks at the base of the pyramid, 5 ft x 8 ft x 12 ft, with an angle of 51 degrees, 51 minutes cut on one of the 12 ft. sides. Each of these stones weighed (5 x 8 x 12)/2000 =
39.9 tons before the face angle was cut. These originally were used for the side casing stones of Step No. 1, in the Pascal computer program. The sizes of all the other blocks were scaled from these two original blocks of the remaining Steps 2 to 201.


One acre = 43,560 sq. ft, or 208.71 feet on a side.
For the pyramid's base, length = width = (square root of 13.097144 acres) x
208.71 feet = 755.321 feet. Or 755.321 x 12 = 9063.85 inches.

Height = (755.321 x tangent 51deg 51 min)/2 = 480.783 feet. Or 480.783 x 12
= 5769.403 inches.

For the cap stone base: length = width = (32.18 x 2)/tangent 51deg 51 min =
50.55 inches.

The average size of a pyramid stone = (5 x 8 x 12)
The average side measurement, at the base = 759.3 ft.
The height used was 201 steps high, or 480 feet. (This is minus the height
of the Capstone, which was one piece in itself. Geometry of the Great Pyramid 

DIMENSIONS of Great Pyramid

The following article is Copyright © 2001-2003 aiwaz.net_institute. 
If the calculations concerning the royal cubit are correct the main dimensions of the pyramid should also prove that. The approximate dimensions of the pyramid are calculated by Petrie according to the remains of the sockets in the ground for the casing stones whose remains are still at the top of the pyramid, and the angle 51° 52' ± 2' of the slopes. The base of 9069 inches is approximately 440 royal cubits (the difference is 9 inches which is not a remarkable difference if we consider the whole dimension and consider that the employed data represent only an estimation of the real values) whereas the calculated height, 5776 inches, is precisely 280 royal cubits. The relation 440:280 can be reduced to 11:7, which gives an approximation of the half value of Pi.

Squaring the Circle

Squaring the Circle

The circle and the square are
united through the circumference:

area of square: 440x440=193600
area of circle:28x28x22/7=246400
sum: 440000


The engagement of Pi value in the main dimensions suggests also a very accurate angle of 51° 52' ± 2' of the slopes which expresses the value of Pi. Another coincidence is the relation between the height of the pyramid's triangle in relation to a half of the side of the pyramid, since it appears to be the Golden Section, or the specific ratio ruling this set of proportions, F = (sqr(5)+1)/2 = 1.618 = 356:220. This ratio, 356:220 = 89:55 is also contained in the first of Fibonacci Series:

1  2  3  5  8  13  21  34  55  89  144 ...

A single composition contains two apparently contradicting irrational numbers P and F, without disrupting each other. This appears to be completely opposed to the classical architectural canon which postulates that in 'good' composition no two different geometrical systems of proportions may be mixed in order to maintain the purity of design. But analysis of other architectural and artistic forms suggested that the greatest masters skillfully juggled the proportional canons without losing the coherent system, for they knew that these systems can be interconnected if the path that links them is found. That is obvious In the case of the Great Pyramid where two different principles are interweaved without interference ruling different angles of the composition, which is most importantly a most simple one, namely 11:7, a most simple ratio obviously signifying such infinite mysteries as the value of P and most 'natural' value of F. In spite of common miss-understanding of architectural composition, the most mysterious and praised compositions are very simple but not devoid of anthropomorphic appeal, since everything is made out of human proportions, just like Vitruvius describing the rations of the human body, very simple and very clean. The numbers 7 and in 11 are successive factors in the second of Fibonacci progressions that approximate geometry of the pentagram:

1   3   4   7   11   18   29   47   76   123   ...

The summary of the selected main mean dimensions is:

b. inch
royal cub.

The Great Pyramid

The main source of all kinds of delusions and speculations about our mythical past for the western man comes of course from Plato. With the myth of Atlantis he planted the necessary seed of mythical Eden, a culture of high intelligence that lived before the known history. If Plato received any wisdom from the ancient Egypt it could perhaps be traced in the canon of numbers that is so latently present throughout his work, but never on the surface. This canon seems to appear in the descriptions of his fantastic cities where everything is most carefully calculated and proportioned. The topic of Plato's Laws is the description of the ideal state called Magnesia which is entirely composed out of the mysterious number 5,040.

The distance* when Earth is closest to Sun (perihelion) is 147x106 km, which is translated into royal cubits 280x109, hinting at the height of the Great pyramid, 
280 royal cubits

The Golden Ratio & Squaring the Circle in the Great Pyramid

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. [Euclid]

The extreme and mean ratio is also known as the golden ratio. 

If the smaller part = 1, and larger part = G, the golden ratio requires that 
G is equal approximately 1.6180

Does the Great Pyramid contain the Golden Ratio?

Assuming that the height of the GP = 146.515 m, and base = 230.363 m, and using simple math we find that half of the base is 115.182 m and the "slant height"  is 186.369 m

Dividing the "slant height" (186.369m) by "half base" (115.182m) gives = 1.6180, which is practically equal to the golden ration! 

The earth/moon relationship is the only one in our solar system that contains this unique golden section ratio that "squares the circle". Along with this is the phenomenon that the moon and the sun appear to be the same size, most clearly noticed during an eclipse. This too is true only from earth's vantage point…No other planet/moon relationship in our solar system can make this claim. 

If the base of the Great Pyramid is equated with the diameter of the earth, then the radius of the moon can be generated by subtracting the radius of the earth from the height of the pyramid (see the picture below).

Also the square (in orange), with the side equal to the radius of the Earth, and the circle (in blue), with radius equal to the radius of the Earth plus the radius of the moon, are very nearly equal in perimeters:

Orange Square Perimeter = 2+2+2+2=8
Blue Circle Circumference = 2*pi*1.273=8

Earth, Radius, Mean = 6,370,973.27862 m *
Moon, Radius, Mean = 1,738,000 m.*
Moon Radius divided by Earth Radius = 0.2728 *

* Source: Astronomic and Cosmographic Data

Let's re-phrase the above arguments **

In the diagram above, the big triangle is the same proportion and angle of the Great Pyramid, with its base angles at 51 degrees 51 minutes. If you bisect this triangle and assign a value of 1 to each base, then the hypotenuse (the side opposite the right angle) equals phi (1.618..) and the perpendicular side equals the square root of phi. And that’s not all. A circle is drawn with it’s centre and diameter the same as the base of the large triangle. This represents the circumference of the earth. A square is then drawn to touch the outside of the earth circle. A second circle is then drawn around the first one, with its circumference equal to the perimeter of the square. (The squaring of the circle.) This new circle will actually pass exactly through the apex of the pyramid. And now the “wow”: A circle drawn with its centre at the apex of the pyramid and its radius just long enough to touch the earth circle, will have the circumference of the moon! Neat, huh! And the small triangle formed by the moon and the earth square will be a perfect 345 triangle (which doesn’t seem to mean much.)

Was the golden ratio intentionally built into the Great Pyramid of Cheops? 
Why would anyone intentionally build the golden ratio into a pyramid, or other structure? What was the significance of to the Egyptians? And did the ancient Egyptians intentionally design the Great Pyramid to square the circle?

The answer to these questions is uncertain since designing the Great Pyramid according to the simple rules explained by the graphic below would give the pyramid automatically (by coincidence? ) all its "magic" qualities. 

The height of the Great Pyramid times 2π exactly equals the perimeter of the pyramid. This proportions result from elegant design of the pyramid with the height equal two diameters of a circle and the base equal to the circumference of the circle.  Click here or on the image below to see larger picture.

For the angle of the Great Pyramid, any theory of the base, combined with any theory of the height, yields a theoretic angle; but the angles actually proposed are the following** :

Angle of casing measured
By theory of 34 slope to 21 base
Height : circumference :: radius to circle
9 height on 10 base diagonally
7 height to 22 circumference
area of face = area of height squared
(or sine) = cotangent, and many other relations)
2 height vertical to 3 height diagonal
5 height on 4 base

51º 52' ± 2' (51.867)
51º 51' 20"
51º 51' 14.3"
51º 50' 39.1"
51º 50' 34.0"
51º 49' 38.3"

51º 40' 16.2"
51º 20' 25"

** Page 184, The Pyramids and Temples of Gizeh 
     by Sir W.M.Flinders Petrie 1883

Comparing the Great Pyramid with the Pyramid of the Sun in Teotihuacan

The Pyramid of the Sun and the Great Pyramid of Egypt are almost or very nearly equal to one another in base perimeter. The Pyramid of the Sun is "almost" half the height of the Great Pyramid. There is a slight difference. The Great Pyramid is 1.03 - times larger than the base of the Pyramid of the Sun. Conversely, the base of the Pyramid of the Sun is 97% of the Great Pyramid's base.

The ratio of the base perimeter to the height:

 Great Pyramid Pyramid of the Sun
 6.2800001... : 1
(deviates by 0.05 % from the 
6.2831853 value for 2 x pi)
 12.560171... :
(deviates by 0.05 % from the 
12.566371 value for 4 x pi)

The Great Pyramid - Metrological Standard

The Great Pyramid is generally regarded as a tomb and as grandiose memorial to the pharaoh who commissioned it.  The opposing view is that of the pyramid being the culminating achievement of those who practiced an advanced science in prehistory.

The Great Pyramid is a repository of universal standards, it is a model of the earth against which any standard could be confirmed and corrected if necessary. 
It is exactly the imperishable standard, which the French had sought to create by the devising of the metre, but infinitely more practical and intelligent. 

From classical times, the Great pyramid has always been acknowledged as having mathematical, metrological and geodetic functions. But ancient Greek and Roman writers were further removed in time from the designers of the Great Pyramid than they are from us. They had merely inherited fragments of a much older cosmology; the science in which it was founded having long since disappeared.

The Concave Faces of the Great Pyramid 

Great Pyramid
Aerial photo by Groves, 1940 (detail).

In his book The Egyptian Pyramids: A Comprehensive, Illustrated Reference, J.P. Lepre wrote:

One very unusual feature of the Great Pyramid is a concavity of the core that makes the monument an eight-sided figure, rather than four-sided like every other Egyptian pyramid. That is to say, that its four sides are hollowed in or indented along their central lines, from base to peak. This concavity divides each of the apparent four sides in half, creating a very special and unusual eight-sided pyramid; and it is executed to such an extraordinary degree of precision as to enter the realm of the uncanny. For, viewed from any ground position or distance, this concavity is quite invisible to the naked eye. The hollowing-in can be noticed only from the air, and only at certain times of the day. This explains why virtually every available photograph of the Great Pyramid does not show the hollowing-in phenomenon, and why the concavity was never discovered until the age of aviation. It was discovered quite by accident in 1940, when a British Air Force pilot, P. Groves, was flying over the pyramid. He happened to notice the concavity and captured it in the now-famous photograph. [p. 65]

This strange feature was not first observed in 1940. It was illustrated in La Description de l'Egypte in the late 1700's (Volume V, pl. 8). Flinders Petrie noticed a hollowing in the core masonry in the center of each face and wrote that he "continually observed that the courses of the core had dips of as much as ½° to 1°" (The Pyramids and Temples of Gizeh, 1883, p. 421). Though it is apparently more easily observed from the air, the concavity is measurable and is visible from the ground under favorable lighting conditions.

creased sides
Ikonos satellite image of the Great Pyramid.
Click to view larger image.

I.E.S. Edwards wrote, "In the Great Pyramid the packing-blocks were laid in such a way that they sloped slightly inwards towards the centre of each course, with a result that a noticeable depression runs down the middle of each face -- a peculiarity shared, as far as is known, by no other pyramid" (The Pyramids of Egypt, 1975, p. 207). Maragioglio and Rinaldi described a similar concavity on the pyramid of Menkaure, the third pyramid at Giza. Miroslav Verner wrote that the faces of the Red Pyramid at Dahshur are also "slightly concave."

concave sides
Diagram of the concavity (not to scale).

What was the purpose for concave Grea the first pyramid should hold true for the others."

concave sides
Three proposed "baselines" of the Great Pyramid (not to scale).

The purpose for the concavity of the Great Pyramids remains a mystery and no satisfactory explanation for this feature has been offered. The indentation is so slight that any practical function is difficult to imagine.

© 2000 by Larry Orcutt,  Catchpenny Mysteries, Reprinted with permission

The Great Pyramid's "Air Shafts"

Shafts and passages of the Great Pyramid at Giza.



Reply  Message 149 of 162 on the subject 
From: BARILOCHENSE6999 Sent: 22/04/2014 00:14

Reply  Message 150 of 162 on the subject 
From: BARILOCHENSE6999 Sent: 29/04/2014 16:30

Número áureo

El número áureo o de oro (también llamado razón extrema y media,[1] razón áurea, razón dorada, media áurea, proporción áurea y divina proporción [cita requerida]) representado por la letra griega φ (phi) (en minúscula) o Φ (Phi) (en mayúscula), en honor al escultor griego Fidias, es un número irracional:[2]

varphi = frac{1 + sqrt{5}}{2} approx 1,61803398874989...

El número áureo surge de la división en dos de un segmento guardando las siguientes proporciones: La longitud total a+b es al segmento más largo a, como a es al segmento más corto b.

También se representa con la letra griega Tau (Τ τ),[3] por ser la primera letra de la raíz griega τομή, que significa acortar, aunque encontrarlo representado con la letra Fi (Φ,φ) es más común.

Se trata de un número algebraico irracional (su representación decimal no tiene período) que posee muchas propiedades interesantes y que fue descubierto en la antigüedad, no como una expresión aritmética sino como relación o proporción entre dos segmentos de una recta; o sea, una construcción geométrica. Esta proporción se encuentra tanto en algunas figuras geométricas como en la naturaleza: en las nervaduras de las hojas de algunos árboles, en el grosor de las ramas, en el caparazón de un caracol, en los flósculos de los girasoles, etc.

Asimismo, se atribuye un carácter estético a los objetos cuyas medidas guardan la proporción áurea. Algunos incluso creen que posee una importancia mística. A lo largo de la historia, se ha atribuido su inclusión en el diseño de diversas obras de arquitectura y otras artes, aunque algunos de estos casos han sido cuestionados por los estudiosos de las matemáticas y el arte.




El número áureo es el valor numérico de la proporción que guardan entre sí dos segmentos de recta a y b (a más largo que b), que cumplen la siguiente relación:

La longitud total es al segmento a, como a es al segmento b.

Escrito como ecuación algebraica: frac{a+b}{a}=frac ab

Siendo el valor del número áureo φ el cociente frac ab

Surge al plantear el problema geométrico siguiente: partir un segmento en otros dos, de forma que, al dividir la longitud total entre la del segmento mayor, obtengamos el mismo resultado que al dividir la longitud del segmento mayor entre la del menor.

Cálculo del valor del número áureo[editar]

Dos números a y b están en proporción áurea si se cumple:

frac{a+b}{a}=frac ab

Si varphi es igual a frac ab entonces la ecuación queda:

 1 +  varphi^{-1} = varphi

Multiplicando ambos miembros por varphi, obtenemos:

 varphi + 1 = varphi^2

Igualamos a cero:

 varphi^2 - varphi - 1 = 0

La solución positiva de la ecuación de segundo grado es:

 frac{1+sqrt{5}}{2}=1	extrm{.}61803398874989ldots

que es el valor del número áureo, equivalente a la relación frac ab.

El número áureo en las matemáticas[editar]

Propiedades y representaciones[editar]

Ángulo de oro[editar]
{frac{360^circ}{varphi+{1}}} approx 137{,}5^circ razón número áureo

Propiedades aritméticas[editar]

varphi^2 = varphi + 1

La expresión anterior es fácil de comprobar:

varphi^2 = frac{ ( 1 + sqrt{5} )^2 }{2^2} = frac{1 + 2sqrt{5} + 5}{2^2} = frac{6 + 2sqrt{5}}{2^2} = frac{3 + sqrt{5}}{2}
varphi + 1 = frac{1 + sqrt{5}}{2} + frac{2}{2} = frac{3 + sqrt{5}}{2}
  • φ posee además las siguientes propiedades:
varphi - 1 = frac{1}{varphi}
varphi^3 = frac {varphi + 1} {{varphi - 1}}
  • Las potencias del número áureo pueden expresarse en función de una suma de potencias de grados inferiores del mismo número, establecida una verdadera sucesión recurrente de potencias.
El caso más simple es: Phi^n = Phi^{n-1}+Phi^{n-2},, cualquiera sea n un número entero. Este caso es una sucesión recurrente de orden k = 2, pues se recurre a dos potencias anteriores.
Una ecuación recurrente de orden k tiene la forma
a_1 u_{n+k-1}+a_2 u_{n+k-2}+...+a_k u_n,,
donde a_i, es cualquier número real o complejo y k es un número natural menor o igual a n y mayor o igual a 1. En el caso anterior es scriptstyle k=2,, scriptstyle a_1 = 1, y scriptstyle a_2 = 1,.
Pero podemos «saltar» la potencia inmediatamente anterior y escribir:
Phi^n = Phi^{n-2} + 2 Phi^{n-3} + Phi^{n-4},. Aquí scriptstyle  k = 4,, scriptstyle  a_1 = 0,, scriptstyle a_2 = 1,, scriptstyle a_3 = 2, y scriptstyle a_4 = 1,.
Si anulamos a las dos potencias inmediatamente anteriores, también hay una fórmula recurrente de orden 6:
Phi^n = Phi^{n-3} + 3 Phi^{n-4} + 3 Phi^{n-5} + Phi^{n-6},
En general:
Phi^n = sum_{i=0}^{	extstyle frac {1}{2} k}{	extstyle   frac{1}{2}kchoose i}Phi^{left [	extstyle n-left(	extstyle frac{1}{2}k+i ight) ight]}	extstyle;k=2jin mathbb{N},	extstyle, nin mathbb{N},	extstyle, iin mathbb{N}.
En resumen: cualquier potencia del número áureo puede ser considerada como el elemento de una sucesión recurrente de órdenes 2, 4, 6, 8,..., 2k; donde k es un número natural. En la fórmula recurrente es posible que aparezcan potencias negativas de Phi,, hecho totalmente correcto. Además, una potencia negativa de Phi, corresponde a una potencia positiva de su inverso, la sección áurea.
Este curioso conjunto de propiedades y el hecho de que los coeficientes significativos sean los del binomio, parecieran indicar que entre el número áureo y el número e hay un parentesco.
  • El número áureo frac{sqrt{5} + 1}{2} es la unidad fundamental «ε» del cuerpo de números algebraicos mathbb{Q}left(sqrt{5} ight) y la sección áurea frac{sqrt{5} - 1}{2} es su inversa, «varepsilon^{-1}». En esta extensión el «emblemático» número irracional sqrt{2} cumple las siguientes igualdades:

Representación mediante fracciones continuas[editar]

La expresión mediante fracciones continuas es:

varphi = 1 + frac{1}{varphi} quad longrightarrow quad varphi = 1 + frac{1}{1 + frac{1}{1 + frac{1}{1 + frac{1}{1 + ...}}}}

Esta iteración es la única donde sumar es multiplicar y restar es dividir. Es también la más simple de todas las fracciones continuas y la que tiene la convergencia más lenta. Esa propiedad hace que además el número áureo sea un número mal aproximable mediante racionales que de hecho alcanza el peor grado posible de aproximabilidad mediante racionales.[5]

Por ello se dice que varphi es el número más alejado de lo racional o el número más irracional. Este es el motivo por el cual aparece en el teorema de Kolmogórov-Arnold-Moser.

El número áureo en la geometría[editar]

El tríangulo de Kepler:
varphi^2 = varphi + 1;

El número áureo y la sección áurea están presentes en todos los objetos geométricos regulares o semiregulares en los que haya simetría pentagonal, que sean pentágonos o que aparezca de alguna manera la raíz cuadrada de cinco.

  • Relaciones entre las partes del pentágono.
  • Relaciones entre las partes del pentágono estrellado, pentáculo o pentagrama.
  • Relaciones entre las partes del decágono.
  • Relaciones entre las partes del dodecaedro y del icosaedro.

El rectángulo áureo de Euclides[editar]

Euclides obtiene el rectángulo áureo AEFD a partir del cuadrado ABCD. El rectángulo BEFC es asimismo áureo.

El rectángulo AEFD es áureo porque sus lados AE y AD están en la proporción del número áureo. Euclides, en su proposición 2.11 de Los elementos, obtiene su construcción.>

 GC = sqrt{5}

Con centro en G se obtiene el punto E, y por lo tanto:


con lo que resulta evidente que

 AE = AG + GE = 1 + sqrt{5}

de donde, finalmente,

frac{AE}{AD} = frac{1 + sqrt{5}}{2}= varphi

Por otra parte, los rectángulos AEFD y BEFC son semejantes, de modo que este último es asimismo un rectángulo áureo.

Generación de un rectángulo áureo a partir de otro.

En el pentagrama[editar]

Los segmentos coloreados del pentagrama poseen proporciones áureas.

El número áureo tiene un papel muy importante en los pentágonos regulares y en los pentagramas. Cada intersección de partes de un segmento se interseca con otro segmento en una razón áurea.

El pentagrama incluye diez triángulos isóceles: cinco acutángulos y cinco obtusángulos. En ambos, la razón de lado mayor y el menor es φ. Estos triángulos se conocen como los triángulos áureos.

Teniendo en cuenta la gran simetría de este símbolo, se observa que dentro del pentágono interior es posible dibujar una nueva estrella, con una recursividad hasta el infinito. Del mismo modo, es posible dibujar un pentágono por el exterior, que sería a su vez el pentágono interior de una estrella más grande. Al medir la longitud total de una de las cinco líneas del pentáculo interior, resulta igual a la longitud de cualquiera de los brazos de la estrella mayor, o sea Φ. Por lo tanto, el número de veces en que aparece el número áureo en el pentagrama es infinito al añadir infinitos pentagramas.

El teorema de Ptolomeo y el pentágono[editar]

Se puede calcular el número áureo usando el teorema de Ptolomeo en un pentágono regular.

Claudio Ptolomeo desarrolló un teorema conocido como el teorema de Ptolomeo, el cual permite trazar un pentágono regular mediante regla y compás. Aplicando este teorema, se forma un cuadrilátero al quitar uno de los vértices del pentágono, Si las diagonales y la base mayor miden b, y los lados y la base menor miden a, resulta que b2 = a2 + ab lo que implica:

{b over a}={{(1+sqrt{5})}over 2},.


El número áureo en el arte y en la cultura[editar]

En la representación del Hombre de Vitruvio Leonardo da Vinci no utiliza el número áureo, sino el sistema fraccionario propuesto por Vitruvio
  • Relaciones en la forma de la Gran Pirámide de Gizeh. La afirmación de Heródoto de que el cuadrado de la altura es igual a la superficie de una cara es posible únicamente si la semi-sección meridiana de la pirámide es proporcional al triángulo rectángulo left( 1,;sqrt{frac{sqrt{5} + 1}{2}},;frac{sqrt{5} + 1}{2} ight), donde 1 representa proporcionalmente a la mitad de la base, la raíz cuadrada del número áureo a la altura hasta el vértice (inexistente en la actualidad) y el número áureo o hipotenusa del triángulo a la apotema de la Gran Pirámide. Esta tesis ha sido defendida por los matemáticos Jarolimek, K. Kleppisch y W. A. Price (ver referencias), se apoya en la interpretación de un pasaje de Heródoto (Historiae, libro II, cap. 124) y resulta teóricamente con sentido, aunque una construcción de semejante tamaño deba contener errores inevitables a toda obra arquitectónica y a la misma naturaleza de la tecnología humana, que en la práctica puede manejar únicamente números racionales.




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