The Bible numerology code number 5

The number 5 in the Bible is significant because his creation, the ‘man’ has five fingers, five senses and five toes. Thus it is the number of God’s grace. There are five great mysteries: Father, Son, Spirit, Creation and Redemption. After the fall of man creation was cursed and it became subject to vanity. So man and creation needed to be redeemed therefore number 5 is the number of God’s grace. The number five may also speak of the inability of man and his weakness as only when ‘man’ is weak does he needs God’s Grace. As only if a man is incapable, he would require God’s Grace.

Number 4 signifies man’s weakness and helplessness and number 5 i.e. 4+1 shows divine strength added to man’s weakness and vanity which of course, means God’s grace. Gematria is the name given to the process of assigning numbers to different words. The gematria of the word Earth would be 296 which, in fact, is a multiple of 4 and the gematria of the word Heaven is 395, which is a multiple of 5. Therefore, Earth, which is the land of the man, is synonymous to his weakness and vanity and Heaven is synonymous of God’s grace. But what do you mean by grace? Grace means favor, an act of kindness and support and favor can be of many types. Favor that is done for the unworthy, favor that is done for someone who doesn’t deserve it, that favor is called grace. God called upon Abram and promised to make his name great. Was there any reason behind it? Did God do that for any purpose? No. He did it without any cause, without any particular reason. This was an instance of God’s grace towards his creation.

God changed Abram’s name to Abraham. This is very significant. Not because God chose Abram or because God endowed upon him his grace, but, the change was made by inserting in between his name the fifth letter of the alphabet h (Hey). Another instance of God’s grace can be seen when David went to fight with the enemy of Israel who was a giant named Goliath. David chose five smooth stones. These five stones were significant of David’s weakness supplemented by a divine power (the number of stones being 4+1). David used only one stone (one with divine power) and not the other four stones to defeat and conquer the mighty enemy.

But God’s Grace should not be confused with merit or warrant. The scriptures tell us again and again through various examples that the grace and merit cannot exist together. However one can’t attain salvation from merit. This is because if the salvation could be achieved through merit then, the there would not be any requirement for God’s Grace. The fifth book of Psalms exemplifies the same great fact. The first Psalm (Psalm 107) magnifies this, and explains to us, how "He sent His word and healed them" (verse 20), and again and again delivered his men out of all their trouble. The importance of number 5 is further exemplified by the Fifth Psalm of the First Book. It has a special reference to the God’ Grace and tells us that God encompasses us all in his grace. The Fifth Book of the Bible (Deuteronomy) magnifies the grace of God, and in it special pains, so to speak, are taken to emphasis the great fact that not for the sake of the people, but for God's own Name's sake had He called, and chosen, and blessed them. In the Bible, there is also a reference to The Stone Kingdom. This kingdom would be the FIFTH kingdom after the world powers when all of the earth’s dominions would be combined to one single kingdom and then the almighty would take over them to rule over the along with his appointed one with glory and with grace. There are also certain phrases which repeat five times in the bible and they speak of God’s Grace "Bless the Lord, O my soul." - All in Psalms 103 and 104, viz., 103:1, 2, 23 and 104:1, 35. The Talmud calls attention to this and says, "As God fills the earth and nourishes it, so He nourishes and blesses the soul."

The various examples stated above tell us a great deal about the significance of the number five in the biblical literature. It tells us through the Gematria that the importance of this number is huge as it signifies the grace of God. Also through the various examples of David, The Fifth Book of Psalms, The fifth kingdom etc that how again and again God has showed Grace to his creation. The number five is the symbol of God’s grace. Grace should never be confused with merit, because merit is only for the worthy. But unlike merit, grace is for everyone without any prejudice and preconception.

Numbers 5 - New International Version - Bible Gateway

Golden ratio

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This article is about the number. For the pop music album, see The Golden Ratio (album). For calendar dates, see Golden number (time).

Line segments in the golden ratio

A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship .

In mathematics and the arts, two quantities are in the golden ratio () if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The figure on the right illustrates the geometric relationship. Expressed algebraically:

where the Greek letter phi () represents the golden ratio. Its value is:

^[1]

At least since the 20th century, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing (see Applications and observations below). A golden rectangle can be cut into a square and a smaller rectangle with the same aspect ratio. Mathematicians since Euclid have studied the golden ratio because of its unique and interesting properties. The golden ratio is also used in the analysis of financial markets, in strategies such as Fibonacci retracement.

The golden ratio is often called the golden section (Latin: sectio aurea) or golden mean.^[2][3][4] Other names include extreme and mean ratio,^[5] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,^[6] golden number, and mean of Phidias.^[7][8][9]

Contents  [hide]

List of numbers Irrational and suspected irrational numbers γ ζ(3) √2 √3 √5 φ ρ δS α e π δ
Binary	1.1001111000110111011...
Decimal	1.6180339887498948482...
Hexadecimal	1.9E3779B97F4A7C15F39...
Continued fraction
Algebraic form
Infinite series

Two quantities a and b are said to be in the golden ratio φ if:

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,

it is shown that

Multiplying by φ gives

which can be rearranged to

Using the quadratic formula gives the only positive solution as

[edit] History

Mathematician Mark Barr proposed using the first letter in the name of Greek sculptor Phidias, phi, to symbolize the golden ratio. Usually, the lowercase form (φ) is used. Sometimes, the uppercase form (Φ) is used for the reciprocal of the golden ratio, 1/φ.^[10]

The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. According to Mario Livio:

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.^[11]

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. The Greeks usually attributed discovery of this concept to Pythagoras or his followers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.

Euclid's Elements (Greek: Στοιχεῖα) provides the first known written definition of what is now called the golden ratio: "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."^[5] Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e. the golden ratio.^[12] Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio.^[13] Some of these propositions show that the golden ratio is an irrational number.

The name "extreme and mean ratio" was the principal term used from the 3rd century BC^[5] until about the 18th century.

The modern history of the golden ratio starts with Luca Pacioli's De divina proportione of 1509, which captured the imagination^{[citation needed]} of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.

Michael Maestlin, first to publish a decimal approximation of the golden ratio, in 1597.

The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as "about 0.6180340," was written in 1597 by Michael Maestlin of the University of Tübingen in a letter to his former student Johannes Kepler.^[14]

Since the twentieth century, the golden ratio has been represented by the Greek letter Φ or φ (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ (tau, the first letter of the ancient Greek root τομή—meaning cut).^[2][15]

[edit] Timeline

Timeline according to Priya Hemenway.^[16]

• Phidias (490–430 BC) made the Parthenon statues that seem to embody the golden ratio.
• Plato (427–347 BC), in his Timaeus, describes five possible regular solids (the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron), some of which are related to the golden ratio.^[17]
• Euclid (c. 325–c. 265 BC), in his Elements, gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio" (Greek: ἄκρος καὶ μέσος λόγος).^[5]
• Fibonacci (1170–1250) mentioned the numerical series now named after him in his Liber Abaci; the ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically.
• Luca Pacioli (1445–1517) defines the golden ratio as the "divine proportion" in his Divina Proportione.
• Michael Maestlin (1550–1631) publishes the first known approximation of the (inverse) golden ratio as a decimal fraction.
• Johannes Kepler (1571–1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers,^[18] and describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." These two treasures are combined in the Kepler triangle.
• Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.
• Martin Ohm (1792–1872) is believed to be the first to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.^[19]
• Édouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
• Mark Barr (20th century) suggests the Greek letter phi (φ), the initial letter of Greek sculptor Phidias's name, as a symbol for the golden ratio.^[20]
• Roger Penrose (b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.

[edit] Aesthetics

De Divina Proportione, a three-volume work by Luca Pacioli, was published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. De Divina Proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.^[2] Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. Containing illustrations of regular solids by Leonardo da Vinci, Pacioli's longtime friend and collaborator, De Divina Proportione was a major influence on generations of artists and architects alike.^{[citation needed]}

[edit] Architecture

Many of the proportions of the Parthenon are alleged to exhibit the golden ratio.

The Parthenon's facade as well as elements of its facade and elsewhere are said by some to be circumscribed by golden rectangles.^[21] Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Midhat J. Gazalé says, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the Elements (308 BC) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties."^[22] And Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 BC, showed how to calculate its value."^[23] Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

A geometrical analysis of the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.^[24] They found ratios close to the golden ratio in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret.

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."^[25]

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.^[26]

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.^[27]

In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque.^[28]

[edit] Painting

The drawing of a man's body in a pentagram suggests relationships to the golden ratio.^[29]

The 16th-century philosopher Heinrich Agrippa drew a man over a pentagram inside a circle, implying a relationship to the golden ratio.^[29]

Leonardo da Vinci's illustrations of polyhedra in De divina proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings.^[30] But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by anything in Leonardo's own writings.^[31]

Salvador Dalí, influenced by the works of Matila Ghyka,^[32] explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.^[2][33]

Mondrian has been said to have used the golden section extensively in his geometrical paintings,^[34] though other experts (including critic Yve-Alain Bois) have disputed this claim.^[2]

A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).^[35] On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.^[36]

[edit] Book design

Depiction of the proportions in a medieval manuscript. According to Jan Tschichold: "Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section."^[37]

According to Jan Tschichold,^[38]

There was a time when deviations from the truly beautiful page proportions 2:3, 1:√3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.

[edit] Finance

The golden ratio and related numbers are used in the financial markets. It is used in trading algorithms, applications, and strategies. Some typical forms include: the Fibonacci fan, the Fibonacci arc, Fibonacci retracement, and the Fibonacci time extension.^[39]

[edit] Industrial design

Some sources claim that the golden ratio is commonly used in everyday design, for example in the shapes of postcards, playing cards, posters, wide-screen televisions, photographs, and light switch plates.^{[40][41][42][43]}

[edit] Music

Ernő Lendvaï analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,^[44] though other music scholars reject that analysis.^[2] In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.^[45] French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position."^[45]

The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section.^[46] Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.^[47] Also, many works of Chopin, mainly Etudes (studies) and Nocturnes, are formally based on the golden ratio. This results in the biggest climax of both musical expression and technical difficulty after about 2/3 of the piece.^{[citation needed]}

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.^[48]

[edit] Nature

A detail of an Aeonium tabuliforme in Trädgårdsföreningen, Göteborg

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.^[49] In connection with his scheme for golden-ratio-based human body proportions, Zeising wrote in 1854 of a universal law "in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form."^[50]

In 2010, the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.^[51]

Several researchers have proposed connections between the golden ratio and human genome DNA.^[52][53][54]

However, some have argued that many of the apparent manifestations of the golden mean in nature, especially in regard to animal dimensions, are in fact fictitious.^[55]

[edit] Optimization

The golden ratio is key to the golden section search.

[edit] Perceptual studies

Studies by psychologists, starting with Fechner, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.^[2][56]

[edit] Golden ratio conjugate

The negative root of the quadratic equation for φ (the "conjugate root") is

The absolute value of this quantity (≈ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, b/a), and is sometimes referred to as the golden ratio conjugate.^[10] It is denoted here by the capital Phi (Φ):

Alternatively, Φ can be expressed as

This illustrates the unique property of the golden ratio among positive numbers, that

or its inverse:

This means 0.61803...:1 = 1:1.61803....

[edit] Short proofs of irrationality

[edit] Contradiction from an expression in lowest terms

If φ were rational, then it would be the ratio of sides of a rectangle with integer sides. But it is also a ratio of sides, which are also integers, of the smaller rectangle obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely, so φ cannot be rational.

Recall that:

the whole is the longer part plus the shorter part;

the whole is to the longer part as the longer part is to the shorter part.

If we call the whole n and the longer part m, then the second statement above becomes

n is to m as m is to n − m,

or, algebraically

To say that φ is rational means that φ is a fraction n/m where n and m are integers. We may take n/m to be in lowest terms and n and m to be positive. But if n/m is in lowest terms, then the identity labeled (*) above says m/(n − m) is in still lower terms. That is a contradiction that follows from the assumption that φ is rational.

[edit] Derivation from irrationality of √5

Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If is rational, then is also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational.

[edit] Alternate forms

approximated of golden mean by infinite continued fractions

The formula φ = 1 + 1/φ can be expanded recursively to obtain a continued fraction for the golden ratio:^[57]

and its reciprocal:

The convergents of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ..., or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive Fibonacci numbers.

The equation φ² = 1 + φ likewise produces the continued square root, or infinite surd, form:

An infinite series can be derived to express phi:^[58]

Also:

These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a pentagram.

[edit] Geometry

Approximate and true golden spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of one square divided by that of the next smaller square is the golden ratio.

The number φ turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles.

There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem). However, a useful approximation results from dividing the sphere into parallel bands of equal area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ ≅ 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.^[59]

[edit] Dividing a line segment

Dividing a line segment according to the golden ratio

The following algorithm produces a geometric construction that divides a line segment into two line segments where the ratio of the longer to the shorter line segment is the golden ratio:

1. Having a line segment AB, construct a perpendicular BC at point B, with BC half the length of AB. Draw the hypotenuse AC.
2. Draw a circle with center C and radius B. This circle intersects the hypotenuse AC at point D.
3. Draw a circle with center A and radius D. This circle intersects the original line segment AB at point S. Point S divides the original segment AB into line segments AS and SB with lengths in the golden ratio.

[edit] Golden triangle, pentagon and pentagram

Golden triangle

[edit] Golden triangle

The golden triangle can be characterized as an isosceles triangle ABC with the property that bisecting the angle C produces a new triangle CXB which is a similar triangle to the original.

If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°-36°-108°.

Suppose XB has length 1, and we call BC length φ. Because of the isosceles triangles XC=XA and BC=XC, so these are also length φ. Length AC = AB, therefore equals φ+1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, and so AC also equals φ². Thus φ² = φ+1, confirming that φ is indeed the golden ratio.

Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to φ, while the inverse ratio is φ - 1.

[edit] Pentagon

In a regular pentagon the ratio between a side and a diagonal is (i.e. 1/φ), while intersecting diagonals section each other in the golden ratio.^[9]

[edit] Odom's construction

George Odom has given a remarkably simple construction for φ involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. This result is a straightforward consequence of the intersecting chords theorem and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer H. S. M. Coxeter who published it in Odom's name as a diagram in the American Mathematical Monthly accompanied by the single word "Behold!" ^[60]

[edit] Pentagram

A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

The golden ratio plays an important role in the geometry of pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is φ, as the four-color illustration shows.

The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.

[edit] Ptolemy's theorem

The golden ratio in a regular pentagon can be computed using Ptolemy's theorem.

The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are b, and short edges are a, then Ptolemy's theorem gives b² = a² + ab which yields

[edit] Scalenity of triangles

Consider a triangle with sides of lengths a, b, and c in decreasing order. Define the "scalenity" of the triangle to be the smaller of the two ratios a/b and b/c. The scalenity is always less than φ and can be made as close as desired to φ.^[61]

[edit] Triangle whose sides form a geometric progression

If the side lengths of a triangle form a geometric progression and are in the ratio 1 : r : r², where r is the common ratio, then r must lie in the range φ−1 < r < φ, which is a consequence of the triangle inequality (the sum of any two sides of a triangle must be strictly bigger than the length of the third side). If r = φ then the shorter two sides are 1 and φ but their sum is φ², thus r < φ. A similar calculation shows that r > φ−1. A triangle whose sides are in the ratio 1 : √φ : φ is a right triangle (because 1 + φ = φ²) known as a Kepler triangle.^[62]

[edit] Golden triangle, rhombus, and rhombic triacontahedron

One of the rhombic triacontahedron's rhombi

All of the faces of the rhombic triacontahedron are golden rhombi

A golden rhombus is a rhombus whose diagonals are in the golden ratio.^{[citation needed]} The rhombic triacontahedron is a convex polytope that has a very special property: all of its faces are golden rhombi. In the rhombic triacontahedron the dihedral angle between any two adjacent rhombi is 144°, which is twice the isosceles angle of a golden triangle and four times its most acute angle.^{[citation needed]}

[edit] Relationship to Fibonacci sequence

The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....

The closed-form expression (known as Binet's formula, even though it was already known by Abraham de Moivre) for the Fibonacci sequence involves the golden ratio:

A Fibonacci spiral which approximates the golden spiral, using Fibonacci sequence square sizes up to 34.

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by Kepler:^[18]

Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:

More generally:

where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when .

Furthermore, the successive powers of φ obey the Fibonacci recurrence:

This identity allows any polynomial in φ to be reduced to a linear expression. For example:

However, this is no special property of φ, because polynomials in any solution x to a quadratic equation can be reduced in an analogous manner, by applying:

for given coefficients a, b such that x satisfies the equation. Even more generally, any rational function (with rational coefficients) of the root of an irreducible nth-degree polynomial over the rationals can be reduced to a polynomial of degree n ‒ 1. Phrased in terms of field theory, if α is a root of an irreducible nth-degree polynomial, then has degree n over , with basis .

[edit] Symmetries

The golden ratio and inverse golden ratio have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations – this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps – they are reciprocals, symmetric about , and (projectively) symmetric about 2.

More deeply, these maps form a subgroup of the modular group isomorphic to the symmetric group on 3 letters, corresponding to the stabilizer of the set of 3 standard points on the projective line, and the symmetries correspond to the quotient map – the subgroup consisting of the 3-cycles and the identity fixes the two numbers, while the 2-cycles interchange these, thus realizing the map.

The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see Alternate forms above). It is, for that reason, one of the worst cases of Lagrange's approximation theorem and it an extremal case of the Hurwitz inequality for Diophantine approximations. This may be the reason angles close to the golden ratio often show up in phyllotaxis (the growth of plants).^{[citation needed]}

The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with φ:

The sequence of powers of φ contains these values 0.618..., 1.0, 1.618..., 2.618...; more generally, any power of φ is equal to the sum of the two immediately preceding powers:

As a result, one can easily decompose any power of φ into a multiple of φ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of φ:

If , then:

When the golden ratio is used as the base of a numeral system (see Golden ratio base, sometimes dubbed phinary or φ-nary), every integer has a terminating representation, despite φ being irrational, but every fraction has a non-terminating representation.

The golden ratio is a fundamental unit of the algebraic number field and is a Pisot–Vijayaraghavan number.^[63] In the field we have , where is the -th Lucas number.

The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is 4 ln φ.^[64]

[edit] Decimal expansion

The golden ratio's decimal expansion can be calculated directly from the expression

with √5 ≈ 2.2360679774997896964. The square root of 5 can be calculated with the Babylonian method, starting with an initial estimate such as xφ = 2 and iterating

for n = 1, 2, 3, ..., until the difference between x_n and x_n−1 becomes zero, to the desired number of digits.

The Babylonian algorithm for √5 is equivalent to Newton's method for solving the equation x² − 5 = 0. In its more general form, Newton's method can be applied directly to any algebraic equation, including the equation x² − x − 1 = 0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself,

for an appropriate initial estimate xφ such as xφ = 1. A slightly faster method is to rewrite the equation as x − 1 − 1/x = 0, in which case the Newton iteration becomes

These iterations all converge quadratically; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute n digits of the golden ratio is proportional to the time needed to divide two n-digit numbers. This is considerably faster than known algorithms for the transcendental numbers π and e.

An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers F ₂₅₀₀₁ and F ₂₅₀₀₀, each over 5000 digits, yields over 10,000 significant digits of the golden ratio.

The golden ratio φ has been calculated to an accuracy of several millions of decimal digits (sequence A001622 in OEIS). Alexis Irlande performed computations and verification of the first 17,000,000,000 digits.^[65]

[edit] Pyramids

A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid's apothem (a), semi-base (b), and height (h); the face inclination angle is also marked. Mathematical proportions b:h:a of and and are of particular interest in relation to Egyptian pyramids.

Both Egyptian pyramids and those mathematical regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios.

[edit] Mathematical pyramids and triangles

A pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semi-base (half the base width) is sometimes called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem. The height of this pyramid is times the semi-base (that is, the slope of the face is ); the square of the height is equal to the area of a face, φ times the square of the semi-base.

The medial right triangle of this "golden" pyramid (see diagram), with sides is interesting in its own right, demonstrating via the Pythagorean theorem the relationship or . This "Kepler triangle"^[66] is the only right triangle proportion with edge lengths in geometric progression,^[62] just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827... degrees (51° 49' 38").^[67]

A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (the source of a large part of modern knowledge of ancient Egyptian mathematics), based on the 3:4:5 triangle;^[68] the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes).^[69] The slant height or apothem is 5/3 or 1.666... times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers,^[70] and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.^[68]

Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 2π times the height, or h:b = 4:π. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the Kepler triangle. This pyramid relationship corresponds to the coincidental relationship .

Egyptian pyramids very close in proportion to these mathematical pyramids are known.^[69]

[edit] Egyptian pyramids

In the mid-nineteenth century, Röber studied various Egyptian pyramids including Khafre, Menkaure and some of the Giza, Sakkara, and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors identified as the Kepler triangle; many other mathematical theories of the shape of the pyramids have also been explored.^[62]

One Egyptian pyramid is remarkably close to a "golden pyramid"—the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the "golden" pyramid inclination of 51° 50' and the π-based pyramid inclination of 51° 51'; other pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° 47')^[68] are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accident remains open to speculation.^[71] Several other Egyptian pyramids are very close to the rational 3:4:5 shape.^[69]

Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ.^[72]

Michael Rice^[73] asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the golden ratio and that it is part of mathematics of the Pyramids, citing Giedon (1957).^[74] Historians of science have always debated whether the Egyptians had any such knowledge or not, contending rather that its appearance in an Egyptian building is the result of chance.^[75]

In 1859, the pyramidologist John Taylor claimed that, in the Great Pyramid of Giza, the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the secant of the angle θ.^[76] The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, Howard Vyse, according to Matila Ghyka,^[77] reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.

http://en.wikipedia.org/wiki/Golden_ratio

1. Y Jesús, seis días antes de la Pascua, vino á Bethania, donde estaba Lázaro, que había sido muerto, al cual había resucitado de los muertos.
2. E hiciéronle allí una cena y Marta servía, y Lázaro era uno de los que estaban sentados á la mesa juntamente con él.
3. Entonces María tomó una libra de ungüento de nardo líquido de mucho precio, y ungió los pies de Jesús, y limpió sus pies con sus cabellos: y la casa se llenó del olor del ungüento. (El nardo tiene fuerte CONTEXTO DE PAREJA HOMBRE-MUJER. Comparar con CANTAR DE LOS CANTARES)
4. Y dijo uno de sus discípulos, Judas Iscariote, hijo de Simón, el que le había de entregar:
5. ¿Por qué no se ha vendido este ungüento por trescientos dineros, y se dió á los pobres?
6. Mas dijo esto, no por el cuidado que él tenía de los pobres: sino porque era ladrón, y tenía la bolsa, y traía lo que se echaba en ella.
7. Entonces Jesús dijo: Déjala; para el día de mi sepultura ha guardado esto; (Observen que el rito de Maria con Jesucristo esta en un CONTEXTO SABATICO E INCLUSO CON EL INGRESO DE NUESTRO SEÑOR JESUCRISTO A JERUSALEM. Recordemos que JERUSALEM ES LA NOVIA DEL CORDERO)
8. Porque á los pobres siempre los tenéis con vosotros, mas á mí no siempre me tenéis.
9. Entonces mucha gente de los Judíos entendió que él estaba allí; y vinieron no solamente por causa de Jesús, mas también por ver á Lázaro, al cual había resucitado de los muertos. (La resurreccion de Lazaro fue en un septimo dia como ya se demostro en este panel)
10. Consultaron asimismo los príncipes de los sacerdotes, de matar también á Lázaro;
11. Porque muchos de los Judíos iban y creían en Jesús por causa de él.
12. El siguiente día, mucha gente que había venido á la fiesta, como oyeron que Jesús venía á Jerusalem, (Jerusalem, la ciudad. El cuarto mandamiento es el unico en que aparece el termino ciudad como fue demostrado en este panel. E incluso en apocalipsis 21 en un contexto sabatico se relaciona a la NUEVA JERUSALEM)
13. Tomaron ramos de palmas, y salieron á recibirle, y clamaban: ¡Hosanna, Bendito el que viene en el nombre del Señor, el Rey de Israel! (Expresion relacionada con EL septimo DIA DE LA FIESTA DE LOS TABERNACULOS EL 21 DE TISHRI, osea OSHANAH RABAH. COMPARAR CON APOCALIPSIS 21 Y SALMOS 118)
14. Y halló Jesús un asnillo, y se sentó sobre él, como está escrito: (El asno tiene relacion con el sabado. El 10 de nissan, la fecha en que Cristo ingreso a Jerusalem tiene fuerte nexo con el shabbat)
15. No temas, hija de Sión: he aquí tu Rey viene, sentado sobre un pollino de asna. (Aqui con la expresion "hija de Sion" se confirma NEXO CON MARIA MAGDALENA QUE SIGNIFICA TORRE. VER MIQUEAS 4:8)
16. Estas cosas no las entendieron sus discípulos de primero: empero cuando Jesús fué glorificado, entonces se acordaron de que estas cosas estaban escritas de él, y que le hicieron estas cosas. (La gloria esta relacionada con el shabbat. Ver Exodo 24 e incluso en las BODAS DE CANA en un septimo dia)
17. Y la gente que estaba con él, daba testimonio de cuando llamó á Lázaro del sepulcro, y le resucitó de los muertos.
18. Por lo cual también había venido la gente á recibirle, porque había oído que él había hecho esta señal;
19. Mas los Fariseos dijeron entre sí: ¿Veis que nada aprovecháis? he aquí, el mundo se va tras de él.
20. Y había ciertos Griegos de los que habían subido á adorar en la fiesta:
21. Estos pues, se llegaron á Felipe, que era de Bethsaida de Galilea, y rogáronle, diciendo: Señor, querríamos ver á Jesús.
22. Vino Felipe, y díjolo á Andrés: Andrés entonces, y Felipe, lo dicen á Jesús.
23. Entonces Jesús les respondió, diciendo: La hora viene en que el Hijo del hombre ha de ser glorificado.
24. De cierto, de cierto os digo, que si el grano de trigo no cae en la tierra y muere, él solo queda; mas si muriere, mucho fruto lleva.
25. El que ama su vida, la perderá; y el que aborrece su vida en este mundo, para vida eterna la guardará.
26. Si alguno me sirve, sígame: y donde yo estuviere, allí también estará mi servidor. Si alguno me sirviere, mi Padre le honrará.
27. Ahora está turbada mi alma; ¿y qué diré? Padre, sálvame de esta hora. Mas por esto he venido en esta hora.
28. Padre, glorifica tu nombre. Entonces vino una voz del cielo: Y lo he glorificado, y lo glorificaré otra vez.
29. Y la gente que estaba presente, y había oído, decía que había sido trueno. Otros decían: Angel le ha hablado.
30. Respondió Jesús, y dijo: No ha venido esta voz por mi causa, mas por causa de vosotros.
31. Ahora es el juicio de este mundo: ahora el príncipe de este mundo será echado fuera.
32. Y yo, si fuere levantado de la tierra, á todos traeré á mí mismo.
33. Y esto decía dando á entender de qué muerte había de morir.
34. Respondióle la gente: Nosotros hemos oído de la ley, que el Cristo permanece para siempre: ¿cómo pues dices tú: Conviene que el Hijo del hombre sea levantado? ¿Quién es este Hijo del hombre?
35. Entonces Jesús les dice: Aun por un poco estará la luz entre vosotros: andad entre tanto que tenéis luz, porque no os sorprendan las tinieblas; porque el que anda en tinieblas, no sabe dónde va. (La luz tiene relacion sabatica como fue demostrado en este panel)
36. Entre tanto que tenéis la luz, creed en la luz, para que seáis hijos de luz. Estas cosas habló Jesús, y fuése, y escondióse de ellos.
37. Empero habiendo hecho delante de ellos tantas señales, no creían en él.
38. Para que se cumpliese el dicho que dijo el profeta Isaías: ¿Señor, quién ha creído á nuestro dicho? ¿Y el brazo del Señor, á quién es revelado?
39. Por esto no podían creer, porque otra vez dijo Isaías:
40. Cegó los ojos de ellos, y endureció su corazón; Porque no vean con los ojos, y entiendan de corazón, Y se conviertan, Y yo los sane.
41. Estas cosas dijo Isaías cuando vió su gloria, y habló de él.
42. Con todo eso, aun de los príncipes, muchos creyeron en él; mas por causa de los Fariseos no lo confesaban, por no ser echados de la sinagoga.
43. Porque amaban más la gloria de los hombres que la gloria de Dios.
44. Mas Jesús clamó y dijo: El que cree en mí, no cree en mí, sino en el que me envió;
45. Y el que me ve, ve al que me envió.
46. Yo la luz he venido al mundo, para que todo aquel que cree en mí no permanezca en tinieblas.
47. Y el que oyere mis palabras, y no las creyere, yo no le juzgo; porque no he venido á juzgar al mundo, sino á salvar al mundo. (La expresion MIS PALABRAS esta resumida en los 10 mandamientos)
48. El que me desecha, y no recibe mis palabras, tiene quien le juzgue: la palabra que he hablado, ella le juzgará en el día postrero.
49. Porque yo no he hablado de mí mismo; mas el Padre que me envió, él me dió mandamiento de lo que he de decir, y de lo que he de hablar.
50. Y sé que su mandamiento es vida eterna: así que, lo que yo hablo, como el Padre me lo ha dicho, así hablo.

GALATAS 4

22 Porque escrito está que Abraham tuvo dos hijos; uno de la sierva, el otro de la libre.


23 Mas el de la sierva nació según la carne; pero el de la libre nació por la promesa. (Es curioso que SIERVA tiene fuerte nexo con el SALMO 119)


24 Las cuales cosas son dichas por alegoría: porque estas mujeres son los dos pactos; el uno ciertamente del monte Sinaí, el cual engendró para servidumbre, que es Agar.


25 Porque Agar ó Sinaí es un monte de Arabia, el cual es conjunto á la que ahora es Jerusalem, la cual sirve con sus hijos. (Jerusalem y el Monte Sinai de Arabia estan practicamente en una misma linea meridional)


http://www.my-forum.org/descripcion....=421269&pag=10
26 Mas la Jerusalem de arriba libre es; la cual es la madre de todos nosotros. (Sara, que es nuestra madre, es un tipo de Jerusalem de arriba. En contexto a JUAN 12 sabemos que MARIA MAGDALENA SIMBOLIZA A JERUSALEM. ¿PABLO ESOTERICAMENTE NOS DICE QUE ELLA TAMBIEN SUBIO AL CIELO Y QUE ES UN TIPO DE LA SUBIDA DE LA IGLESIA ?. Compare con Genesis 17:15,16)

4:27 Porque está escrito: Regocíjate, oh estéril, tú que no das a luz; Prorrumpe en júbilo y clama, tú que no tienes dolores de parto; Porque más son los hijos de las desolada, que de la que tiene marido. (Hay un obvio nexo con el JUBILEO/LIBERTAD)
4:28 Así que, hermanos, nosotros, como Isaac, somos hijos de la promesa. (El mensaje de ISAAC NEWTON, EN EL CONTEXTO A LA MANZANA Y EN UN CONTEXTO MERIDIONAL, EN EL CODIGO DA VINCI tiene relacion esoterica con este capitulo) 
4:29 Pero como entonces el que había nacido según la carne perseguía al que había nacido según el Espíritu, así también ahora.
4:30 Mas ¿qué dice la Escritura? Echa fuera a la esclava y a su hijo, porque no heredará el hijo de la esclava con el hijo de la libre.
4:31 De manera, hermanos, que no somos hijos de la esclava, sino de la libre.