Phi in the Bible

hat are Fibonacci numbers? Discovered by Leonardo Fibonacci (1170-1250 A.D.) who was born in Pisa, Italy, the Fibonacci sequence is an infinite sequence of numbers, beginning: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... where each number is the sum of the two preceding it. Thus: 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13 and so on. For any value larger than 3 in the sequence, the ratio between any two consecutive numbers is 1:1.618, or the Golden Ratio.

Known by the Greek letter phi, the Golden Ratio is an irrational number (i.e. one that cannot be expressed as the ratio or fraction of two whole numbers) with several curious properties. We can define it as the number that is equal to its own reciprocal plus one: phi = 1/phi + 1, with its value commonly expressed as 1.618,033,989. It's digits were calculated to ten million places in 1996, and they never repeat. It is related to Fibonacci numbers in that if you divide two consecutive numbers in the Fibonacci sequence, the answer is always an approximation of phi.

Also known as the Divine Proportion, the Golden Mean, or the Golden Section, this ratio is found with surprising frequency in natural structures as well as man-made art and architecture. where the ratio of length to width of approximately 1:1.618 is seen as visually pleasing. It is the point dividing a distance into two segments where the proportion of the smaller segment to the larger segment is the same as the larger to the whole.

Leaves On A Stem

If we look down on a plant, the leaves are often arranged so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.

The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one. If we count in the other direction, we get a different number of turns for the same number of leaves. One estimate is that 90 percent of all plants exhibit Fibonacci numbers in their leaf patterns.
Some common trees with their Fibonacci leaf arrangement numbers are:

1/2 elm, linden, lime, grasses
1/3 beech, hazel, grasses, blackberry
2/5 oak, cherry, apple, holly, plum, common groundsel
3/8 poplar, rose, pear, willow
5/13 pussy willow, almond

The numerator is # of turns and denominator is # of leaves til we arrive a a leaf directly under the starting leaf. Each leaf is this fraction of a turn after the last leaf. Or, there are so many turns (numerator) for every so many leaves (denominator).

Cactus's spines often show the same spirals as we see on pine cones, petals and leaf arrangements, but they are much more clearly visible.

Plants illustrate the Fibonacci series in the numbers and arrangements of petals, leaves, sections and seeds. Plants that are formed in spirals, such as artichokes, pinecones, pineapples, daisies and sunflowers, illustrate Fibonacci numbers.  Many plants produce new branches in quantities that are based on Fibonacci numbers.

Calla Lilies have one petal. Euphorbia have two petals. Lilies have three. Buttercups, Roses, Pinks and Geraniums have five, Delphiniums and Bloodroot have 8 petals, Marigolds and Ragworts have 13, Asters and Black-eyed Susans 21, Pyrethrum have 34, and most Daisies have 34, 55, or 89. You don't find any other numbers very often. (The main exceptions are when those same numbers appear doubled, or when the so-called anomalous series appears: 3, 4, 7, 11, 18 and so on.). The five-pointed star is unusually common in nature. Not only in flowers, but also in the starfish, sand dollar, apple, etcetera. Why? Because Phi is the ratio of the side of a pentagon to its diagonal (see picture).

The Sunflower, Pinecone, Daisy and Nautilus Shell

Furthermore, the SUNFLOWER has 21 spirals on its head in one direction and 34 going in the other -- consecutive Fibonacci numbers. Or 34 and 55, or 55 and 89, or 89 and 144. The precise numbers depend on the species of SUNFLOWER.The same is true with DAISIES. The outside of a PINECONE has spirals that run clockwise and counterclockwise, and the ratio of the number of spirals is sequential Fibonacci values. In the elegant curves of a NAUTILUS shell, each complete new revolution will be a ratio of 1:1.618, when compared to the distance from the center of the previous spiral. If the divergence angle of a SUNFLOWER varies from 137.5 degrees (seen in the pattern below) to either 137.4 or 137.6 degrees, much of the beauty of the spiral is lost as well as its most compact arrangement. The SUNFLOWER therefore is in a precise state of perfection. ROMANESQUE BROCCOLI/CAULIFLOWER is Fibonacci spiral upon spiral. The same can be said of any other plant or animal containing this Fibonacci sequence. Everything in nature is already in a state of perfection -- contrary to the theory of Evolution..Where are the transitional stages? Where are the vestigial organs? Where are the imperfect spirals? Who told a DAISY to have the same Fibonacci sequence as a NAUTILUS, or a PINECONE, or a SUNFLOWER, or a RAM'S HORN or even the appearance of a BREAKING WAVE? None of them have a brain. None of them are in any way related to the other. Design demands a Designer. Beauty demands a Master Artist.

The Human Head

Human beauty is based on the Divine Proportion. Many experiments have been carried out to prove that the proportions of the top models' faces conform more closely to the Golden Ratio than the rest of the population. The human face is based entirely on Phi .The head forms a golden rectangle with the eyes at its midpoint.  The mouth and nose are each placed at golden sections of the distance between the eyes and the bottom of the chin.  The ear reflects the shape of a Fibonacci spiral. Even the dimensions of our teeth are based on phi. The front two incisor teeth form a golden rectangle, with a phi ratio in the heighth to the width. The ratio of the width of the first tooth to the second tooth from the center is also phi. The ratio of the width of the smile to the third tooth from the center is phi as well.

The Human Finger Joints and Hand

Each segment of each finger, from the tip to the base of the wrist, is larger than the preceding one by about the Fibonacci ratio of 1.618. In other words, the smallest segment is 2, the next 3, the next 5, and the back of your hand is 8. By this scale, your fingernail is 1 unit in length. You also have 2 hands, each with 5 digits, and your 8 fingers are each comprised of 3 sections.  All Fibonacci numbers! Your hand creates a golden section in relation to your forearm, as the ratio of your forearm to your hand is also 1.618, the Divine Proportion. Furthermore, your hand and forearm together are a golden section in relation to the entire length of your arm.

The Human Body

The ventricles of the human heart reset themselves at the Golden Ratio point of the heart's rhythmic cycle.

Leonardo Da Vinci found that the total height of the human body, from toes to top of head, and the height from the toes to the navel depression are in Golden Ratio. Also the height from toes to navel and from the navel to the top of the head are in Golden Ratio. This has been confirmed by measuring 207 students at the Pascal Gymnasium in Munster Germany, where the perfect value of 1.618 was obtained for the first ratio average and 1.619 was obtained for the second. This value held for both girls and boys of similar ages ("Golden Mean of the Human Body" by T. Antony Davis and Rudolf Altevogt).

But there is more. From the top of the head to the bottom of the feet is the body's total height A. From the top of the head to the bottom of the fingertips is a golden section B of the total height A. From the top of the head to the navel and elbows is a golden section C of the golden section B. From the top of the head to the pectorals and armpits, the width of the shoulders, the length of the forearm and shinbone is a golden section D of the golden section C. From the top of the head to the base of the skull and the width of the abdomen is a golden section E of the golden section D. The width of the head and half the width of the chest and the hips is the golden section F of the golden section E.

The human body is also based on the number 5: The torso has 5 appendages, in the arms, legs and head., In turn, each of these has five appendages, in the fingers and toes and 5 openings on the face. and 5 senses in sight, sound, touch, taste and smell. The golden section is also based on 5, as the number phi, or 1.6180339 is computed from the square root of 5 multiplied by .5 plus .5 = Phi.

Relative Planetary Distances

The relative planetary distances average to Phi. The average of the mean orbital distances of each successive planet in relation to the one before it approximates phi: The mean distance is in million kilometers per NASA. If Mercury is 1, these are the relative mean distances Mercury 57.91 (1.00000) Venus 108.21 (1.86859) Earth 149.60 (1.38250) Mars 227.92 (1.52353) Ceres (the largest asteroid) 413.79 (1.81552) Jupiter 778.57 (1.88154) Saturn 1,433.53 (1.84123) Uranus 2,872.46 (2.00377) Neptune 4,495.06 (1.56488) and Pluto 5,869.66 (1.30580) The total is 16.18736 and so if we divide by 10, we get the average which is1.61874. Phi is1.61803.

The Fibonacci Series also predicts accurately the distances of the moons of Jupiter, Saturn and Uranus from each one's respective parent planet. Individual offsets can be attributed to planetary densities. Jupiter has 12 moons. Saturn has 9 moons and Uranus has 5 moons. For actual plotted graphs of these amazing results refer to the Fibonacci Quarterly October 1970, vol. 8, Number 4.

The Greatest Human Achievements Include the Golden Ratio

The dimensions of the King's Chamber of the Great Pyramid in Giza, Egypt are based upon the Golden Ratio. The Great Pyramid's vertical height and the width of any of its sides are in Golden Sections. The architect, Le Corbusier designed his Modulor system around the use of the ratio; the painter Mondrian based most of his work on the Golden Ratio; Leonardo Da Vinci included it in many of his paintings and Claude Debussy used its properties in his music. So did Bela Bartok. The Golden Ratio also is found in widescreen televisions, postcards, credit cards, and photographs which all commonly conform to its proportions. Like the brilliant Pythagoras before him, Leonardo had made an in-depth study of the human figure, showing how all of its major parts were related to the Golden Ratio. The Mona Lisa's face fits inside a perfect Golden Rectangle. The Golden Rectangle is found in his painting called The Last Supper also. Many other Renaissance artists did the same. After Leonardo, artists such as Raphael and Michelangelo made great use of the Golden Ratio to construct their works. Michelangelo's beautiful sculpture of David conforms to the Golden Ratio, from the location of the navel with respect to the height and placement of the joints in the fingers. The builders of the medieval and Gothic churches and cathedrals of Europe also made these structures conform to the Golden Ratio. The Golden Rectangle is one whose sides are in the proportion of the Golden Ratio. This means the longer side is 1.618 times longer than the shorter side. The front of the Parthenon in Athens, Greece can be framed within a Golden Rectangle. Da Vinci's drawing of the Vitruvian Man has the outlines of a rectangle based on the head, one on the torso, and another over the legs.

The Golden Ratio In The Bible

The Ark of the Covenant is a Golden Rectangle. 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.

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.

Exodus 27:1-2 mentions the dimensions of the altar -- constructed according to phi: "Build an altar of acacia wood, three cubits high; it is to be square, five cubits long and five cubits wide."

Furthermore, the location of Jerusalem is 31 degrees 45 minutes north of the equator. God said Jerusalem is "the city which I have chosen to put my name there" (1 Kings 11:36). Why did God select this location? First build a rectangular building in Jerusalem with sides that exhibit the golden rectangle ratio. The longer two sides (1.618) must run from east to west. The shorter two sides (1) must run from north to south. Then at each of the four corners place a flag pole. Make the roof flat. Remember that the sun rises in the east and sets in the west. Now if you draw one diagonal line from the northwest to the southeast corner, and another diagonal line from the northeast to southwest corner, you will create angles of 31 degrees 45 minutes with respect to the straight line joining the two southern corners of the building. At the summer solstice, the sun is over the Tropic of Cancer, 23.5 degrees north of the equator. At this point the shadows will point most easterly and westerly. At the winter solstice, the sun is over the Tropic of Capricorn, 23.5 degrees south of the equator. At this point the shadows will point most northerly. On the spring (March 21) and fall (Sept. 23) equinoxes, when the sun is directly over the equator, the shadows created by sunrise and sunset will fall exactly on these diagonal lines. This method can be used at any latitude with rectangles of different proportions. But a rectangle with the proportions of "the golden section " will only give this result at the exact latitude of Jerusalem. It is important to know when the equinoxes occur each year in order to celebrate God's festivals. We are to take the first new moon on or after the spring equinox as New Year's Day. No complex postponement rules or calculated calendar is necessary. Babylon is about one degree further north than Jerusalem. The pyramids of Gizeh are further south and Mecca still further south.

Fibonacci Numbers In The Musical Scale

The musical scales are based on Fibonacci numbers. The piano keyboard scale of C to C has 8 white keys in an "octave" and 5 black keys, making 13 keys total. The 5 black keys are divided into groups: of 2 keys and 3 keys. A scale is comprised of 8 notes, of which the 5th and 3rd notes create the basic foundation of all chords, and are based on whole tone which is 2 steps from the root tone, that is the1st note of the scale. Although there are only 12 notes in the scale, if you don't have a root and octave, a start and an end, you have no means of calculating the gradations in between, so this 13th note as the octave is essential to computing the frequencies of the other notes.  The word "octave" comes from the Latin word for 8, referring to the eight whole tones of the complete musical scale, which in the key of C are C-D-E-F-G-A-B-C.

In a scale, the dominant note is the 5th note of the major scale, which is also the 8th note of all 13 notes that comprise the octave.  This provides another instance of Fibonacci numbers in key musical relationships.  Interestingly, 8/13 is .61538, which approximates phi.  What's more, the typical three chord song in the key of A is made up of A, its Fibonacci & phi partner E, and D, to which A bears the same relationship as E does to A. This is analogous to the "A is to B as B is to C" basis for the golden section, or in this case "D is to A as A is to E."

Musical frequencies are based on Fibonacci ratios Notes in the scale of western music have a foundation in the Fibonacci series, as the frequencies of musical notes have relationships based on Fibonacci numbers: A440 Hertz is an arbitrary standard.  The American Federation of Musicians accepted the A440 as standard pitch in 1917.  It was then accepted by the U.S. government its standard in 1920 and it was not until 1939 that this pitch was accepted internationally.  Before recent times a variety of tunings were used.  It has been suggested by James Furia and others that A432 be the standard.  A432 was often used by classical composers and results in a tuning of the whole number frequencies that are connected to numbers used in the construction of a variety of ancient works and sacred sites, such as the Great Pyramid of Egypt.  The controversy over tuning still rages, with proponents of A432 or C256 as being more natural tunings than the current standard.

Posted on viernes, 15 de enero de 2010 20:55:47 by truthfinder9

Interesting. From this book (sorry, not all of the formulas converted right):

Mathematical Signatures in Nature: A Sign of Design?

“[The Universe] is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it...” – Galileo Galilei1

Math is the universal language, but it is not a human construct. Sure, we create symbols for numbers and mathematical computations, but math itself is more fundamental. 2 + 2 = 4 is universally true, universal in the sense that everywhere in the universe it is true. It does not matter what symbols we create to communicate “2 + 2 = 4” — the equation always remains true. In other words, what we call math is simply a way to describe or visualize the order that is the foundation of the universe’s structure and mechanics.

As we have discussed at length, such order cannot be produced by chance. The level of order is too sophisticated for a random cause. The patterns that are often revealed are too precise. Only intelligence produces such things. Let us look at some of the mathematical constructs that we have uncovered in nature.

Phi [M]

Phi (M = 1.6180339887...) is an irrational number like pi (p = 3.141592653...). Phi and pi are both ratios defined by particular Euclidean geometries, with phi being the division of a line “so that the ratio of the lesser part to the greater part is the same as the ratio of the greater part to the whole.2”

Phi’s abundance in the universe as earned it names such as the Golden Section, the Divine Proportion, the Golden Ratio and the Golden Mean. These names stem from the fact that phi can be found in many natural constructs such as in human and animal proportions (i.e. the arrangement of physical features). Phi relationships can be found in DNA, among the planets of the Solar System (as in Kepler’s Laws), and so on. Even in fractal geometry (used for the irregular geometries found in nature) we find phi in everything from coastlines to crystal formation.

Many argue that phi is also used by humans in such things as art, architecture and music for the balance it produces in designs. However, there are some caveats to this. For example, some claim one can find phi — not surprisingly — in the structure or design of the pyramids. Is phi intentional in the pyramids or merely the result of what is good geometrical design? It is probably the latter. Architecture is often designed with balance and stability which necessitates geometries that contain phi, though these geometries are not always necessary. Phi may be intentional in some structures, but it is accidental or inadvertent in others.

While there is always an amount of subjectivity in how one designs a structure, there is much more of this in artwork. Phi can be seen in certain art pieces, but many more do not exhibit phi. Art that uses geometry as a basis will naturally converge on phi (with phi not necessarily “encoded” in the painting by the artist). We can purposely incorporate phi or inadvertently do so through our use of the geometries we have discovered.

Either way we (intelligent designers) are using precise constructs. How do such precise relations appear repeatedly in nature?

Fibonacci Series

This is the Fibonacci series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. . .etc. It is very simply explained as each number in the series being the sum of the previous two numbers. The ratio of each successive pair of numbers (starting with 3/2) in the series approximates phi (i.e. 8 divided by 5 is 1.6). The accuracy of these ratios’ approximations to phi increases considerably as you go through the series. Phi can also be used to estimate any number in the Fibonacci series: fn = Mn / 5½.

Why is this important? Because the Fibonacci series precisely describes the spiral patterns common in nature: In shells, hurricanes, whirlpools, spiral galaxies, DNA and plant life. Phi is all around us. For example: The ratio of scales in the opposing spirals around a pine cone is 5:8; bumps on a pineapple are 8:13; seeds in a sunflower are 21:34. All of these ratios are adjacent pairs in the Fibonacci series.

Biblical Indications of Phi?

Exodus 25:10 writes: “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.” Here we find that “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.3”

God instructed Noah in Genesis 6:15 to build an ark this way: “This is how you are to build it: The ark is to be 450 feet long, 75 feet wide and 45 feet high.” Hence, 75 by 45 feet is also in the ratio of 5 to 3, or 1.666..., another “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.4”

These indications of phi may or may not have been intentional. However, they are more evidence of a trend for man to use the same logic embedded in the universe. As Gary Meisner writes, “The pervasive appearance of phi throughout life and the universe is believed by some to be the signature of God, a universal constant of design used to assure the beauty and unity of His creation.5”

Pi [p]

Pi is the ratio of the circumference of a circle to its diameter (p = 3.141592653...). Pi is just as fundamental as phi in the universe, but because it is more familiar, its presence does not seem as startling. Yet mathematicians have spent millennia computing its numbers and looking for patterns in the pi sequence. Perhaps not surprisingly, pi can be related to phi. One way is through trigonometric relations: 2 H cos (p / 5) = M and 2 H sin (p / 5) = Ö(3 - M). There are other relations as well.6

Pi in the Bible?

1 Kings 7:23 writes: “He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.” We know that circumference equals pi times diameter (c = pd). So from 1 Kings 7:23 we produce 30 = p10. Solving for p we get 3. Since p actually equals 3.141592653..., some people reason that the Bible is wrong. Is it?

First we must recognize that the Bible commonly uses rounded figures. These are descriptions, not architectural blueprints. Secondly, it has been shown by some that since the Hebrew does not have digits — all letters are also numbers — the relevant Hebrew in this passage can be calculated to find pi.7 The calculation comes out to 3.14150943... This is only a difference of 0.0000832 with actual pi, making the Bible’s description of pi the most accurate in antiquity.

Perhaps the Hebrews did not specifically calculate pi, but they managed to come very close “accidentally.” Another caveat is that many have tried to abuse the Hebrew and find endless “codes” hidden in the Bible. It can be shown how and why these codes are nonexistent8 or at least cannot be attributed the variety of meanings that some people claim the codes produce. While certain numbers, sequences, etc., in the Bible and elsewhere have been attributed particular meanings and usages, there is a big difference between what can be defined as pseudoscientific numerologies and scientific patterns. The latter are constant and repeatable (pi and phi) the former are inconsistent and variable (“Bible codes”).

Universal Language

So it seems mathematical patterns are inherent to the very structure of the universe. While we use our language of mathematics to describe these patterns and all of the precise fine-tuning found in the universe, the mathematical ratios themselves seem to be design evidences: A universal language, unchanging throughout time or place; too precise and nonnrandom to be products of chance.

We have also seen how biblical writings show examples of these mathematical constructs, both in usage and understanding. Once again the Bible shows a level of knowledge and accuracy that matches and surpasses other cultures contemporary to the Hebrews.

1. Mario Livio, The Golden Ratio (New York, NY: Broadway, 2003), pp. 241-242.

2. Miranda Lundy, Sacred Geometry (New York. NY: Walker, 1998), pp. 24-25.

For more on how phi is defined, see the websites “Phi: The Golden Number” at www.goldennumber.net by Gary Meisner, “The Golden section ratio: Phi” at www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html by Ron Knott and “Phi: That Golden Number” at jwilson.coe.uga.edu/emt669/Student.Folders/Frietag.Mark/Homepage/Goldenratio/goldenratio.html by Mark Freitag.

3. - 5. Gary Meisner’s “Phi in the Bible” at goldennumber.net/bible.htm.

6. Gary Meisner, “Pi, Phi and Fibonacci Numbers” at goldennumber.net/pi-phi-fibonacci.htm. More on the Fibonacci series in Ron Knott’s “Fibonacci Numbers and the Golden Section” at www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html.

7. Jochen Katz, “Pi in the Bible” at answering-islam.org.uk/Religions/Numerics/pi.html.

8. Randall Ingermanson, Who Wrote the Bible Code? A Physicist Probes the Current Controversy. Colorado Springs, Colorado: Waterbook, 1999. The author maintains additional information and appendices to his book on-line at www.rsingermanson.com. More on the Bible codes can be found at www.reasons.org/resources/apologetics/biblecode.shtml?main and at www.answering-islam.org.uk/Religions/Numerics/.

http://www.freerepublic.com/focus/f-religion/2429476/posts

from Mid-Atlantic Geomancy
 

If we want to talk with God/dess, experience has shown that it helps to be in the right environment. Spiritual seekers from Mayans through Christians, Native Americans, Egyptians and Hindus to the Neolithic builders of the stone rings in Britain and Ireland (and many more) found that by constructing their sacred places using certain geometrical ratios - just a small handful of them - they could more easily connect with their Maker.

Yes, it is possible to speak with our Creator anytime. However, sacred geometry makes this easier, and different ratios make different connections easier. The ratios have to do with different spiritual activities like healing, foretelling the future, long-distance communication, levitation and, most important, heightened ability to communicate with our Maker. These ratios help us to vibrate at the appropriate frequency to aid us in accomplishing the particular spiritual activity we have in mind.

Nearly every ancient archaeological site predating recorded history, from the Pyramids of Egypt and Mexico, to Stonehenge and beyond, employs mysterious mathematical alignments throughout their design.

These architectural formulas, rarely used today, are considered sacred and have also been found in the way they're arranged relation to each other and, most inexplicably, in the Monuments of Cydonia, and the Face on Mars.

When one looks at sacred enclosures globally, there is a group of five mathematical ratios that are found all over the world from Japan's pagodas to Mayan temples in the Yucatan, and from Stonehenge to the Great Pyramid. These ratios are:

• Square Root of Two   = 1.414...

• Square Root of Three = 1.732...

• Square Root of Five   = 2.236...

• Phi = 1.618...
  Phi is the Golden Section of the Greeks. It was said to be the first section in which the One became many.

• Pi   = 3.1416...
  Pi is found in any circle. If the diameter is 1, the circumference is 3.1416 (C = D).

These are all irrational numbers. Pi can be taken to 1500 decimal places with no discernable pattern to it (is that Chaos?). Let's take a closer look at each of these special numbers, and see how we can find them in the sacred geometry used by geomancers around the world. All five of these numbers gain their meaning only when beaten against the One. They are all ratios of x:1. The One is where it begins.

Pi - 3.1416 : 1 - the Circle

Square Root of Two - 1.414 : 1 - the Square

Square Root of Three - 1.732 : 1 - Vesica Pisces

The Vesica Pisces is created by two identical intersecting circles, the circumference of one intersecting the center of the other. The vulva-shaped space thus created is called the Vesica Pisces.

Square Root of Five - 2.236 : 1 - the Double Square

Another place where a double square is found is the Calendar II underground chamber site in central Vermont in the USA. It measures ten feet by twenty feet.

Phi - 1.618:1 - Ø - the Phi Rectangle

3:5 : : 5:8. This ratio indicates that it is part of this series: 1 . 2 . 3 . 5 . 8 . 13 . 21 . 34 . 55 . 89, and so on. This is called the Fibonacci Series. Start anywhere in the series, add the number below, and you get the next number (for example, 21 + 13 = 34). As one ascends up the series, any number in the series, when divided into the next one up, gets closer and closer to (but never hits exactly) 1.618, phi, the Golden Section.

Squaring the Circle - The Great Pyramid

Squaring the Circle - The Earth & the Moon

The Math:
 

Sacred Geometry
Geomancers are interested in sacred geometry because this is the study of the way that spirit integrates into matter - by echoing and amplifying the geometry of nature and planetary movements, we help to align the resonance of body/mind/spirit with the harmonic frequencies of the above and the below.

Geomancers are interested in sacred geometry because it has been found that certain spaces, with particular ratios, enable the participant to resonate or vibrate at the appropriate rate that maximizes the possibility of connection to the One.

A violin isn't built out of a cigar box! It is built with the proper wood with the proper shape and ratios, so that it resonates correctly for the notes/frequencies it is expected to produce. These same principles are applied to sacred spaces to maximize the possibility that whatever is being done there on spiritual levels will succeed.

Definitions
 

Two Dimensions
I've been a student of sacred geometry for over twenty-five years. While there has been recent interest in three-dimensional sacred geometry based on the Platonic Solids and in sacred sites themselves, most sacred geometrical documents I've read talk in only two dimensions - height and width.

Obviously there is a fourth dimension and others beyond it that are much more complex and sophisticated. But why does the record left to us from geomancers of the past come primarily in two dimensions?

Two is closer to the One than three is. It's less complex. I think one of the biggest mistakes Western geomancers have made was to take something that is very simple and make it much more complex. The Chartres Labyrinth strikes me as being an example of this. This stuff is simple. If you really gnow (that is, know both rationally and intuitively) a handful of irrational ratios - pi (), phi (Ø) and the square roots of two, three and five, you've basically got it all.

Three-dimensional sacred geometry just builds on this basic handful.

Numbers
One aspect of Sacred Geometry is that it works with irrational numbers. To go to the spiritual, one must go beyond the rational, and it appears that some of these ratios and numbers can lead us there. By being inside a sacred space that has been constructed using one of a handful of these sacred geometrical ratios, the resonance that has been set up can enhance the possibility of your making the spiritual connection you want to make.

So, what are these irrational numbers? Let's begin with the rational.

Rational Numbers
A rational number is a number which can be expressed as the ratio of two integers (whole numbers), such as 1/3 or 37/22. All numbers which, when represented in decimal notation, either stop after a finite number of digits or fall into a repeating pattern, are rational numbers.

Irrational Numbers
An irrational number is one that cannot be represented as a ratio of any two whole-number integers, and consequently it does not fall into a repeating pattern of any sort when written in decimal notation.

All of the Sacred Geometry ratios we will be working with, the square roots of two (1.414), three (1.732) and five (2.238), phi (1.618) and pi (3.1416), are all irrational numbers.

Transcendental Numbers
There are certain kinds of irrational numbers that are called transcendental numbers. Just like irrational numbers, they are defined by what they are not (they aren't rational numbers), yet transcendental numbers are so identified because they are not another sort of number, known as an algebraic number.

Any number which is a solution to a polynomial equation is an algebraic number. A polynomial equation is a sum of one or more terms involving the same variable raised to various powers, for example:
 

7 (x5) + 5 (x3) + x = 137

Any X for which any such equation is true is an algebraic number. Because the square root of two is a solution to the polynomial equation,

x2 = 2

it is an algebraic number.

A transcendental number requires an infinite number of terms to be defined exactly. That's one way of thinking of God/dess. There are special equations to derive transcendental numbers where the terms get smaller and smaller as you go along, so you can keep adding them together to reach any level of accuracy you need, but the true number cannot be reached exactly. That is the beauty of transcendental numbers!

Pi (= 3.1416...) is such a transcendental number. It is the only one we will be using here with Sacred Geometry. One infinite equation which relates to the value of pi () is this:

Pi / 4 = 1 - (1/3) + (1/5) - (1/7) + (1/9) - (1/11) + (1/13) - (1/15) + ...and so on into infinity. 
 http://www.bibliotecapleyades.net/geometria_sagrada/esp_geometria_sagrada_3.htm

Connections within the Mathematics and Calendrical Systems

of Ancient Mesoamerica

(Rev 12:8) And²⁵³² prevailed²⁴⁸⁰ not;³⁷⁵⁶ neither³⁷⁷⁷ was their⁸⁴⁸ place⁵¹¹⁷ found²¹⁴⁷ any more²⁰⁸⁹ in¹⁷²² heaven.³⁷⁷²

(Rev 12:9) And²⁵³² the³⁵⁸⁸ great³¹⁷³ dragon¹⁴⁰⁴ was cast out,⁹⁰⁶ that old⁷⁴⁴ serpent,³⁷⁸⁹ called²⁵⁶⁴ the Devil,¹²²⁸ and²⁵³² Satan,⁴⁵⁶⁷ which deceiveth⁴¹⁰⁵ the³⁵⁸⁸ whole³⁶⁵⁰ world:³⁶²⁵ he was cast out⁹⁰⁶ into¹⁵¹⁹ the³⁵⁸⁸ earth,¹⁰⁹³ and²⁵³² his⁸⁴⁸ angels³² were cast out⁹⁰⁶ with³³²⁶ him.⁸⁴⁶

In verse 7 we have the Archangel Michael contending in spiritual warfare against the dragon and his angels even as we see below in Daniel 10. In verse 9 we find four names of the adversary [dragon, old serpent, the Devil, and Satan], referring to four different aspects of the enemy’s spiritual dominion over the darkness of this world. We also see a fifth name-deceiver, which is a common trait in the other four names of the enemy. The Devil must deceive people into following his ways, because nobody in their right mind would ever voluntarily follow him if they knew his true nature. Thus deception is a universal trait of the enemy’s kingdom, and his mode of operation. This shows how the enemy has marshaled his spiritual forces of fallen angels in direct Opposition to the Godly order, reflected both in the curtains of Moses’ Tabernacle, and in the symbols of the Cardinal Directions. We have just seentwo examples of how the serpentine aspects of the enemy’s spiritual dominion are celestially represented in the Dragon-Draco and the Hydra. We will now look at Serpens the serpent, followed later by Cetus the sea monster.



One aspect of Serpens that arrests our attention is the recurring theme of the dragon’s head and tail, which we discussed above in connection with the serpentine asterism of Draco. This alerts us to the idea that, as the 4D symbolism of the Cardinal Directions paints a picture of composite truth, so their shadows do also. Just as each of the enemy’s 4 names in Rev. 12:9, indicate different functions in how the adversary opposes God’s purposes, so these serpents in the sky indicate varying aspects of his activity, which when taken together, provide us a detailed picture of how he works his evil designs in his world kingdom. Therefore we should consider these four serpentine signs both individually and collectively, as shadows of the cardinal points, to glean the fuller lesson of their message. As Robert Burnham states:

”Serpens is the only star group completely divided into two portions: Serpens Caput [head] to the west of Ophiuchus, and Serpens Cauda [tail] to the east.”¹³

In support of this structure, the stars within this Constellation follow suit. Burnham points out that “the Beta star in Serpens marks the head… while the Theta star marks the end of the serpent’s tail.”¹⁴ [ppg.1764-65] He goes on to highlight an intriguing link to Roman myth, where “Serpens is a symbol of the divine healer Aesculapius who became the god of medicine, and who appears in the sky as Ophiuchus the serpent bearer.” ¹⁵