"All the World's a Stage We Pass Through" R. Ayana

Friday 23 March 2012

Sacred Geometry: How to Create a Multiverse

Sacred Geometry
How to Create a Multiverse

Sacred Geometry graphic by Craig Nicholson.

Mathematics is a wonderfully clean subject. It is not subject to opinion, nor does it rely on heated debates as to what the correct answer is. And geometry specifically is pure beauty! It is a great synthesizer, merging the linear, rational aspect of math through the left-side of the brain with the graphical, artistic aspect of pattern and beauty through the right brain.

It unites the mind and the heart (called the "intelligence of the heart" by the ancient Egyptians), spirit and matter, science and spirituality. These are all apparently separate halves of the Whole.

Plato said in the Republic (VII, 527 d, e) that it is through geometry that one purifies the eye of the soul, "since it is by it alone that we contemplate the truth." Mother nature uses geometry everywhere you look, from the spirals of the nautilus shell, sunflower centre and spiral galaxies to the hexagon symmetry of snowflakes, flower petals and honeycomb.

         Geometry in nature arranges the shapes of the molecules and crystals that make up our bodies and the physical cosmos. It is the key to the creation of the universe. A good primer on this subject is Michael Schneiders interesting book A Beginner's Guide to Constructing the Universe. In his introduction, he says: "It's a shame that children are exposed to numbers merely as quantities instead of qualities and characters with distinct personalities relating to each other in various patterns. If only they could see numbers and shapes as the ancients did, as symbols of principles available to teach us about the natural structure and process of the universe and to give us perspective on human nature."

         Studying and meditating on geometry helps us to better understand the laws, patterns and blueprints within nature. As we are part of nature, it can help us key in to and become more of our true selves, the magnificent spiritual beings who we really are. But if you are not into that, just studying the wonders and beauty of geometry is a healthy and thoroughly enjoyable past time anyway!

Sacred geometry has many diverse and extremely interesting aspects. We start with Pythagoras of Samos, who helped put sacred geometry "on the map", and then go logically through the process of creation into more specific areas of sacred geometry.


         People have understood the spiritual significance of geometry for ages, from the ancient Egyptians to the Vedas in early India. These were studied by Pythagoras, who helped immensely in putting sacred geometry "on the map" i.e. making it more generally known. Born around 580 BC in Samos, this amazing Greek philosopher, mathematician, and spiritual leader formulated principles that influenced the thought of Plato and Aristotle and contributed to the development of mathematics and Western rational philosophy. He founded the Pythagorean brotherhood in Cortona, a mystical school developed as the archetype of a golden-age educational community.

        They uncovered many of nature's secrets, and held that at its deepest level, reality is mathematical in nature. (Pythagoras once declared: "All is number." This theme has been repeated many times since, including by Bruce L. Cathie in his many books on harmonics, like The Bridge to Infinity.) They studied the mathematical laws behind the rhythm and harmony of music. They also pondered the intangible, not only by the logic of the mind but by the intuitive faculties of the heart. Pythagoras believed in life after death and reincarnation, and taught that the soul can transcend the cycles of rebirth and rise to union with the divine. Their community followed an ethical and moral code, and strove to put their spiritual knowledge into practical everyday use.

         John Strohmeier and Peter Westbrook say the following on page 66 of their book Divine Harmony: The Life and Teachings of Pythagoras (Berkley Hills Books, 1999): "For Pythagoras, mathematics was a bridge between the visible and invisible worlds. He pursued the study of mathematics not only as a way of understanding and manipulating nature, but also as a means of turning the mind away from the physical world, which he held to be transitory and unreal, and leading it to the contemplation of eternal and truly existing things that never vary. He taught his students that by focusing on the elements of mathematics, they could calm and purify the mind, and ultimately, through disciplined effort, experience true happiness."

         Unfortunately for Pythagoras (and for all of us since) there was much jealousy directed towards him and his community, plus resistance to his enlightened ideas. He and many of his followers were eventually killed, and his teachings largely destroyed. So much of what we know about him is from third-hand accounts, some of it being aimed at discrediting him. But his legacy and genius live on, and many after him have been inspired to study and elaborate on this wonderful subject.

Geometry and creation.


         The role of geometry in the process of creation is covered well by Robert Lawlor, in his fascinating book Sacred Geometry: Philosophy and Practice (Thames and Hudson, London, 1982). In it he says the following: "The perspective of volume offers yet another metaphor for the original and ever-continuing creative act of the materialization of Spirit and the creation of form. The very ancient creation myth coming from Heliopolis in Egypt gives an example of this mode of envisioning. Nun, the Cosmic Ocean, represents pure, undifferentiated spirit-space, without limit or form. It is prior to any extensive, any specificity, any god. It is pure potentiality. By the seed or will of the Creator, who is implicit within this Nun, the undifferentiated space is impelled to contract or coagulate itself into volume. Thus Atum, the creator, first creates himself or distinguishes himself from the undefinable Nun by volumizing, in order that creation might begin."

The Five Platonic Solids.

"What form, then, might this first volume have? What indeed are the most essential volumetric forms? There are five volumes which are thought to be the most essential because they are the only volumes which have all edges and all interior angles equal. They are the tetrahedron, octahedron, cube, dodecahedron and the icosahedron, and are the expressions in volume of the triangle, the square and the Pentagon: 3, 4, 5. All other regular volumes are only truncations of these five. These five solids are given the name 'Platonic' because it is assumed that Plato has these forms in mind in the Timaeus, the dialogue in which he outlines a cosmology through the metaphor of planar and solid geometry."

In this dialogue, which is one of the most thoroughly 'Pythagorean' of Platos works, he establishes that the five elements of the world are earth, air, fire, water and aether (prana). Lawlor continues: "Plato's fabricator of the universe created order from the primordial chaos of these elements by means of the essential forms and numbers. The ordering according to number and form on a higher plane resulted in the intended disposition of the five elements in the physical universe. The essential forms and numbers then act as the interface between the higher and lower realms. They have in themselves, and through their analogues with the elements, the power to shape the material world.

         "As Gordon Plummer notes in his book The Mathematics of the Cosmic Mind, the Hindu tradition associates the icosahedron with the Purusha. Purusha is the seed image of Brahma, the supreme creator himself, and as such this image is the map or plan of the universe. The Purusha is analogous to the Cosmic Man, the Anthropocosm of the western esoteric tradition. The icosahedron is the obvious choice for this first form, since all the other volumes arise naturally out of it."

The Hindu tradition associates the icosahedron with Purusha, the seed image of Brahma, the supreme creator himself.
The icosahedron generates the dodecahedron, representing Prakriti, the feminine power of creation of the matter (mother) universe. The star born within it's pentagon (below) produces the Golden Ratio, the energy of resurrection and symbol of the Christ.

The star born within the pentagon:

Patterns created from the moving 3-D Icosahedron lattice above:

The Golden Ratio

The golden ratio is the mathematical proportion 1:1.618034...ad infinitum. Often called by its Greek name phi, it is also known as the Divine Proportion, the Golden Ratio, the Golden Section and the Golden Mean. It is indelibly inscribed in the heart of nature and man. Trace the thread of this "golden mean" from daisy to pyramid, from pine cone to Parthenon. Or from seashell to spiral nebulae as described above. The cosmos weaves its garments with perfect integrity, laced by the golden ratio. It is so omnipresent that many philosophers, artists, mathematicians, and scientists have considered it an essential component of beauty, perhaps integral to life itself. Plato considered it the key to the physics of cosmos. The Egyptians thought it more than a number and believed it symbolized the creative process and the fire of life.

        This magical proportion is found by dividing a line (AC) at a particular point to yield two unequal sections, where the smaller one (AB) is proportionate to the larger one (BC) as the larger one (BC) is to the entire line (AC). The ratio is expressed as AB/BC = BC/AC. There is only one such point (B) on any line. It is easy to produce this ratio geometrically. One of many ways is the five-pointed star, as drawn within a pentagon above. Every outside line on the pentagon is in a phi ratio to any line of the star. Also, every line on the star has 3 segments. One side segment is in a ratio of 1:1.618 to the rest of that line. This ratio is what was used in the architecture of the great pyramids of Egypt, and results in the focusing of universal energy (a.k.a. prana) at a particular point, which helps in rejuvenation of life (or the preservation of dead bodies). This resurrection energy and ratio is the mark of the Christ, the victor over death and rejuvenating principle in nature.

Phi and the Great Pyramid.


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 (related to the Pythagorean Theorem). Using the 345 triangle is also the easiest way to draw a squared circle.

Visualizing crystal structures for higher consciousness.

If you really want to get into the spirituality and personal growth aspect of sacred geometry, it is recommended to meditate on the geometric structure of crystals. David Tame, in his fascinating book The Secret Power of Music, (Destiny Books, 1984) says the following: "On the subject of crystals, modern esotericists have often recommended crystals, pictures of crystals, and models of the molecular structure of crystals as a subject for meditation. Contemplation upon their geometry is said to provide a route by which the consciousness of man can attune itself to the various qualities of the Consciousness of the Supreme."

         Here is a related and rather challenging exercise. It is a message from the Ascended Lady Master Leto through the messenger Elizabeth Clare Prophet, April 15, 1976, found on an audio tape (#B7629). "You have been told .... to meditate upon the crystal and its structure. Now I ask you ... to select images of crystals and then to diagram the molecular structure of crystals and to begin to memorize these structures and to endow them with flame, and to realize that your altar is a place of change .... And therefore consider that the crystal, especially of rare gems and their molecular structure - and I meditate now with you upon the structure of the emerald - is a means of contacting the plane of Spirit. For these emanations of the mind of God manifest in matter are parallel to the energies manifest in Spirit. They are time-space coordinates of the coalescing of energy in infinity.

         "If you can therefore meditate upon, and endow your meditation with the flaming fire, this is what will take place: You will trace from effect to cause, to first cause, the very components of creation. And if you will study the treatise on alchemy by beloved Saint Germain, you will find that your visualizations will bring into manifestation elements of the Christ consciousness, abundance, the ability to expand a mandala for the community of the Holy Spirit."
(Quoted with permission. Copyright © Summit University Press. All rights reserved. Website: www.tsl.org)

         It is not that easy to find the molecular structure of crystals. To help you on your way, here is a link to an excellent website by the Institut Laue-Langevin (ILL) in France on The atomic structure of materials. This site allows you to obtain a 3D VRML model of structures by clicking on flagged links. If you do not have a 3D viewer on your computer, click the Help link at the top of their page. Once you have one, click on the “Gems and Minerals” link. Clicking on flagged links will allow you to navigate WITHIN and around these 3 dimensional crystal structures. Very cool! PS: If you have any ideas on how to “endow these structures with Flame”, please share them with us. Thanks.

Cymatics and Creation by Sound.

Now we come to one of nature's greatest mysteries, the creation of matter by sound. "And God SAID, let there be light." "In the beginning was the WORD, ... and the Word was God." Jesus and his followers obviously studied the ancient Hindu Vedas, written long before his time. They state: "In the beginning was Brahman with whom was the Word. And the Word was Brahman." The sacred texts of India teach that sound holds the key to the mysteries of the universe, to the creation and constant changing of our world. They put much emphasis on mantras, spoken prayers and sacred music. We in the West, constantly being bombarded by discordant sounds and music, are largely unaware of its tremendous power.

         The book Cymatics: A Study of Wave Phenomena & Vibration by Hans Jenny sums up the phenomena of the effect that sound has on matter. He developed techniques to show that every frequency and musical note has a particular geometric pattern that it forms, like this colour image on the book's cover. 
Cymatics book coverHe built on the work of Chladni, who created sand patterns on a metal disc through sound. You can find more on Cymatics at this AlphaOmega website. More insight into this subject is given by David Tame in his fascinating book The Secret Power of Music mentioned above. In his chapter The Physics of the Om, he says: "Complex and meaningful patterns are even more apparent in Jenny's sound-affected substances when viewed at the microscopic level. Then are revealed beautiful and mathematically-precise mandala-structures looking like groupings of microscopically-viewed snowflakes. The stress-interactions created in substances by their exposure to sound frequencies always result in formations replete with meaningful numerological, proportional and symmetrical qualities."

         He goes on to discuss the works of several researchers who prove that the whole-number ratios of musical harmonics correspond to an underlying numerical and geometric framework existing in nature and matter. This sheds light on Goethe's statement "Geometry is frozen music." The book also covers research on the effects of sound and music on the human body and psyche. In general, the findings show that classical type music promotes physical and mental health, whereas rock music has the opposite effect…

From Wholesome Balance  @ http://geometry.wholesomebalance.com/index.html & http://geometry.wholesomebalance.com/Sacred_Geometry_2.html

Sacred Geometry


Sacred Geometry Introductory Tutorial


by Bruce Rawles


In nature, we find patterns, designs and structures from the most minuscule particles, to expressions of life discernible by human eyes, to the greater cosmos. These inevitably follow geometrical archetypes, which reveal to us the nature of each form and its vibrational resonances. They are also symbolic of the underlying metaphysical principle of the inseparable relationship of the part to the whole.

It is this principle of oneness underlying all geometry that permeates the architecture of all form in its myriad diversity. This principle of interconnectedness, inseparability and union provides us with a continuous reminder of our relationship to the whole, a blueprint for the mind to the sacred foundation of all things created.


The Sphere


(charcoal sketch of a sphere by Nancy Bolton-Rawles)

Starting with what may be the simplest and most perfect of forms, the sphere is an ultimate expression of unity, completeness, and integrity. There is no point of view given greater or lesser importance, and all points on the surface are equally accessible and regarded by the center from which all originate. Atoms, cells, seeds, planets, and globular star systems all echo the spherical paradigm of total inclusion, acceptance, simultaneous potential and fruition, the macrocosm and microcosm.


The Circle


The circle is a two-dimensional shadow of the sphere which is regarded throughout cultural history as an icon of the ineffable oneness; the indivisible fulfillment of the Universe. All other symbols and geometries reflect various aspects of the profound and consummate perfection of the circle, sphere and other higher dimensional forms of these we might imagine.

The ratio of the circumference of a circle to its diameter, Pi, is the original transcendental and irrational number. (Pi equals about 3.14159265358979323846264338327950288419716939937511…) It cannot be expressed in terms of the ratio of two whole numbers, or in the language of sacred symbolism, the essence of the circle exists in a dimension that transcends the linear rationality that it contains. Our holistic perspectives, feelings and intuitions encompass the finite elements of the ideas that are within them, yet have a greater wisdom than can be expressed by those ideas alone.


The Point


At the center of a circle or a sphere is always an infinitesimal point. The point needs no dimension, yet embraces all dimension. Transcendence of the illusions of time and space result in the point of here and now, our most primal light of consciousness. The proverbial “light at the end of the tunnel” is being validated by the ever-increasing literature on so-called “near-death experiences”. If our essence is truly spiritual omnipresence, then perhaps the “point” of our being “here” is to recognize the oneness we share, validating all “individuals” as equally precious and sacred aspects of that one.

Life itself as we know it is inextricably interwoven with geometric forms, from the angles of atomic bonds in the molecules of the amino acids, to the helical spirals of DNA, to the spherical prototype of the cell, to the first few cells of an organism which assume vesical, tetrahedral, and star (double) tetrahedral forms prior to the diversification of tissues for different physiological functions. Our human bodies on this planet all developed with a common geometric progression from one to two to four to eight primal cells and beyond.

Almost everywhere we look, the mineral intelligence embodied within crystalline structures follows a geometry unfaltering in its exactitude. The lattice patterns of crystals all express the principles of mathematical perfection and repetition of a fundamental essence, each with a characteristic spectrum of resonances defined by the angles, lengths and relational orientations of its atomic components.


The Square Root of Two


The square root of 2 embodies a profound principle of the whole being more than the sum of its parts. (The square root of two equals about 1.414213562…) The orthogonal dimensions (axes at right angles) form the conjugal union of the horizontal and vertical which give birth to the greater offspring of the hypotenuse. The new generation possesses the capacity for synthesis, growth, integration and reconciliation of polarities by spanning both perspectives equally. The root of two originating from the square leads to a greater unity, a higher expression of its essential truth, faithful to its lineage.

The fact that the root is irrational expresses the concept that our higher dimensional faculties can’t always necessarily be expressed in lower order dimensional terms – e.g. “And the light shineth in darkness; and the darkness comprehended it not.” (from the Gospel of St. John, Chapter 1, verse 5). By the same token, we have the capacity to surpass the genetically programmed limitations of our ancestors, if we can shift into a new frame of reference (i.e. neutral with respect to prior axes, yet formed from that matrix-seed conjugation. Our dictionary refers to the word matrix both as a womb and an array (or grid lattice). Our language has some wonderful built-in metaphors if we look for them!


The Golden Ratio


The golden ratio (a.k.a. phi ratio a.k.a. sacred cut a.k.a. golden mean a.k.a. divine proportion) is another fundamental measure that seems to crop up almost everywhere, including crops. (The golden ratio is about 1.618033988749894848204586834365638117720309180…) The golden ratio is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion. As such, it symbolically links each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage.

The golden ratio (phi) has some unique properties and makes some interesting appearances:

  • phi = phi^2 – 1; therefore 1 + phi = phi^2; phi + phi^2 = phi^3; phi^2 + phi^3= phi^4; ad infinitum.

  • phi = (1 + square root(5)) / 2 from quadratic formula, 1 + phi = phi^2.

  • phi = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/…)))))

  • phi = 1 + square root(1 + square root(1 + square root(1 + square root(1 + square root(1 + …)))))

  • phi = (sec 72)/2 =(csc 18)/2 = 1/(2 cos 72) = 1/(2 sin 18) = 2 sin 54 = 2 cos 36 = 2/(csc 54) = 2/ (sec 36) for all you trigonometry enthusiasts.

  • phi = the ratio of segments in a 5-pointed star (pentagram) considered sacred to Plato and Pythagoras in their mystery schools. Note that each larger (or smaller) section is related by the phi ratio, so that a power series of the golden ratio raised to successively higher (or lower) powers is automatically generated: phi, phi^2, phi^3, phi^4, phi^5, etc.

  • phi = ratio of adjacent terms of the famous Fibonacci Series evaluated at infinity; the Fibonacci Series is a rather ubiquitous set of numbers that begins with one and one and each term thereafter is the sum of the prior two terms, thus: 1,1,2,3,5,8,13,21,34,55,89,144… (interesting that the 12th term is 12 “raised to a higher power”, which appears prominently in a vast collection of metaphysical literature)

The mathematician credited with the discovery of this series is Leonardo Pisano Fibonacci and there is a publication devoted to disseminating information about its unique mathematical properties, The Fibonacci Quarterly

Fibonacci ratios appear in the ratio of the number of spiral arms in daisies, in the chronology of rabbit populations, in the sequence of leaf patterns as they twist around a branch, and a myriad of places in nature where self-generating patterns are in effect. The sequence is the rational progression towards the irrational number embodied in the quintessential golden ratio.

This most aesthetically pleasing proportion, phi, has been utilized by numerous artists since (and probably before!) the construction of the Great Pyramid. As scholars and artists of eras gone by discovered (such as Leonardo da Vinci, Plato, and Pythagoras), the intentional use of these natural proportions in art of various forms expands our sense of beauty, balance and harmony to optimal effect. Leonardo da Vinci used the Golden Ratio in his painting of The Last Supper in both the overall composition (three vertical Golden Rectangles, and a decagon (which contains the golden ratio) for alignment of the central figure of Jesus.

The outline of the Parthenon at the Acropolis near Athens, Greece is enclosed by a Golden Rectangle by design.


The Square Root of 3 and the Vesica Piscis


The Vesica Piscis is formed by the intersection of two circles or spheres whose centers exactly touch. This symbolic intersection represents the “common ground”, “shared vision” or “mutual understanding” between equal individuals. The shape of the human eye itself is a Vesica Piscis. The spiritual significance of “seeing eye to eye” to the “mirror of the soul” was highly regarded by numerous Renaissance artists who used this form extensively in art and architecture. The ratio of the axes of the form is the square root of 3, which alludes to the deepest nature of the triune which cannot be adequately expressed by rational language alone.




This spiral generated by a recursive nest of Golden Triangles (triangles with relative side lengths of 1, phi and phi) is the classic shape of the Chambered Nautilus shell. The creature building this shell uses the same proportions for each expanded chamber that is added; growth follows a law which is everywhere the same. The outer triangle is the same as one of the five “arms” of the pentagonal graphic above.




Rotating a circle about a line tangent to it creates a torus, which is similar to a donut shape where the center exactly touches all the “rotated circles.” The surface of the torus can be covered with 7 distinct areas, all of which touch each other; an example of the classic “map problem” where one tries to find a map where the least number of unique colors are needed. In this 3-dimensional case, 7 colors are needed, meaning that the torus has a high degree of “communication” across its surface. The image shown is a “birds-eye” view.




The progression from point (0-dimensional) to line (1-dimensional) to plane (2-dimensional) to space (3-dimensional) and beyond leads us to the question – if mapping from higher order dimensions to lower ones loses vital information (as we can readily observe with optical illusions resulting from third to second dimensional mapping), does our “fixation” with a 3-dimensional space introduce crucial distortions in our view of reality that a higher-dimensional perspective would not lead us to?


Fractals and Recursive Geometries


There is a wealth of good literature on this subject; it’s always fascinating how nature propagates the same essence regardless of the magnitude of its expression…our spirit is spaceless yet can manifest aspects of its individuality at any scale.


Perfect Right Triangles


The 3/4/5, 5/12/13 and 7/24/25 triangles are examples of right triangles whose sides are whole numbers. The graphic above contains several of each of these triangles. The 3/4/5 triangle is contained within the so-called “King’s Chamber” of the Great Pyramid, along with the 2/3/root5 and 5/root5/2root5 triangles, utilizing the various diagonals and sides.


The Platonic Solids



The 5 Platonic solids (Tetrahedron, Cube or (Hexahedron), Octahedron, Dodecahedron and Icosahedron) are ideal, primal models of crystal patterns that occur throughout the world of minerals in countless variations. These are the only five regular polyhedra, that is, the only five solids made from the same equilateral, equiangular polygons. To the Greeks, these solids symbolized fire, earth, air, spirit (or ether) and water respectively. The cube and octahedron are duals, meaning that one can be created by connecting the midpoints of the faces of the other. The icosahedron and dodecahedron are also duals of each other, and three mutually perpendicular, mutually bisecting golden rectangles can be drawn connecting their vertices and midpoints, respectively. The tetrahedron is a dual to itself.

From April, 2000 through (at least) December, 2003, we conducted a poll on which Platonic Solid folks liked best; here’s the results from that period. If there’s interest, I’ll start another poll to see if the preferences has shifted!

Here are some animations of counter-rotating polyhedra and images of the Platonic solids showing their relationships as duals.

Here are fold-up patterns for the Platonic Solids.

The Archimedean Solids


There are 13 Archimedean solids, each of which are composed of two or more different regular polygons. Interestingly, 5 (Platonic) and 13 (Archimedean) are both Fibonacci numbers, and 5, 12 and 13 form a perfect right triangle.


Stellations of The Platonic Solids and The Archimedean Solids


This is a stellation of a dodecahedron where each pentagonal face is capped with a pentagonal pyramid composed of 5 golden triangles, a sort of 3-dimensional 5-pointed star.

Here are more images of polyhedra (Platonic and Archimedean Solids.)


Metatron’s Cube


Metatron’s Cube contains 2-dimensional images of the Platonic Solids (as shown above) and many other primal forms.


The Flower of Life


Indelibly etched on the walls of temple of the Osirion at Abydos, Egypt, the Flower of Life contains a vast Akashic system of information, including templates for the five Platonic Solids. The background graphic for this page is a repetitive hexagonal grid based on the Flower of Life.


From The Geometry Code @ http://www.geometrycode.com/sacred-geometry/

The Fibonacci Sequence

edieval mathematician and businessman Fibonacci (Leonardo Pisano) posed the following problem in his treatise Liber Abaci (pub. 1202):

How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?

It is easy to see that 1 pair will be produced the first month, and 1 pair also in the second month (since the new pair produced in the first month is not yet mature), and in the third month 2 pairs will be produced, one by the original pair and one by the pair which was produced in the first month. In the fourth month 3 pairs will be produced, and in the fifth month 5 pairs. After this things expand rapidly, and we get the following sequence of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

This is an example of a recursive sequence, obeying the simple rule that to calculate the next term one simply sums the preceding two:

F(1) = 1
F(2) = 1
F(n) = F(n – 1) + F(n – 2)

Thus 1 and 1 are 2, 1 and 2 are 3, 2 and 3 are 5, and so on.

This simple, seemingly unremarkable recursive sequence has fascinated mathematicians for centuries. Its properties illuminate an array of surprising topics, from the aesthetic doctrines of the ancient Greeks to the growth patterns of plants (not to mention populations of rabbits!). Consider, for example, the following diagram:

Here we have taken squares with sides whose lengths correspond to the terms of the Fibonacci sequence, and arranged them in an “outwardly spiraling” pattern. Notice that the rectangles which result at each stage are all roughly the same shape, that is, that the ratio of length to width seems to “settle down” as we build the pattern outward. Notice also that the ratio of length to width is at every step the ratio of two successive terms of the Fibonacci sequence, that is, the ratio of the greater one to the lesser. These ratios may be thought of as forming a new sequence, the sequence of ratios of consecutive Fibonacci numbers:

This sequence converges, that is, there is a single real number which the terms of this sequence approach more and more closely, eventually arbitrarily closely. We may discover this number by exploiting the recursive definition of the Fibonacci sequence in the following way. Let us denote the nth term of the sequence of ratios by xn, that is,

Then using the recursive definition of F(n) given above, we have:

Now supposing for the moment that the sequence converges to a real number x (a fact which requires proof, but we'll leave that aside), we may observe that both xn and xn - 1 have the same limit, that is,

Consequently, the real number x to which the sequence of ratios converges must satisfy the following equation:

This is a simple equation to solve for x: it is really a quadratic equation, and its positive root is the value we are looking for:

This number was known to the ancient Greeks and was called by them the Golden Mean. It is usually denoted by the Greek letter f (phi), and sometimes by m (mu). They believed that the proportion f:1 was the most most pleasing, indeed the aesthetically perfect proportion, and all of their artwork, sculpture, and especially architecture made use of this proportion. A rectangle whose sides had this proportion was called the Golden Rectangle. (And that is the shape being more and more closely approximated by our “spiralling rectangles” above.)

Whether or not you agree with the Greeks’ aesthetic judgment, it's a safe bet that Nature herself does:

The growth of this nautilus shell, like the growth of populations and many other kinds of natural “growing,” are somehow governed by mathematical properties exhibited in the Fibonacci sequence. And not just the rate of growth, but the pattern of growth. Examine the crisscrossing spiral seed pattern in the head of a sunflower, for instance, and you will discover that the number of spirals in each direction are invariably two consecutive Fibonacci numbers.

The Fibonacci sequence makes its appearance in other ways within mathematics as well. For example, it appears as sums of oblique diagonals in Pascal’s triangle:

From Math Academy @ http://www.mathacademy.com/pr/prime/articles/fibonac/index.asp

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  1. This will be the 3rd time I try to publish my thoughts, interesting 3 times 3 sides.. I have a long interesting story how I found your blog and why I stayed. The answer to Lady Leto's Flame, is your Qi energy. Meditate, focus on a specific crystal structure, and imbue it with your Flame. Some monk's chant into Chi or Qi state to presumably float or focus self-structural points of energy, so if you can generate Qi, you focus not out but in. Well seemed like the answer, and I found answers here, so thank you

  2. http://www.academia.edu/8991727/Phenomenal_World_as_an_Output_of_Cognitive_Quantum_Grid_Theory_of_Everything_using_Leibniz_Kant_and_German_Idealism

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