It is the abstraction that obscures the beauty, both from musical notation, and more pertinently, from mathematics, two subjects inextricably entwined in the fabric of our reality. In the distant past the two subjects were considered part of the same science of nature, and for many years, certainly until the 17^th Century, music occupied its rightful place as a branch of mathematics, with both for example being taught in the earliest of English schools, as branches of the same fundamental discipline.

Pythagoras was the first recorded human to investigate music in a mathematical way, being extremely interested in the dependence of the length to the resulting pitch emitted by a vibrating string. He found that many of the most pleasing and harmonious sounds are produced by strings vibrating in simple ratios, such as 2:3:4, or 4:5:6. The latter ratio corresponding to the major triad, whose accord was explained by Helmholtz as the presence or absence of "beats" between the overtones of these notes. Pythagoras found that a string can be balanced by dividing it into two unequal parts whereby plucking each part produces the same note as the other, but in a different pitch, being an octave apart. This balancing point was determined to be two-thirds the length of the string, and if the whole string is first plucked, and then the two-thirds portion, the tones have a certain intrinsic balance that has been called the `Chord of Triumph'. This sound has a universal appeal to the human soul, but is a reflection not of the tones themselves, but of the relationship, or ratio between them. If we continually subdivide each two-thirds section of the string into further two-thirds, we produce a series of proportionately shorter strings and higher frequency notes, based upon the same harmonious sound.

Extrapolation of this effect produced the first musical scales, which were designed around a principle of `mediation', whereby two tones were bound together by a third placed in harmonious balance. For the Pythagoreans such harmony was not limited to musical intervals, but was intrinsically related to the world of nature, and the whole cosmos. They theorised that the planets in their celestial movements emitted harmonious sounds, which they termed `The harmony, or music of the Spheres', and taught that music is a reflection of the mathematics found in nature, expressed in whole number ratios. For the Pythagoreans then, the whole of nature was the visible manifestation of hidden number; and order and harmony, a balance of opposites that could be expressed as ratios.

But it is not the concurrence of these ratios that is the important point; it is that the universal appeal of them speaks of some deep psychological impulse, or effect within human consciousness. The mathematician H. E. Huntley believed that emotionally charged archaic memories were roused on a subconscious level by music, finding expression in the subconscious mind, as opposed to the conscious mind, which busied itself in language and gesture.  He further, almost intuitively, believed that the subconscious appreciation of ratio was not limited to music, but also found expression in mathematics, both accounting in part for our universal appreciation of beauty. More radically he theorised that the subconscious mind may become articulate in mathematics as well as in music.¹

A far-fetched theory some would say, and yet the subconscious correlation between mathematics and music would appear to be supported by empirical evidence. Researchers using brain-imaging techniques have tested various professional musicians and mathematicians, who were tasked with doing a variety of simple and complex procedures associated with their separate disciplines. Comparison of the brain images showed similar images, leading to the conclusion that both expert musicians and mathematicians, appear to be using the same brain circuitry. However when amateurs were tested, the same did not always hold true. ² There was something fundamentally different about the way expert mathematicians did maths, which most of us cannot replicate. Keith Devlin in `The Maths Gene' argues that we are all capable of appreciating higher mathematics, but are constrained by the abstraction of the notation, which requires a different type of reasoning, which in the majority lies dormant.

Whether this is in fact the case, is not germane to the issue, but there are however certain patterns in both mathematics and music which require only a minimum of artistic appreciation to stimulate some emotional response, being simple geometric shapes and patterns, together with simple musical intervals. Both are a reflection of beauty, with music finding expression in the harmonies engendered, whilst mathematical beauty is found in patterns. In both cases beauty is not intrinsic to the individual parts, which make up the whole, rather it is the interplay of the parts within the whole that accounts for its appeal. Music is often considered to be dynamic and mathematics static, but this is a false conclusion born of our ill developed `maths gene'. Mathematical notation is as static as Musical notation isolated from its reality. Both are mankind's way of describing changeable conditions in a fixed and immutable manner.

Mathematical symbols and Musical notes are both abstract notations divorced from time. By themselves they have no meaning, and only come to the fore within the mind of humanity, which converts them to a dynamic force by the application of periodicity. In the case of music, the estimation of pitch involves a mental measurement of time intervals referenced to the bodies internal clock or rhythm. The physical process involves pressure waves striking the eardrum to produce a stream of nerve signals, whose interval is interpreted by the brain. Middle `c' for example has a frequency of 256 cycles per second, and the octave above, exactly half the time interval of the fundamental characteristic of `c'.

Musical reality then, exists in our minds, but its almost universal appeal implies some deeper hidden truth of Pythagorean complexity. Some have suggested that mathematical patterns strike a similar accord in the human brain, working in a similar manner to the brains interpretation of music. If this were so then the appreciation of beauty in patterns would in some way involve the measurement of time. Experiments on number comparison support this idea in a broad way, having identified that almost all humans either consciously or subconsciously possess a mental number line with which they compare numbers. Put simply our minds view numbers as being positions on a line, usually, but not exclusively, horizontal, with 1 on the left and each successive number positioned to the right. Significantly these numbers are not positioned uniformly; the further we go along the `number line' the closer the spacing of the numbers. This positioning has a scientifically proven effect on our number reasoning ability. For example it has been demonstrated that subjects can decide that 5 is larger than 4 much faster than 53 is larger than 52. The explanation for this is that the gap between the smaller numbers is larger on our mental line enabling a quicker response.³

The minds' differentiation of number in certain cases would appear to be based on a comparison of length, which others theorise is directly comparable to musical appreciation, being a similar comparison of time intervals. This theory can be applied to geometric shapes and patterns, which are the building blocks of our reality. For example, to observe a rectangle, our brains make an unconscious comparison of the width to length, which sets in our mind the general shape and area of the rectangle, as a ratio of proportions. This of course takes a varying amount of time, with the mind taking longer to determine the length of the longest side. More pleasing shapes are formed, when the comparison of lengths has an almost `musical' resonance, or rhythm.

Musical notes and geometric shapes would both seem to impress themselves on the brain as a ratio of time intervals, and whilst this view is by no means universally accepted, it is strikingly similar to what the ancient Greeks believed.  To the Greek Pythagorean and Platonist geometers, rhythmically repeated proportions produced symmetry in all consciously composed plans. To the Greeks moreover, symmetry was not merely the balance of disposition either side of an axis of symmetry, it was a harmonious and rhythmic blending of all aspects of the design. The Roman commentator Vitruvius summarises this situation admirably. "Symmetry resides in the correlation by measurement between the various elements of the plan, and between each of these elements and the whole. .. . it proceeds from proportion—the proportion which the Greeks called analogia—(it achieves) consonance between every part and the whole . .. When every important part of the building is thus conveniently set in proportion by the right correlation between height and width, between width and depth, and when all these parts have also their place in the total symmetry of the building, we obtain eurhythmy." ⁴

For the Greeks, rhythm and eurhythmy, were general concepts dominating aesthetics, music and metaphysics, and cadence and number were parts of the same whole. Ideal design was as in nature, a repetition of the basic building blocks put together in a balanced and harmonious way, giving consonance of similar shapes. For them architecture was living music, whose notation was the ratio of the walls to one another, and the balance of enclosed areas. Periodicity and proportion only exist under the aegis of time, and thus the appreciation of ancient architecture can only be attempted in a fluid manner. There is no rhythm or canon of proportions in the observation that buildings are a collection of rectangles; concurrence comes from an appreciation of their dynamism and interplay. A situation summarised by Matila Ghyka who believed that there are rhythms in time and space that are produced by the dynamic action of proportion on a uniform beat or recurrence, that was exemplified by the Platonic conception of Proportion, Rhythm, and Beauty.⁵

Now to the Greeks, certain shapes had more dynamism and artistic appeal to the human psyche than others. One of the more pleasing forms is a rectangle of ratio 8 to 5, a fact demonstrated by the German psychologist Gustav Fechner, in a series of experiments in 1876, that have since been repeated and verified.⁶ Fechner made thousands of ratio measurements of common rectangular objects, such as windows, playing cards, book covers, etc and found that the average came close to an 8 to 5 ratio. He also established that the majority of people preferred a rectangle that lay between a square and double square in size, which pointed once more to this ratio. Why this particular rectangle is preferred is not truly understood, and because no seemingly rational explanation can be found, the subject is often dismissed as fantasy. It is difficult to believe however that the universal attraction of this rectangle is entirely coincidental and without substance, and as one scholar ably reminds us, the difficulty of accounting for phenomena does not invalidate their reality. Strikingly, it has also been demonstrated that the musical interval that gives the greatest pleasure to the most people, is the major sixth, whose frequency ratio is approximately 8:5. In both cases the brain interprets the two time intervals, one for the sound, the other for a stream of visual consciousness that appears to invoke some effect in that part of the brain that handles both music and mathematical appreciation. Whether this is by association, or some other effect, is not relevant, the point is that ratios of around 8 to 5 appear to have some inherent attraction for the human brain.

Just as the ancient Greeks, believed that the design of a musical scale involved the mediation of opposites by a third tone placed harmoniously between the two; they also believed that the same theory underpinned social harmony and discord. In seeking to find the perfection of harmony and the balancing of opposites, it has been theorised that the Greeks experimented with ratios until they found nature's perfect ratio, which approximates to 8:5. In quantifying, and qualifying this ratio we are drawn inexorably into the web of wonder that is the mathematical concept of Phi, for it is just this ratio that fascinated the ancient Greeks.

The Greeks found that there is one, and only one way of dividing a line so that the length of the entire line is proportioned to the large part in the same ratio that the large part is to the small part. Such a division is the perfection of harmony, and is called amongst other things, The Golden Mean. Nowadays it is usually called The Golden Cut, but in ancient times the terminology was more obscure, and its more common description was `Division in extreme and mean ratio .' It can be visually represented as a line, such as that drawn below.

If we divide the line AB at C in such a manner that the ratio of AB:AC is the same as the ratio AC:CB, we have divided the line in `extreme and mean ratio', (DEMR). The length AB in such ratios is termed Phi, and AC represents unity. Now changing a quantity by adding an amount results in arithmetic proportion, doing so by multiplying by some amount results in geometric proportion. In the above case the resulting proportion is both arithmetic and geometric and thereby represents perfection in proportion.

In strict mathematical terms phi + 1 = phi * phi.

Another property of Phi helps to explain its uniqueness in that it is the only number, which when diminished by unity, becomes its own reciprocal. ( phi – 1 =1/phi)

No other number demonstrates these relationships, which in part accounts for Phi's uniqueness, and importance within mathematics. The key point at the moment is that although nowadays we tend to think of Phi as an incommensurable mathematical number, (1.618033989…), at its most basic, it is also a geometric concept, the derivation of which is illustrated below.

That the ancient Greeks knew of the Golden Mean seems in little doubt, as two ancient Greek statements demonstrate. The first attributed to Cleobulus of Lindus, one of the Seven Wise Men of Ancient Greece, who lived in the sixth Century BC, quite succinctly states: `Avoid extremes—keep the Golden Mean'. <sup>7</sup> Another, attributed to Plato is coached in Platonic obscurity, but clearly refers to the same division: `Two things cannot be rightly put together without a third; there must be some bond of union between them. And the fairest bond is that which makes the most complete fusion of itself and the things which it combines; and proportion is best adapted to effect such a union.'⁸

Even if we disregard these ancient sources as being open to interpretation, then it seems to be of little doubt that DEMR was known to Euclid (c330 – 275 BC), as the mathematician Roger Herz-Fischler ably demonstrates in his book, `The mathematical History of Phi'⁹ Now whether we think of Phi as a number, a ratio, or a geometric construct, it has a remarkable ability to pop up in the most unexpected of places and permeates the very fabric of nature. Indeed it is so ubiquitous that it is hardly any wonder that the ancients were in awe of it; it would have been extraordinary had they not have been.

Take for example a simple addition series formed according to the rule that each term is the sum of the two preceding terms, whatever the first two terms may be. Such a sequence is called a Lucas sequence and can be represented as 

6, 3, 9, 12, 21, 33, 54, 87, 141, 228, …               from which we may determine approximations to Phi. of

...  33/21 =1.571  then 54/33 =1.636  then 87/54 =1.611  then 141/87 =1.6207  then 228/141 =1.61702 etc

This process gets closer and closer to the value of Phi, with each successive value oscillating above and below the actual value of Phi.¹⁰ Any sequence of numbers whatsoever , if formed in this manner will approximate to Phi. The simplest and therefore the purest of Lucas sequences is that formed by 0 and 1.

                                0, 1,1, 2, 3, 5, 8, 13, 21, 34, 55

Eduard Lucas called this sequence the Fibonacci sequence in 1877 in honour of the 13^th Century Italian merchant and mathematician Leonardo di Pisa, or Fibonacci. You will readily observe that two of the terms of this series, 5 and 8 have already been identified as having a special place in the mind of humanity, being the ratio of the most pleasing rectangle and the frequency ratio of the major sixth.

However it should be equally apparent that any two successive integers in this sequence would provide a rectangle whose ratio approximates to 8 to 5. It is not therefore the ratio of 5 to 8 that is the most pleasing rectangular shape; it is rather the hidden exactitude, which underpins it. To illustrate this consider the following demonstration of an old geometric fallacy provided by H. E. Huntley. First construct a square with length equal to the sum of two consecutive Fibonacci numbers. In this case 8 and 5 have been chosen. The square is divided up into the sections indicated, and then fitted together to form a rectangle. The areas of the square and rectangle differ by 1 unit, with the square giving the larger area. Choose the next two numbers, 8 and 13, and the result would have made the rectangle bigger by 1 unit. (The explanation of this paradox is that the fit along the diagonals of the rectangles is not exact. Sometimes there is a gap of 1 square unit and sometimes an overlap of 1 square unit).

There is however one series, and only one, that produces an exact fit, which makes use of the Golden Section, and is called the Golden Series. Using  as a symbol for phi it is written

                              1, , 1+1, 1+2, 2+3, 3+5, 5+8

Or it may also be written as

From this it can be seen that the rectangle, which most people find to have the most pleasing shape is any that approximates to two successive numbers in the Golden Series. Such a ratio inevitably becomes 1 to Phi , or 1 to 1.618033989…and this shape is called a Golden Rectangle, which is depicted below.

Many architects through the ages have made use of its properties, and strange appeal to the human psyche. The most famous example is contained within the Parthenon of Ancient Greece, which was begun in 447 BC on the orders of Pericles. His architects meticulously planed every facet of this strange structure, which is the best-executed example of an optical illusion in architecture. Even then, the Greeks were well aware of the frailties of the human eye, which `imagines' that a straight column is narrower in the middle than at the ends. The architects of the Parthenon, Ictinus, and Callicrates, designed each column with a slight central bulge, to counteract this. Further visual enhancements were built into the columns themselves, which almost imperceptibly angled inwards, and were designed to meet around two kilometres upwards, if extended into the sky. The horizontal `straight' lines are similarly bowed, and the whole effect is to mirror the perfection of the gods, by compensating for the frailty of the human mind. The least obvious manifestation of this frailty is that there are no straight lines in the Parthenon, with every line being curved almost imperceptibly to enhance the visual experience.

The Parthenon is one of a number of temples that make up the Acropolis. Its name means the `abode of the virgin', and was built as the home of the goddess Athena, whose 40 foot gilded statue, crafted by Phidias, stood in the inner sanctum. In fact the very name Phi, is taken from the Greek letter , the initial letter of Phidias's name, and was done so by convention early in the last Century, in honour of this great man. The entire Acropolis was conceived and designed around the Greek belief in sacred geometry, whose overt manifestation was elucidated by numerical symbolism. A central, east-west sun line runs through the centre of the Acropolis, and through a point east of the entrance structure, called the Propylea. From this point, lines of site based on the Greek twelve-part system serve to position all the structures on the Acropolis hill. This twelve-part system was based on a 360° circle divided into twelve equal 30° parts.  From this locus position, an observer would view the maximum possible side surface area of the Parthenon, which was positioned on the hill, to produce just this effect. The principal structures of the Acropolis according to Professor Demetrios Papageorgiou, of Imperial College, are further delimited by an equilateral triangle, which was believed by the Pythagoreans to be sacred to Athena. The Greeks were thereby continuing a tradition that I believe, was begun by the architects of the Egyptian Old Kingdom.

The following diagrams depict the plan, and frontal views of the Parthenon, and clearly show root three geometry that we have seen at Giza. The position of the sanctum walls, and offering tables, bear the stamp of geometric design.

The frontal view shows how the column height, and the entablature above, is determined by a clever combination of

Further indication that the Greeks were following the example of the ancient Egyptians, in utilizing root numbers in temple design.

It might be more convenient to dismiss the golden section as a man-made artificial construct, with only limited appeal to the mathematically obscure, and yet its presence in that most pleasing of rectangular shapes argues against it. Furthermore its mathematical elegance as a number has led many mathematicians to postulate that it is best considered as a constant of nature, like pi, c, the velocity of light in a vacuum, h, Planck's constant, and e, a letter denoting both the fundamental electric charge, and the base of natural logarithms. Such constants appear in the fabric of space-time and lie in a reality divorced from ours, and it is their effects and inter-relationships in our reality that we observe. Just as matter has now been rendered to a flow of activity by quantum mechanics, and matter its manifestation born of electrostatic charge, so these fundamental aspects of our underlying reality lie outside of our realm, only finding expression in patterns formed within our minds.

Such a view leads to the implication that beauty may well be both fundamental and a subject defying classification, being at its simplest, a construction of patterns by the application of time. The universal appeal of certain shapes may then be interpreted as common reaction to some inbuilt human biorhythm that quantifies pleasing shapes in an almost musical fashion. Such theories may account for the universality of beauty, being a common inbuilt construction process of the human brain. What these theories don't account for is the appearance of the golden section in nature, whose presence supports the belief that Phi is a universal constant, which nature uses as a building block of reality.

Many examples exist of the Fibonacci sequence in nature. It appears in the leaf and branch spacing patterns of many plants, the reproductive cycle of rabbits, and the genealogy of the drone bee. It appears in hidden disguise in the stock markets of the world, where one analyst discovered that the market unfolds in cyclical waves of 5 upward and 3 downward, making a complete cycle of 8 waves. It appears in music, being found in Bartok sonatas, Bach fugues, and Gregorian chants, and it appears frequently in poetry.¹² The most sophisticated example analysed to date is Virgil's Aeneid, which was analysed in the 1960's by Professor G. E. Duckworth of Princeton University, who discovered that the Fibonacci sequence was fundamental to its vocalisation.¹³

In my book The Geometry of The Sun God, I explain how the Kings Chamber height, and the Queen's Chamber

While in the Kings Chamber this value equates to half the diagonal of the plan view of the chamber, no similar arrangement applies in the Queens Chamber.

mathematical derivation. As there is little geometric logic to the builders' having used half of the square root of five to determine the Queens Chamber height, we must conclude that there is a strong probability that the pyramid builders were aware of the value and implications of Phi. If we now consider the cross sectional area of the Great Pyramid, as depicted below we can see that by using Phi, (once again denoted by the Greek letter ), we obtain a meaningful Pythagorean triangle that follows the general principle of the other two pyramids.

If we were to reconsider for a moment the question of Pi, then taking the base value as being 440 cubits, the pyramid height would need to be 280.113 cubits to provide an accurate value for Pi. The variation between these two differently produced heights is less than six inches, which has undoubtedly fuelled the debate over the presence of Pi in the Great Pyramid, as this is far less than that value given by Petrie as a tolerance. We can therefore see that by using Phi, we have a cohesive theory of cross sectional areas that fits all three pyramids, and provides us with another key to the geometric and mathematical puzzle that is the Giza Plateau.

So if we accept that the design of the three pyramids was geometric, expanded to 1000 times the square root of three, we are left with the problem of interpreting and explaining the individual sizes of the three pyramids. Adoption of such a geometric hypothesis precludes the possibility that the pyramids were randomly sized, but there is an infinite number of base and height combinations that can add up to 1000 times root three. What factors influenced their sizing? We have seen for example that the base length of the Great pyramid is around 440 cubits, but whether it was 438 or 442, or any other number close to this value, we just do not know.

Yet mathematically, at least two of the pyramids had to have had their base dimensions fixed, in order to calculate the base dimension of the third pyramid. You cannot randomly select three sets of base dimensions, apply the individual cross sectional triangle to obtain the height, and hope to get anywhere near the combination of 1000 root three. So which two pyramids were selected to calculate the third, and what was the design criterion behind their selection? To my mind it is marginally more probable that the first and the third pyramids were mathematically encoded, if for no other reason than they stand at the vertices of a

We must therefore look for a mathematical model that determines either the base lengths, or the heights of these pyramids.

The most obvious consideration is that the perimeter of the Great Pyramid is close to the value of one half minute of latitude of the Earth. But to accept this conclusion we have to conclude that the Egyptians first deduced that the Earth was round, and then devised a method of determination. This is not so simple as it sounds, but a clue to the feasibility of this assertion may lie in the nature of ancient Egyptian civilisation. Many ancient commentators, Herodotus and Plato being two of the more famous, have commented that the Egyptian priesthood were the most consummate of record keepers.

Time and again we find references to the earlier records of their ancestors, with the unspoken assumption being that there was a continuous unbroken line of detail stretching back into antiquity. It is known that at some early date in their history the Egyptians switched from a lunar calendar to a solar one, and it is certainly suggestive that the obelisks they erected would cast long and easily measurable shadows. Had the Egyptian priests systematically recorded the lengths of the shadows cast by obelisks of the same height, in different parts of the country, it would have been easily possible for anyone scrutinizing such records to notice that noon shadow lengths become progressively longer the further north you went.

Neugebauer in his History of Ancient Mathematical Astronomy suggests that the Greeks may have deduced that the Earth was spherical by rationalizing the variation of the Celestial Pole with this variation in the sun's shadow.¹⁴ If the Greeks could do it with their limited experience of practical astronomical observation, then both the Egyptians, and the Babylonians would have been better suited to such a deduction. However deducing the Earth is round is a long way from calculating its size. To do this you require the ability to accurately measure the angular distances between two points, and their separation in a direct north-south line. Various people have suggested methods, by which this could have been done, but the proof is not in the hypothesis, it rests on the measure obtained, if any. The fact that the perimeter of the Great Pyramid provides a reasonably good estimate of the circumference of the Earth does not necessarily indicate this was the intention.

In fact there is a mathematical relationship that whether by accident or design, can be used to predict the sizing of the great Pyramid, and has the added benefit that it requires only two variables, the square root of two, and Phi, (represented by the Golden Series), to produce the intended dimension. Moreover as this model is a combination of geometric progression and mathematics, two subjects that we have seen the Egyptians had a talent for; its probability is greater than the Earth circumference hypothesis.

Because of the necessity to meet the requirements of the triangular arrangement depicted in the cross-sectional diagram of Khufu, there is only one combination of values that satisfies both this equation, and the triangular cross sectional area.

These values are 440.09148 cubits for the base and 279.90249 cubits for the height. So if this mathematical relationship of the square root of two to the pyramids volume is not just mere coincidence, we have just determined the exact design dimensions of the Great Pyramid.¹⁵ However I would still urge caution in accepting this solution, as the model cannot be applied in the same format to either of the other two pyramids. However we have seen that it was also necessary to size one of the other two pyramids, in order to calculate the dimensions for the remaining pyramid. The clue to solving this enigma is once again Phi, which leads us into new and interesting territory.

1 The Divine Proportion, A Study in Mathematical Beauty, H. E. Huntley, Dover Publications, NY, 1970, p 79

² The Maths Gene, Keith Devlin, Phoenix, 2001, p 77

³ The Maths Gene, Keith Devlin, Phoenix, 2001, p 57

⁴ The Geometry of Art and Life, Matila Ghyka, Sheed & Ward, 1966, p 4

⁵ The Geometry of Art and Life, Matila Ghyka, Sheed & Ward, 1966, pp 5, 149

⁶ They were repeated by Witmar (1894), Lalo (1908) and Thorndike (1917). The results were similar in each case.

⁷ A Beginner's Guide to Constructing the Universe, The mathematical archetypes of Nature, Art, and Science, Michael S. Schneider, Harper Perennial, 1995, p 109

⁸ A Beginner's Guide to Constructing the Universe, The mathematical archetypes of Nature, Art, and Science, Michael S. Schneider, Harper Perennial, 1995, p 121

⁹ A mathematical History of the Golden Number, Roger Herz-Fischler, Dover Publications, New York, 1998, p 1

¹⁰ The Divine Proportion, A Study in Mathematical Beauty, H. E. Huntley, Dover Publications, NY, 1970, p 37

¹¹The Divine Proportion, A Study in Mathematical Beauty, H. E. Huntley, Dover Publications, NY, 1970, p 50

¹² Fibonacci number sequences appear throughout the world of natural phenomena, such as the shape of snail shells and the heads of sunflowers, and in phyllotaxis, which amongst other things, describes the angular separation of leaves on the stems of plants. Fibonacci sequences are very important in many branches of mathematics, particularly equation theory, the study of genetics, electronics, and statistical data handling. The stock market appearance of the Fibonacci series is more commonly known as the Elliot Wave Principle, which states that the market unfolds in a 5 to 3 ratio, which is interpreted as a rhythmic alteration of human optimism and pessimism. The Italian merchant and mathematician Leonardo di Pisa, who published his influential Liber abaci, in 1202, provided a graphic example of the Fibonacci sequence. The following description is taken from Mathematics From The Birth of Numbers, Jan Gullberg, W. W. Norton & Co, New York, London, 1997, p 286, and relates the problem.

A pair of newly born rabbits, male. And female, were. placed in a hutch. In two months these rabbits began their breeding cycle and produced one pair of rabbits, one male and one female. 'The original rabbits and their offspring continued to breed in this manner, that is, the first pair of offspring appearing at the parental age of two months and then a new pair every month thereafter - always one male and one female. All rabbits survived the first year.

What then is the total number of pairs of rabbits at the beginning of each month during the first year?

The original pair had their first pair of offspring at beginning of the third month, and another pair at beginning of the fourth month. At the beginning of the fourth month, 2 pairs were fertile, resulting in 2 pairs of offspring the beginning of the fifth month, and so on:

A look at the resulting sequence,

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

reveals that, from the third term onward, every successive term is the sum of the two immediately preceding terms in a clear Fibonacci sequence.

¹³ A. F. Horadam, "Further Appearance of the Fibonacci Sequence," Fibonacci Quarterly 1 (December 1963): 42., as cited in Fascinating Fibonaccis, Trudi Hammel Garland, Dale Seymour Publications, 1987, p 40

¹⁴ A History of Ancient Mathematical Astronomy, Vol 2, Neugebauer, p576 as cited on www.sover.net

Remembering that this Golden Series is essentially Geometric Proportion, it follows that its use will proportionately increase all parameters with which it is associated. If we multiply root two by each number in the Golden Series, we obtain a corresponding pyramid volume from which we may calculate the height and length in such a manner that they obey the triangular arrangement of

Therefore by using the Golden Series and multiplying this by root two, the pyramid builders achieved a perfection of geometric form that has no parallel in human history.

The results of this geometric proportion are set out in tabular form below.