Fibonacci Rules! Purpose and Structure in Science and Philosophy

The starting point for this consideration is my lifetime love of botany.  I have long been fascinated by the myriad forms of plant species, their names, their anatomy, and their family relationships.  This passion for structure was encountered in my high school botany class with two challenging and opposing concepts.  The first was the account of Harold Urey and Stanley Miller’s research into the seemingly random creation of amino acids, the building blocks of life. The second was the statement, abundantly documented, that “the basic form of life is the spiral.”  It can be seen in the DNA molecule, ram’s horns, snail shells, pine cones, tree branching, and much more.  Blind chance and a pervasive evidence of design were presented to us side by side, with no requirement that we reconcile them or choose between them.  They were just there and years later they still are.  It is my task in the post to address them both. 

Early in the 1950s Harold Urey speculated that the early terrestrial atmosphere was composed of ammonia, methane, and hydrogen. One of his Chicago graduate students, Stanley L. Miller, showed that, if such a mixture be exposed to electric sparks and to water, it can interact to produce amino acids. The Miller–Urey experiment was a chemical experiment that simulated the conditions thought at the time (1952) to be present on the early Earth and tested the chemical origin of life under those conditions.  In a 1996 interview, Stanley Miller recollected his lifelong experiments following his original work and stated: Just turning on the spark in a basic pre-biotic experiment will yield 11out of 20 amino acids.  

So we have a random accident as the source of life and, from there, evolution.  Does this deny the presence of structure and purpose?  To some it may seem to be so, paving the way for the forces of evolution to the exclusion of a notion of order.  But this is not the end of the story.  Evolution versus creation by design is for me is not an either/or proposition, nor should it be for anyone else.  There is more to bear on the problem.  A touchstone of philosophical and scientific thought which has served as valuable resource lies in the writings of Lecomte Du Nouy in his influential book Human Destiny (New York, Mentor Books, 1949).

There is a philosophical back story to this argument.  It lies is the prevalence of what is called  positivism in modern thought.  This philosophy denies the possibility of any metaphysics or ontology (the branch of metaphysics dealing with the nature of being) and is also known as logical empiricism and logical positivity.  The core of its teaching maintains that all statements purporting to be true fall into one or two kinds.  Statements of the first kind assert something about the world that can be tested by experience.  Statements of the second kind assert that certain logical or mathematical relations hold between the terms of the assertions of the rules or conventions governing the use of these terms. [Sidney Hook, Contemporary Philosophy, 1968]  Although primarily applied by its originator, August Comte, to science, its method was also applied to sociology (of which Comte was the founder) and was later to be incorporated into the writings of Karl Marx with his assertion that societies develop according to the laws of nature.

To my thinking positivism is the child of skepticism, which challenges the adequacy or reliability of claims, asking what principles they are based upon or what they actually establish. The original Greek meaning of skeptikos was “an inquirer,” someone who was unsatisfied and still looking for truth.  This is a healthy attitude for all who seek knowledge and understanding to have. There is, however, a difference between skepticism and denial.  The refusal to allow what lies beyond one’s power to reason is also a liability.  Allowing for the admission of scientific evidence, there is no reason that Lecomte Du Nouy’s research would not provide ample facts to modify the evolutionary view, even in this the midst of its positivistic environment.

I cannot give you a complete survey of Lecomte Du Nouy’s work, but can provide a few pertinent keys to the philosophy he brings to bear on the problem.  To begin, a very brief synopsis is presented in the editor’s introduction : “…divine intelligence has given a purpose and direction to evolution and…man is the goal and end-product of biological evolution.”

Scientific thought constrained by positivism can only be countered by additional scientific and rational thinking.  This provides the thrust for the argument and a motivation to do more than has thus far been provided.  He writes, “…materialism spread not only among technicians but, alas, in the masses.  Rational thinking should have been employed to fight this disease of reason.  A mathematical argument can only be fought by other mathematical arguments, a scientific reasoning can only be destroyed by a reasoning of the same kind…..We must use the right key if we want to fight paralyzing skepticism and destructive materialism which are by no means the inevitable consequences of the scientific interpretation of nature, as we have been led to believe.” [p. 11]

Next, there is a key difference between physio-chemical reactions (such as those in Urey and Miller’s chamber) and those of biological life.  To ignore this difference is to eliminate vital evidence pertaining to the process: “As long as we do not know the relationships between a physio-chemical phenomenon and the phenomena pertaining to life and psychology which may accompany it, we shall not be able to say that we know its whole significance.” [p. 23]  It is simply not the whole story.

Thus there is a key difference between life as a whole and the more limited and insufficient theory of evolution from inorganic compounds, which, in Du Nuoy’s words, “is rather pathetic, inasmuch as inorganic evolution is essentially a recent human concept hatched in man’s brain, while the evolution of life is a reality supported by an immense series of observable facts such as the fossils…..any attempt to subordinate biological evolution to inorganic evolution cannot, a priori, be considered as scientific or philosophic.” [p. 40]

Philosophically, the argument is to be carried by “a teleological hypothesis, that is, a finalism with a very ultimate goal, a ‘telefinalism,’ if we may be allowed to coin a new word.” [p. 47]   The history of teleology (from Greek telos, “end,” and logos, “reason”) as a philosophic discipline dates back to Aristotle, when he declared that a full explanation of anything must consider its final cause as well as its “efficient, material and formal causes” which provide the form or pattern.  Nature, indeed, has a pattern, which we shall discover in the work of Fibonacci.

Evolution of life, then, is “dominated by a finality, a precise and distant goal.  If we do not accept the reality of this orienting pole, not only are we forced to recognize that evolution is rigorously incompatible with our laws of matter….The laws of evolution are teleological, whereas those of the transformation of each species simply tend toward a state of equilibrium with the surrounding medium.” [pp. 66-67]

This points to a change in natural selection as it is commonly understood.  What follows is a somewhat involved argument but it is necessary if we are to unravel some of the assumptions of  evolutionary thinking.  Consider this explanation of the dynamics involved in what we have commonly regarded as “survival of the fittest”:  “…when this equilibrium representing perfect adaptation has been attained, the animal naturally ceases to transform itself as long the external conditions are not sufficiently modified to make a new adaptation necessary.  This strain then constitutes one of those countless fixed branches which for thousands and thousands of centuries have dotted the history of living beings are no longer anything but memories of a vanished past.  The actual fauna of the world often represent the masterpieces of adaptation, but only the ‘leftovers’ of evolution….It is not the being best adapted to his environment who contributes to evolution.  He survives but his better adaptation eliminates him…from the ascendant progression and only contributes in increase the number of more or less stagnant species which people the earth…. Adaptation then works to neutralize its own anterior efforts, and natural selection tends to eliminate those it had heretofore protected.” Lecomte Du Nuoy concludes, “…the evolution of living beings, as a whole, is in absolute contradiction to the science of inert matter.” [p. 157]

Why is this viewpoint important?  Because earnest, inquisitive young students of science are forced to choose between a creation myth and a pathetically lacking theory of random chance which narrowly limits its argument to the point that, not unlike creationism, it must be taken on faith.  Not all who study the evolution of life are so constrained, but in the face of such a persuasive argument it is surprising that after more than seventy years the argument for telefinalism  is not more further advanced than it is.

The evidence for purpose has been made.  To accompany it, a pattern, a rule, for structure would then serve as a powerful tool.  As it turns out, a coherent structural theory was already in place.  Italian mathematician Leonardo Fibonacci (1170-1250), provides us with powerful evidence.  There will not be much math here, but the key lies in “the golden ratio”, Phi (φ)= 1.618.  It is said that God is a mathematician and, if so, Fibonacci is God’s mathematician.  

The Fibonacci sequence is based on the fact that the ratio between any number and the previous one in the series tends towards a well-defined value: 1.618… This is the golden ratio, Phi, which  frequently occurs in nature.  The Fibonacci sequence began, humorously enough, with a question about reproduction of rabbits: How many pairs of rabbits will be born in a year, starting from a single pair, if each month each pair gives birth to a new pair which becomes reproductive from the second month?  The solution to this problem is the famous “Fibonacci sequence”: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…, a sequence of numbers in which each member is the sum of the previous two.  Despite its monumental implications, to Fibonacci this was a solution to a “recreational mathematics” problem and he did not give it particular importance.   It was only in 1877 that  the mathematician Édouard Lucas published a number of important studies on this sequence, which, in the honor of the author, he called “Fibonacci sequence”. So many further studies were made, and so many applications of this sequence were discovered that since 1963 a journal exclusively dedicated to it, The Fibonacci quarterly, has been published.


Observing the geometry of plants, flowers or fruit, it is easy to recognize the presence of recurrent structures and forms. The Fibonacci sequence, for example, plays a vital role in phyllotaxis, which studies the arrangement of leaves, branches, flowers or seeds in plants, with the main aim of highlighting the existence of regular patterns. The various arrangements of natural elements follow surprising mathematical regularities: D’arcy Thompson observed that the plant kingdom has a curious preference for particular numbers and for certain spiral geometries, and that these numbers and geometries are closely related.  We can easily find the numbers of the Fibonacci sequence in the spirals formed by individual flowers in the composite inflorescences of daisies, sunflowers, cauliflowers and broccoli.  Each species contains a number which corresponds to a number in the Fibonacci series.

It was Kepler who noted that on many types of trees the leaves are aligned in a pattern that includes two Fibonacci numbers. Starting from any leaf, after one, two, three or five turns of the spiral there is always a leaf aligned with the first and, depending on the species, this will be the second, the third, the fifth, the eighth or the thirteenth leaf.

Consider the incredible manifestation of the Fibonacci series in nature:

1. As previously stated, the number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five, the chicory’s 21, the daisy’s 34, and so on. Phi appears in petals on account of the ideal packing arrangement as selected by Darwinian processes; each petal is placed at 0.618034 per turn (out of a 360° circle).  This allows for the best possible exposure to sunlight and other factors which facilitate growth.

2. The seed head of a flower is also subject to Fibonaccian processes. Typically, seeds are produced at the center, and then migrate towards the outside to fill all the space. Sunflowers provide a great example of these spiraling patterns.  In some cases, the seed heads are so tightly packed that total number can get quite high — as many as 144 or more. And when counting these spirals, the total tends to match a Fibonacci number.

3.  Similarly, the seed pods on a pine cone are arranged in a spiral pattern. Each cone consists of a pair of spirals, each one spiraling upwards in opposing directions. The number of steps will almost always match a pair of consecutive Fibonacci numbers. For example, a 3-5 cone is a cone which meets at the back after three steps along the left spiral, and five steps along the right.

4. Likewise, similar spiraling patterns can be found on fruits.  To give you an example, look at pineapples and cauliflower.

5.  The Fibonacci sequence can also be seen in the way tree branches form or split. A main trunk will grow until it produces a branch, which creates two growth points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems.  Branching typically progresses from the tree base as 1-2-3-5-8-13, and onward. Root systems and even more primitive forms such as algae exhibit this pattern.

6.  In shells, their shape represents a ratio of the sides a and b which is equal to the golden mean (Phi).  This can result in a nesting process that can be repeated into infinity — and which takes on the form of a spiral.  It’s called the logarithmic spiral, and it abounds in nature.  Snail shells and nautilus shells follow the logarithmic spiral, as does the cochlea of the inner ear. It can also be seen in the horns of certain goats, and the shape of certain spider’s webs. To give you a better idea of the pattern, here is the arrangement of seeds in a sunflower head:

7. Not surprisingly, spiral galaxies also follow the familiar Fibonacci pattern. The Milky Way has several spiral arms.  Each arm spirals at an angle of about 12 degrees. As early as 1925, astronomers realized that, since the angular speed of rotation of the galactic disk varies with distance from the center, the radial arms should become curved as galaxies rotate. The spiral arms should start to wind around a galaxy, but they don’t.  This unique trait of the cosmos helps to preserve its shape.

8.  The spiraling of hurricanes exhibits the same logarithmic pattern found in shells, horns, webs, and spiral galaxies.  

9. Finally, the material of our genetic makeup is not immune to Fibonacci. The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. These numbers, 34 and 21, are numbers in the Fibonacci series, and their ratio 1.6190476 closely approximates Phi, 1.6180339.

I have often been amazed at how more modern concepts are sometimes encoded in ancient mythology, religion, and philosophy.  The spiral is no exception, and has spiritual implications.  It was anticipated by Hinduism, with its expansive Sanskrit vocabulary.  Here is a quote from Jill Purce in The Mystic Spiral: “…the transcendent One, who is the non-manifest point, the seed syllable Om(Aum), the germ Bindi bursting with latent creativity, which unfolds as the windings of the feminine Shakti down through the layers of being.  It is the task of the Yogi to return in consciousness on the upward spiral, through all these layers, elements and sounds within his individual self. She is coiled 3 ½ times round a vertical  lingam, within the lowest of man’s seven centres, the root chakra or fundamental support and pole of the body: She is Shakti, the creative energy of activity of the actionless god Shiva, who is the point Bindu by virtue of which she exists, and around which she, as manifestation, revolves like the coils of the dragon around the Axis Mundi or World Tree (Ashvattha)….Conceptually, she is spiraling from the point Bindu out to the circumference, which she thereby creates; but because she is also spiraling back towards her union with the One, Shiva, she is, while going outwards and unfolding, simultaneously going in towards the centre, Shiva, with whom she will ultimately fuse.  This activity may therefore be conceived as a whole when seen on the spherical vortex.”

Yes, we exist in an ocean of causality and are subject to forces and circumstances beyond our control; nothing can change that.  But underlying the randomness of action and reaction is something far more substantial and permanent, an overriding purpose and a structural pattern.  Du Nuoy leaves us with this observations:  “The agnostic and the atheist do not seem to be in the least disturbed by the fact that our entire organized, living universe  [my emphasis] becomes incomprehensible without the hypothesis of God.” [p.134] 

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