Fibonacci, Fractals, and Inorganic Teleology

ImageAmong the most philosophically challenging scientific data of the last half century are those relating to the physical constants  of the universe (listed below) which allow it to be a cosmos instead of utterly disordered chaos. These constants were “finely tuned” to their present values when the universe came into existence out of absolutely nothing roughly 13.7 billion years ago. You cannot derive their values from something more basic; they simply occurred as “givens” from the first second of our universe’s existence. These values did not develop but were present full-blown at singularity. They did not evolve: they simply were.
Cosmologists Barrow and Tipler wondered what would happen if they were slightly different. Tinker ever so slightly with the values of any of the basic physical constants, and life would have been impossible, not just life of our kind, but life of any kind that involves complexity.

Because of their highly ordered nature, random origin of the constants has been widely conceded to be effectively zero probability (cf. physicist Donald Page has calculated the odds as 1 in 10,000,000^124. By comparison, there are 10^18 seconds since the creation of the universe and around 10^80 atoms in the observable universe).

Mathematician Emile Borel affirmed that anything with odds of happening less than one in 10^50 is impossible (Borel is best known for creating the the first effective theory of measuring sets of points beginning the modern theory of functions of a real variable). Random origin of the constants is well beyond this threshold -by orders of magnitude; selection by lottery would only overcome this statistical obstacle if there were an infinite number of unobservable universes from which ours was selected, yet a universe generating “machine” would also have to be exceptionally highly ordered too, and contemporary physicists have recently suggesting that multiple universes would be clones of one another rather than infinitely variable as the infinite unobservable multi-universes lottery selection theory requires.

Many physicists and philosophers have been attracted to similar arguments in the last thirty years (during which the ramifications of the delicately balanced physical constants first came to our attention; cf. the lecture by Dr. Francis Collins (PhD, & MD), first and long-time director of the Human Genome Project, here.  Collins’ PhD is in Quantum Mechanics, though his focus now is on genetics). It was this issue which former leading atheist and world famous philosopher Antony Flew cited as convincing him to abandon atheism for belief in God (many atheists claim it was rather because Flew must have become senile!).

The general failure of a one universe hypothesis to escape this conundrum has become a powerful reason among many naturalists to postulate a hypothetical infinite number of unobservable universes from which a highly ordered one could have been a random occurrence. Contemporary atheists who hold this option find themselves in the paradoxical, and at present largely fideistic (or at best inductive or abductive) position of arguing for  actuality of an essentially zero-probability event by postulating something in principle infinitely immeasurable and humanly unobservable (non-scientific/metaphysical).

“‘I can’t believe that!’ said Alice. ‘Can’t you?’ the Queen said in a pitying tone. ‘Try again: draw a long breath, and shut your eyes.’ Alice laughed. ‘There’s no use trying,’ she said: ‘one can’t believe impossible things.’ ‘I daresay you haven’t had much practice, said the Queen. ‘When I was your age, I always did it for half-an-hour a day. Why, sometimes I believed as many as six impossible things before breakfast.'” -Lewis Carroll, Through the Looking Glass

The slightest alteration of the following physical constants would result in a universe incapable of supporting life -not just life of our kind, but life of any kind that involves complexity, and in a universe that would be chaos rather than cosmos:

Gravitational Coupling Constant
Strong Nuclear Force Coupling Constant
Weak Nuclear Force Coupling Constant
Electromagnetic Coupling Constant
Ratio of Protons to Electrons
Ratio of Electron to Proton Mass
Expansion Rate of the Universe
Entropy Level of the Universe
Mass of the Universe
Uniformity of the Universe
Stability of Protons
Fine Structure Constants
Velocity of Light
Distance Between Stars
Rate of Luminosity of Stars
8Be, 12C, and 4He Nuclear Energy Levels.

An infinite number of unobservable universes -even were it the case- would not, of course, necessarily “belong” to our atheist friends who need it so badly to account for zero probability of random origin of the universe’s physical constants  at singularity; in fact it would be a perfect case scenerio of the ancient Augustinian cosmological theodicy of pleroma, which posited all possible varieties and ranges of entities might actually exist; we will leave that subject for a  possible future post; let us now move along to consider the central topic of this essay.


In what follows, we will explore one of the most astonishing and brilliant discoveries of the last century. Although initially a mathematical discovery, its implications can be seen to permeate our cosmos from the microsphere to the macrosphere to a degree that is nothing less than mind-boggling, in a manner that was utterly unknown just a few decades ago.

“This is how God created a system that gave us free will. It’s the most brilliant maneuver in the universe, to create something in which everything is free! How could you do that?! …exploring this set I certainly never had the feeling of invention. I had never the feeling that my imagination was rich enough to invent all the extraordinary things. I was discovering them; they were there although no one had ever seen them before. It’s marvelous! A very simple formula describes all of these very complicated things. Who could have dreamed that such an incredibly simple equation could have generated images of literally infinite complexity? We’ve all read stories of maps that revealed the location of some hidden treasure. In this case the map is the treasure!” -Benoit Mandelbrot, in Fractals: the Colors of Infinity (Arthur Clarke Documentary).

Why do certain patterns (Fibonacci/fractal patterns) constantly reappear throughout nature (organic and inorganic!) in phenomena as incredibly diverse as neuron firing patterns, trees and flowers, lightning, networks of veins in body, crystal structures, the inner structure of lungs, hearts, and other organs, hurricanes, spiral galaxies, viruses, most formations of plant life, cellular microtubules, chemical structures (e.g. platonic solids), family trees, snowflakes, even thoughts? (if you are not a mathematician or scientist, you will be able to more fully appreciate what all the things in this essay –indeed most things in our universe- have in common after you watch the video which follows

II. FRACTALS IN HUMAN PHYSIOLOGY (Yale University Biology Dept.)

“Some of the most visually striking examples of fractal forms are found in physiology: The respiratory, circulatory, and nervous systems are remarkable instances of fractal architecture, branches subdividing and subdividing and subdividing again. Nice pictures are provided in Goldberger, Rigney and West. Although no clear genetic, enzymic, or biophysical mechanism yet have been shown to be responsible for this fractal structure, few doubt this. Careful analysis of the lungs reveal fractal scaling, and it has been noted that this fractal structure makes the lungs more fault-tolerant during growth. The heart is filled with fractal networks: the coronary arteries and veins, the fibers binding the valves to the heart wall, the cardiac muscles themselves, and the His-Purkinje system, etc.

“In addition to falut-tolerance during growth, fractal branching makes available much more surface area for absorption and transfer in bronchial tubes, capallaries, intestinal lining, and bile ducts. Kalda has proposed a fractal model of the blood vessel system that achieves a homogeneous oxygen supply throughout the body. Also, the redundancy of fractal structures make them robust against injury. For example, the heart can continue to function even after the His-Purkinje system has suffered considerable damage. From his work on the ability of fractal drums to damp vibrations, Bernard Sapoval deduced another advantage of the fractal character of the circulatory system: “the fractal structure of the human circulatory system damps out the hammer blows that our heart generates.” “The heart is a very violent pump, and if there were any resonance in blood circulation, you would die.” Fractals may save our lives every minute. Here are some casts of animal lungs. Finally, we note the body exhibits dynamical fractals. For example, it is well-known that healthy heartbeats are chaotic rather than regular. A careful plot of heart rates over several time scales reveals self-similar scaling (Goldberger, Rigney and West). ; On the fractal nature of the human circulatory system, see here.


Blood cell dynamics are also governed by Fibonacci ratios: “This paper demonstrates that the pattern of lipid spicules that emerge on the surface of red blood cells in the classic ‘Discocyte to Echinocyte’ shape change is a generative spiral, and presents a qualitative, fluid- driven mechanism for their production, compatible with the work of Douady and Couder. Implications for the dynamics of cell growth, plant cell phyllotaxy, programmed cell death and gravity sensitivity are explained in terms of a new qualitative model of cellular fluid dynamics.”


Plant Phylotaxis is the arrangement of plant elements as primordia on the shoot apex, e.g. branches, leaves, petals, stamens, sepals, florets, etc.).] The angle subtended at the apical center by two successive primordia is equal to the golden angle (137.5 degrees) in more than 90% of all plants studied worldwide. Such an arrangement allows incorporate maximum packing efficiency for fruits and seeds, maximal access to sunlight by leaves, etc. See further at


Information encoding in the brain is also a Fibonacci process, as explained by Harald Weiss and Volkmar Weiss, “The golden mean as clock cycle of brain waves” in Chaos, Solitons & Fractals Volume 18, Issue 4, November 2003, Pages 643-652; cf. abstract and full article here; cf. also Weiss, Volkmar, “Memory Span as the Quantum of Action of Thought,” Cahiers de Psychologie Cognitive 14 (1995) 387-408 here.

The fine structure constant and the structure of space are also according to Fibonacci. See Carlos Castro, “Fractal Strings as the Basis of Cantorian-Fractal Spacetime and the Fine Structure Constant,” Chaos Solitons Fractals 14 (2002) 1341-1351 here.

All fractal self-similar structures (whereupon parts resemble the whole object in shape) incorporate in their geometrical design numbers which are functions of the golden mean.

In case the reader is unfamiliar with fractal geometry, the relationship between Fibonacci and fractals is illustrated simply in the photo above which shows phi (which represents the average of the Fibonacci series) in the Mandelbrot set. The Mandelbrot set is used to generate fractals and fractal art. As anyone knows who has viewed this type of art, amazingly complex and at the same time strikingly NATURE-LIKE and LIFE-LIKE PATTERNS (like insects, leaves, flowers, rock formations, faces, etc,) are quite common with the Mandelbrot “algorithm” (google fractal art!). Random generation is not contra “structure” given an algorithm which “guides” the unfolding; hence chance itself may be ‘governed’ in such a way as the gorgeous symphony of life may still be to a degree “orchestrated”

It is critically important to note that the “algorithm” which generates the ‘similarity in infinite diversity’ we see in nature is not a specific number (phi), but a range of Fibonacci ratios, e.g. from 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 32/21 and so on infinitely. Phi, the average of these ratios, is not a ‘magic number,’ but merely the average of all such ratios. In nature it is not so much phi that is important as it is the SPECTRUM OR RANGE OF VALUES which phi may embody as an average. It is the fact of a Fibonacci range rather than a specific uniform value that especially allows infinite diversity within the algorithmic similarity to be maximized. Thanks to God not all flowers or faces look alike! It is the startling diversity of our universe, in spite of its unfolding within the fine-tuned confines of specific physical constants etc. that never fails to take our breath away.


“Also Mandelbrot curves have been discovered in cross-sections of magnetic field borders, implying there is a 3-D mandelbrot equivalent that is closely tied to electromagnetism and therefore a deep structural and fundamental aspect of life, and physical space/time…

“…we see the Mandelbrot relationship to the period-doubling ‘chaos’ equation which is used to describe population expansion, plant growth, weather instability and a host of other physical processes. This relationship also has a habit of popping up unexpectedly in other dynamic non-linear equations (fractals made from Newtons method of deriving a cube-root being the most obvious)

“Think about that for a moment – Take any slice of the magnetic field of the earth, sun, a plant, the data on audio or video tape, and there is our old familiar Buddha looking mandelbrot! This suggests an unknown, yet-to-be-clarified fundamental importance of the Mandelbrot Set in many physical processes. Clearly this is something far more significant than a means of generating visually pleasant mathematical abstractions.

“Many things previously called chaos are now known to follow subtle fractal laws of behavior. So many things turned out to be fractal that the word “chaos” itself (in operational science) had to be formally defined as following inherently unpredictable yet generally deterministic rules based on nonlinear iterative equations. Fractals are unpredictable in specific details yet deterministic when viewed as a total pattern – in many ways this reflects what we observe in the small details & total pattern of life in all it’s physical and mental varieties, too….” -


Physicists, Stéphane Douady and Yves Couder from the Laboratory for Statistical Physics in Paris in 1992 demonstrated tiny magnetized ferrofluid droplets in pool of silicone oil ultimately form spirals described by the golden angle (see the video here; an abstract in Physical Review (Douady, S. and Y. Couder. 2002. “Phyllotaxis as a physical self-organized growth process,” Physical Review Letters 68 (March 30):2098-2101) can be found here).

We have already seen the amazing extent to which Fibonacci inform us about our neural structure, thought structure, organ structure, blood dynamics, circulatory system, etc. etc.; the golden ratio also explains human perceptions of beauty to the degree that standard practice for plastic surgeons and dentists is to carefully employ these ratios in reconstruction of the face and teeth, branching patterns and proportions of skeletal structure, and many other applications beyond the scope of this brief essay.


In March of 2006, an incredible news story was released about a double helix nebula found near the center of the Milky Way. “We see two intertwining strands wrapped around each other as in a DNA molecule,” said Mark Morris, a UCLA professor of physics and astronomy, and lead author. “Nobody has ever seen anything like that before in the cosmic realm. Most nebulae are either spiral galaxies full of stars or formless amorphous conglomerations of dust and gas — space weather. What we see indicates a high degree of order.” See the original article at the UCLA website here.

The double helix, mysteriously mirrored in the stars and the DNA of living creatures, is another Fibonacci structure:


Intriguing also is the presence of this precise ratio in the Tabernacle of Israel. If it was only there once, it would be more reasonable to attribute its presence to chance, and we are reminded of the exhortation to Moses to ensure he adhered exactly to the heavenly pattern (cited below). This “signature literally written all across the face of the cosmos” appears in two critically important structures which have different dimensions, in the most important artifacts known to ancient Judaism: the Holy Ark of the Covenant, and the Brazen Altar where all Israel’s sacrificial offerings were brought.

2.5/1.5 = 1.6666667 (=5/3)
Exodus 25:10:”And they shall construct an ark of acacia wood two and a half cubits long, and one and a half cubits wide, and one and a half cubits high.”
5/3 = 1.6666667
Exodus 27:1: “And you shall make the altar of acacia wood, five cubits long and five cubits wide; the altar shall be square, and its height shall be three cubits”
Exodus 25:40: “And see that you make them after the pattern for them, which was shown to you on the mountain.”

If one already considers Christian theism of reasonable warrant it becomes difficult not to see the very signature of God in almost every direction one can possibly look -from the heavens to the mirror.

From Constantine, David, “They Look Alike, but There’s a Little Matter of Size” (New York Times, Aug 15, 2006): “One is only micrometers wide. The other is billions of light-years across. One shows neurons in a mouse brain. The other is a simulated image of the universe. Together they suggest the surprisingly similar patterns found in vastly different natural phenomena. Mark Miller, a doctoral student at Brandeis University, is researching how particular types of neurons in the brain are connected to one another. By staining thin slices of a mouse’s brain, he can identify the connections visually. The image (below left) shows three neuron cells on the left (two red and one yellow) and their connections. An international group of astrophysicists used a computer simulation last year to recreate how the universe grew and evolved. The simulation image (below right) is a snapshot of the present universe that features a large cluster of galaxies (bright yellow) surrounded by thousands of stars, galaxies and dark matter (web).” (Source by Mark Miller, Brandeis University; Virgo Consortium for Cosmological Supercomputer Simulations;

How can the large scale structure of the universe and a mouse neuron have the same structure? The degree to which the Fibonacci ratio is present in both inorganic structures and the bodies of living beings suggests there is something more than genetics and selection alone involved in the latter. Despite a stunning difference in scale there a key similarity: neural networks, like the fine structure of the universe, are (and the large scale structure of the universe) are fractal (


“What does it take to build a world? This is the central question of my research. My overarching goal is the creation from first algorithmic principles of an entire planet, well-defined everywhere and at all scales, with visual complexity, appearance, and beauty similar to Earth, and to bring that model to real-time performance. Needless to say, this undertaking subsumes a large number of interesting and challenging elements. These include developing our capabilities in visual realism, models of natural phenomena, computational efficiency in such models, and algorithmic art. I am confident that there are enough challenges involved to keep me busy for the rest of my days.

A planet, at the scales of ordinary human experience, is defined by its landscapes. Landscapes are in turn defined by the form of the land, the lighting, the current state of the atmosphere, and by the life forms found within it. My research encompasses the first three, terrain, lighting, and atmospherics; peculiarities of taste and predilection lead me to eschew modeling life forms, leaving them to others to perfect. There is no accounting for taste, and “I love landscapes!” …All successful synthetic terrain models for computer graphics are fractal: That is, they feature complexity resulting from the repetition of form over a variety of scales. The complexity resulting from this repetition of form over many scales leads to the odd idea of fractal dimension: a spatial dimension which is intermediate between the familiar integer-valued (i.e., 1, 2, and 3) dimensions we’re used to dealing with.”

-Professor Ken Musgrave, Fall 1994 George Washington U. EECS Newsletter; Dr. Musgrave works with Benoit Mandelbrot, the discoverer of the Mandelbrot set.


All informed observers concede the undeniable and astonishing ubiquity of Fibonacci and fractals in nature. Even some basic natural phenomena which at first glance seem to depart from mathematization according to *phi* (an average of the ratios of the Fibonacci series) can be found, upon closer analysis to contain more Fibonacci than is first apparent to the uncritical eye. As we have emphasized, it is the Fibonacci series which is so prevalent in nature rather than *phi* (the average of said series) per se: phi is a mathematical abstraction of the range of things we see, it is but an average, a human methodological contrivance. There is an additional surprising point to consider regarding phenomena like seashells and planetary orbits, which are Fibonacci on the main, but with variations:  the manner in which variation from Fibonacci occurs in e.g. seashells and planetary orbits actually itself algorithmic according to precise Fibonacci exponents (see


There are some rather revolutionary conclusions from the recent revolution in fractal geometry stemming from Mandelbrot’s great discovery. First, “chance” itself is “governed” algorithmically to an extent entirely unsuspected just a few decades ago. Yet this algorithmic “governance” is such that infinite variety is not precluded, but rather enabled! The centuries-old philosophical presupposition that chance and governance are opposite has been effectively demolished by Mandelbrot’s amazing discovery. It is also interesting that the discovery was initially within the realm of pure mathematics, rather than the empirical (empirical, i.e. depending on sensory observation); as a video presented below emphasizes, some philosophers of science who prefer to methodologically restrict the “scientific” to the directly observable in a manner similar to that of the Logical Positivists of the early 20th century resisted acknowledging the vast extent to which nature embodies this newly discovered fractal geometry (the demarcation criterion between science and non-science is obviously of critical importance for the philosophy of science; significantly this issue has not been resolved and may be irresolvable in principle (cf. the Kuhn-Popper debate and beyond). Such skepticism, however, could not long survive decades of careful measurement.

Arthur C. Clark produced a fabulous series on fractals titled The Colors of Infinity which documents and describes all this and more, with interviews with such luminaries as Benoit Mandelbrot (who discovered fractal geometry), Stephen Hawking, and many other great pioneers of fractal mathematical applications (the videos are found below). We’ll close this brief essay with some fascinating quotations from some of these men taken from part 5:

“This is how God created a system that gave us free will. It’s the most brilliant maneuver in the universe, to create something in which everything is free! How could you do that?!”

“…Albert Einstein refused to accept the idea of a dice-playing deity. He wrote a letter to Max Borne in which he said ‘you believe in a dice-playing God; I believe in complete law and order.’ So he obviously felt that chance and deterministic laws were not compatible and he preferred the deterministic laws. Now what the Mandelbrot set and Chaos and related things have done for us is to show that you can have both at the same time. So it is not whether God plays dice that matters, it’s how God plays dice.” “I [Benoit Mandelbrot] can tell you exploring this set I certainly never had the feeling of invention. I had never the feeling that my imagination was rich enough to invent all the extraordinary things. I was discovering them, they were there although no one had ever seen them before.” It’s marvelous! A very simple formula describes all of these very complicated things…” “Who could have dreamed that such an incredibly simple equation could have generated images of literally infinite complexity? We’ve all read stories of maps that revealed the location of some hidden treasure. In this case the map is the treasure!” -from Fractals: The Colors of Infinity

© 2010 by (text)

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8 Responses to Fibonacci, Fractals, and Inorganic Teleology

  1. sarisa adami says:

    i need to get more knowledge day by day

  2. august 18, 1987 says:

    Fucking amazing blog, man…. Wow thank you :), this has seriously MADE my DAY (morning!).

    Thank you for the links to the academic resources rather than just talking about it. And for this topic……. of topics. 🙂

  3. Christian says:

    Thanks so much! This is absolutely fantastic!

    It’s really difficult being a believer and a scientist sometimes, since a lot of scientists regard it as a gross incompatibility. Articles like this make me feel a lot better about my position and give me great material to cite/explain in arguments. Thanks again!

  4. Reblogged this on gospel(s) and commented:
    I’ve often had murky thoughts, and strands of pipedreams along the lines of the mystery that this post just absolutely nails in its first section. It just brought a smile to my face recognizing a kindred spirit out there fussing and fiddling with this same mystery.

  5. kristy mapp says:

    I’ve just found your blog, and wow! I will be visiting often, but I have a question. I noticed you have an older set of blogs with lots of old testament things (Genesis for example) and was wondering if you ever merged them here? It would be great to have it all in one place 🙂

    Thank you for the truth you present!!

  6. ±ÝÇü says:

    Fantastic site. Lots of useful information here. I am sending it to a few buddies ans also sharing in delicious.
    And obviously, thank you to your sweat!

  7. abookandart says:

    Reblogged this on A Book & Art and commented:
    Fantastic Fibonacci and so glad I’ve found this blog site. very happy today.

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