Golden Ratio

The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[3][4][5] Other names include extreme and mean ratio,[6] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[7] and golden number.[8][9][10]

 

Check pic

Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.[11] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.

Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.

Ernő Lendvai analyzes Béla Bartók‘s works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[53] though other music scholars reject that analysis.[3] French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy‘s Reflets dans l’eau (Reflections in Water), from Images (1st series, 1905), in which “the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position”.[54]

The musicologist Roy Howat has observed that the formal boundaries of Debussy’s La Mer correspond exactly to the golden section.[55] Trezise finds the intrinsic evidence “remarkable”, but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[56]

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.[57]

Though Heinz Bohlen proposed the non-octave-repeating 833 cents scale based on combination tones, the tuning features relations based on the golden ratio. As a musical interval the ratio 1.618… is 833.090… cents (pastedGraphic.pngPlay (help·info)).[58]

Nature

pastedGraphic_1.png

Detail of Aeonium tabuliforme showing the multiple spiral arrangement (parastichy)

Main article: Patterns in nature

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of parts such as leaves and branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these patterns in nature he saw the golden ratio operating as a universal law.[59][60] In connection with his scheme for golden-ratio-based human body proportions, Zeising wrote in 1854 of a universal law “in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.”[61]

In 2010, the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.[62]

Since 1991, several researchers have proposed connections between the golden ratio and human genome DNA.[63][64][65]

However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.[66]

Optimization

The golden ratio is key to the golden-section search.

Perceptual studies

Studies by psychologists, starting with Gustav Fechner, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have

The Golden Ratio: Structure or process ?

Introduction 

Plato, in the Timaeus 31b-32a, makes it clear that the geometric mean is nature’s primary bond: 

«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 (analogia) is best adapted to effect such a union». 

In this perspectve,  the golden ratio appears to produce the simplest and most powerful universal geometric standard. When the golden ratio is used as the basis for an additive series, it uniquely is both additive and multiplicative. No other number has these properties. 

In fact, the prevalence of the golden ratio in the sciences sug- gests that it is the fundamental source of everything in Nature, including all elementary particles, and even in the proportions of dark matter and energy relative to visible matter and energy. It is evident in the structure and growth functions of plants and animals and it can be found in the physiological functions of humans. Below a few examples

2. Biology

Phyllotaxis is one of the growth functions of plants and refers to the patterns of leaf spacing on a stem. These recurring patterns have been one of the most fruitful areas for research into the role of the golden ratio in nature. Leonardo Da Vinci (1452-1519) had noticed that the spacing of leaves on plants was often spiral in arrangement. Johannes Kepler (1571-1630) later noted that the majority of wild flowers are pentagonal, and that Fibonacci numbers occur in leaf ar- rangement. Observation by naturalists of the spiral patterns such as florets in the head of a daisy or scales of a pinecone led to the devel- opment of the 19th century field of research called Phyllotaxis, literally ‘leaf arrangement’. The Bravais brothers (1837) discovered the crystal lattice and the ideal divergence angle of phyllotaxis: Research has found that phyllotaxis leads to a leaf ar- rangement that is most effective and efficient for the plant in receiving maximum sunlight, moisture and pollination. The suggestion herein is that this is an example of aesthetics based in the efficiency of function itself. 

In the foreword to their monumental treatise, Symmetry in plants, Jean and Barabe place the golden ratio front and center of the mystery when they write: «Daisies and sunflowers are the emblems of phyllotaxis: all the problems of phyllotaxis are summarized there- in. The presence of particular numbers (e.g. Fibonacci numbers, an angle of 137.5o, the golden number) 

(e.g. logarithmic spirals) in their capituli, and shoot apices, demands an explanation, and has served as a spur to the human intellect. […] It is in phyllotaxis that symmetry in plants is most striking and puz- zling» (Jean and Barabe 1998, vii). 

Architect Oleg Bodnar, a Ukrainian Professor at Lvov National Academy of Arts, published his discovery of a new geometrical theory of phyllotaxis based upon hyperbolic rotation using golden hyperbol- ic functions. He called it The law of spiral bio-symmetry transforma- tion, and discussed it in his book The golden section and Non- Euclidean geometry in science and art (1994). Non-Euclidean hyper- bolic geometry had been used by Herman Minkowski to help visual- ize the curvature of space-time in Einstein’s theory of Special Relativ- ity. But Non-Euclidean hyperbolic geometry had only found applica- tion at very high speeds in outer space. Now, for the first time, the same geometry was found to be applicable throughout nature itself. 

2. Heart 

Russian biologist V.D. Tsvetkov published his findings on the role of the golden ratio in the cardiac activity in humans and other mammals in his 1997 book Heart, the golden section and symmetry. Tsvetkov found that if we take the middle blood pressure in the aorta as the measurement unit, then the systolic blood pressure approaches 0.382…, and the diastolic pressure approaches 0.618…, that is, their 

ratio corresponds to the golden ratio (0.618… : 0.382… = 1.618…). The golden ratio can also be observed in a cardiogram, where the comparison is made of two time intervals of different duration cor- responding to the heart’s systolic (t1) and diastolic (t2) activity. Tsvetkov discovered that there is an optimal (or golden) frequency for man and other mammals; here, the durations of systole, diastole and the full cardiac cycle (T) are in golden mean proportion, that is, T : t2 = t2 : t1. It means that cardiac performance in the timing cycles and blood pressure variations is optimized by the law of the golden ratio. According to this principle, nature has constructed the heart in such a way that it performs its function with minimal expenditures of energy, blood, muscle and vascular tissue. Again, another example of aesthetics being found in optimal function. 

2.3. Golden division of biological cells 

C.P. Spears and M. Bicknell-Johnson demonstrated that an extension of Fibonacci numbers, based on the Pascal triangle, can model the growth of biological cells. They conclude that «binary cell division is regularly asymmetric in most species. Growth by asymmetric binary division may be represented by the generalized Fibonacci equation. […] Our models, for the first time at the single cell level, provide a rational basis for the occurrence of Fibonacci and other recursive phyllotaxis and patterning in biology, founded on the occurrence of the regular asymmetry of binary division» (Spears and Bicknell- Johnson 1998). 

2.4. Golden human genome 

M.E.B. Yamagishi and A.I. Shimabukuro (2008, 643) found that nuc- leotide frequency in the human genome can be accurately approx- imated and predicted using Fibonacci numbers. It is interesting to note the solution was arrived at as part of an optimization problem – closely mimicking nature. 

J.C. Perez (2010) discovered that codon populations in single- stranded whole human genome DNA are fractal and fine-tuned by the golden ratio or 1.618…. This discovery was found by first comparing CG major and minor codons as well as TA major and minor codons. Then comparing the CG ratios with the TA ratios produces the result of 0.2618844228 which is within thousandths of a percent of match- 

. Perez also discovered that there appears to be two binary code attractors for the human genome. The top state matches the Lesser golden ratio ( 1/   0.618…). The bottom state 

matches one half the Lesser golden ratio ( / 2  0.309…). These two 

states create a self-organizing bistable binary code. Amazingly these two states are in a perfect octave of one another, the top state being exactly twice the bottom state. Here an example of beauty through musical harmonics being reflected in the function of DNA. 

3.  chemistry
3.1. Quasicrystals
Daniel Shechtman was awarded the 2011 Nobel Prize for his discov- ery of a unique form of substance called quasicrystals. On April 8, 1982, while studying an aluminum and manganese compound, Shech- tman observed crystals with 10 points. But this meant they had pen- tagonal symmetry, which of course is replete with the golden ratio. This seemed impossible according to the prevailing paradigm and established ‘laws’ of crystallography as to how crystals can form. Shechtman even remarked to himself in Hebrew: «Eyn chaya kazo!» («There can be no such creature!»). But there they were, though changing the paradigm would take time. 

Shortly after the discovery, quasicrystals were synthesized in labs, ultimately around the world. They have regular patterns, but they never repeat. Because of their uneven structure, having no ob- vious cleavage planes, they are extremely strong and durable. They lack surface friction, don’t react with anything, and do not oxidize or become rusty. And then in 2009 the first naturally occurring quasi- crystals were discovered. They are found in some of the most durable forms of steel, and are now used, for example, in razor blades and thin steel needles for eye surgery. 

Quasicrystals had been anticipated earlier by Johannes Kepler who drew quasicrystal-like patterns in Mysterium cosmographicum. Some even felt that Sir Roger Penrose should have been awarded the Nobel Prize because of his discovery of non-periodic pentagonal tiling patterns, which are a two-dimensional analog directly relevant to Shechtman’s three-dimensional discovery. Shechtman admitted to Penrose that he may have been unconsciously influenced by the Pe- nrose tiling discovery. But the gracious Penrose acknowledged that he may himself have been unconsciously influenced by the drawings of Kepler in his own discovery. 

These discoveries were actually anticipated by early Islamic pentagonal pattern mosaics in mosques. They are evident in the Al- hambra Palace in Spain and the Darb-I Imam Shrine in Iran. An anal- ogy or link between the beauty of the quasicrystal and pentagonal architectural form. 

3.2. C60, the fullerene 

Another golden ratio related discovery led to the 1996 Nobel Prize in chemistry for Harry Kroto of the University of Sussex, and Richard Smalley and Robert Curl of Rice University. Kroto (along with Cana- dian radio-astronomers) had discovered long linear carbon chains in interstellar space and was certain these long flexible molecules had been created in red giant stars rich in carbon. In 1985, using graphite heated with a laser-supersonic cluster beam, they replicated the car- bon chains and also discovered a perplexing set of stable spheres, particularly in the C60 range and some in the C70 range. These spheres formed incredibly strong stable structures which they called a ‘wadge’ – British for ‘a handful of stuff’. Smalley began calling C60 the ‘mother wadge’, and Kroto called it the ‘God wadge’. 

The C60 is a truncated icosahedron, an icosahedron in which the corners have been ‘truncated’ or cut off (Figure 4). It consists of 12 pentagons (like a dodecahedron) and 20 hexagons. If the edge of a truncated icosahedron is equal to one or unity, the distance across to an opposing edge is exactly 3 . 

Therefore, the inner scaffolding of C60 is made up of three per- pendicular intersecting 3x 1 rectangles (Figure 5). And in C60 the carbon atoms are located at each of the 60 vertices (corners, Figure 6). 

Smalley succeeded in building a model by interspersing penta- gons amongst the hexagons. Not knowing what it was, he later re- ported: ‘I was ecstatic and overtaken with its beauty!’. The group pre- sented it to Bill Veech of the Rice Mathematics Department asking him what it was. Veech responded: ‘I could explain this to you in a number of ways, but what you got there, boys, is a soccer ball’. They decided to call the C60 a Buckminsterfullerene or Buckyball for short. The entire group of spheres was named Fullerenes. Though not ac- cepted until 1990 when scientists at both the Max Planck Institute for Nuclear Physics in Germany and the University of Arizona succeeded in synthesizing a sufficient amount of C60 to study its structure. In 1992 C60 was found in Russia in a family of minerals called Shungites. Lastly huge fullerenes have been discovered made up of hundreds and sometimes thousands of atoms, yet still maintaining the «same beautiful latticework shells» (American Chemical Society 2010). 

3.3. Buckyballs in outer space 

In 2010 NASA’s orbiting Spitzer infrared telescope recorded Buck- yballs, the largest molecules ever found in space, in the nebula around a distant, white dwarf star. Jan Cami reported: ‘When we saw these whopping spectral signatures, we knew immediately that we were looking at one of the most sought-after molecules’. Bernard H. Foing responded that he and P. Ehrenfreund: ‘measured previously 

[in 1994] the spectral fingerprint signature of buckyballs in many directions in the galaxy. […] This C60+ feature has been observed in many lines of sight and seems ubiquitous in the interstellar medium’. 

3.4. Fibonacci periodic table 

Recently, Ukrainian researcher Sergey Jakushko analyzed Men- deleev’s Periodic System of elements in terms of Fibonacci numbers (Jakushko 2010). If the atomic mass of each element in a period is divided by the atomic mass of the last element in the period, i.e. a noble gas, the results can be graphed thereby obtaining an average line for the period. Performing this for each period allows for a com- parison of the slopes of each average line. It turns out that the slopes, starting with the first period up to the last period, change by the fol- lowing law: 

This numerical sequence is simply the inverse of the Fibonacci num- bers: 1, 1, 2, 3, 5, 8, 13…. Thus, Jakushko uncovered a new Fibonacci regularity underlying the Periodic Table of Elements, linking it to the aesthetics of the golden ratio as expressed in nature. 

4. Golden physics
4.1. Nested vibration, chaos border and winding number Mohamed El Naschie recognized that because the golden ratio is the most unique of all constants, being the simplest continued fraction (Figure 1), simplest nested radical (Figure 2), and the most irrational of all irrational numbers, it is the primary candidate to give stability through what is called a ‘nested vibration’.
El Naschie wrote: «It is the simplest realistic unit from which a Hamil- tonian dynamics can start developing a highly complex structure, a so-called nested vibration». And the chaos border between order and chaos, called «the KAM Theorem (after Kolmogorov, Arnold and Mos- er), asserts that the most stable periodic orbit is that which has an irrational ratio of resonance frequencies. Since the Golden Mean is the most irrational number, the corresponding orbit is the most sta- ble orbit». And «in the view of string theory, particles are vibrating strings. Therefore, to observe a particle, the corresponding vibration 

pastedGraphic_2.png pastedGraphic_2.png pastedGraphic_3.png pastedGraphic_4.png pastedGraphic_5.png pastedGraphic_4.png pastedGraphic_6.png 

must be stable and that is only possible in the [aforementioned] KAM interpretation [known as the VAK Cantorian theory of vacuum fluctu- ation]… when the winding number corresponding to this dynamics is equal to the Golden Mean!» (quoted in Olsen 2006, 57). 

4.2. Fine structure constant, quark masses, “harmonic musical ladder”
El Naschie discovered that the standard model of particle physics, when viewed through the eyes of E-Infinity Theory, appears to be ‘a cosmic symphony’ with incredible states of harmonic resonance. In effect, he may have found the Pythagorean Music of the Spheres present in the world of physics. And from his E-Infinity Theory he deduced the fine structure constant of nature (204= 137.0820…), the masses of quarks – the most elementary known constituents of matter, as well as the masses of the electron, proton, neutron and the other sub-atomic particles! El Naschie stated that: «the quark masses are in excellent agreement with the majority of the scarce and diffi- cult to obtain data about the mass of quarks. It takes only one look at these values for anyone to realize that they form a harmonic musical 

pastedGraphic_7.png 

ladder!» (Olsen 2006, 57). This raises a very interesting issue regard- ing the Higgs Boson which is said to give quarks their masses. This is a direct example of aesthetics resulting from the golden modular of nature itself. 

4.3. Entanglement, nonlocality and  5 

In 1993, Lucien Hardy wrote a paper titled Nonlocality for two par- ticles without inequalities. This was a masterful piece of work, a thought experiment, in which Hardy demonstrated that entangle- ment occurs with the probability of 9.017%. Unfortunately Hardy rounded off the actual calculation, and as a result, at least initially, El Naschie and Penrose (two of the few in the world who would have recognized it) missed it. The result suggests that in roughly 1 out of 11, or 9 out of 100 trials, entanglement of two particles will be ob- served. This also is tantamount to 9 instances in 100 trials where a quantum particle will be in two different locations at exactly the same time! 

Eventually El Naschie realized that Hardy had rounded off his result. The actual calculation was 0.09169945… or written as a per- centage, should be 9.0169945…%. Hardy apparently did not realize it, but he had discovered that entanglement occurs at the ‘Lesser’ golden ratio,  , raised to the 5th power, i.e. 5  (0.6180339…)5  0.09169945 . 

El Naschie found that Hardy’s result was perfectly consistent with what he derived through E  Theory employing ‘fractal’ Cantor sets. 

Thus, even the quantum mystery of entanglement is guided by the aesthetics of the golden ratio. 

4.4. Solving quantum paradoxes 

The great insight of El Naschie was the realization that fractal Cantor sets (driven by the golden ratio) allow one to not only capture the fractal nature of quantum theory, but resolve its difficult paradoxes, including the wave-particle duality paradox. When using Cantor sets, the quantum particle state is a ‘zero measure Cantor set’ of (0; ) 

equal to 0.6180339… and the quantum wave state is the ‘empty 

measure Cantor set’ of (-1;2) equal to 0.3819660…. The wave and 

particle are separated by (i.e. related through) what is beginning to appear to be the Modular of Nature itself:  ≈ 1.6180339…. Hence, 

56 pastedGraphic_8.pngScott Olsen 

multiply a wave by  and it transforms into a particle. Divide a par- ticle by  and it transforms into a wave. This in fact mimics (or is mimicked by) the interplay of adjacent Fibonacci numbers in nature. Multiply a Fibonacci number by the Modular≈ 1.6180339… and you will get an approximation to the next Fibonacci number. Of course, divide a Fibonacci number by the Modular  and you will get an approximation to the previous Fibonacci number. 

What is lurking behind the scenes here, and most importantly for our deeper interests, is the fact that these ‘fractal Cantor sets’ – guided by the golden ratio – allow for nonlocality. When there is no spatial or temporal separation at the most fundamental level, as Bohm and others had always maintained with quantum mechanics, there is immediate contact. This has a tremendous bearing on all is- sues related to consciousness research, morphic resonance and PSI phenomena. 

4.5. Dark energy 

El Naschie concludes that the energy given by Einstein’s famous for- mula E = mc2 consists of two parts. «The first part is the positive energy of the quantum particle modeled by the topology of the zero set [ ≈ 0.618…]. The second part is the absolute value of the negative 

energy of the quantum Schrödinger wave modeled by the topology of the empty set [2 ≈ 0.381…]. We reason that the latter is nothing else 

but the so-called missing dark energy [inclusive of dark matter] of the universe which accounts for 94.45% of the total energy, in full agreement with the WMAP and Supernova cosmic measurement which was awarded the 2011 Nobel Prize in physics. The dark energy of the quantum wave cannot be detected in the normal way because measurement collapses the quantum wave» (El Naschie 2013, 591). 

4.6. Quantum mechanics and golden symmetry 

On January 8, 2010, researchers in Germany (Helmhotz-Zentrum Ber- lin) working with colleagues in England (Oxford and Bristol Universi- ties, and the Rutherford Appleton Laboratory) reported that they had discovered the golden ratio in quantum mechanics. Applying a mag- netic field to cobalt niobate (CoNb2O6), their press release stated that they: «have for the first time observed a nanoscale [golden ratio] 

symmetry hidden in solid state matter. […] By tuning the system and artificially introducing more quantum uncertainty the researchers observed that the chain of atoms acts like a nanoscale guitar string». Dr. Radu Coldea of Oxford University, principal author of the paper in the journal «Science», later explained: «The tension comes from the interaction between spins causing them to magnetically resonate. For these interactions we found a series (scale) of resonant notes: The first two notes show a perfect relationship with each other. Their fre- quencies (pitch) are in the ratio of 1.618…, which is the golden ratio famous from art and architecture. […] It reflects a beautiful property of the quantum system – a hidden symmetry. Actually quite a special one called E8 by mathematicians, and this is its first observation in a material» (Coldea 2010). 

5. Consciousness 

In 1995, University of Oxford mathematics professor Sir Roger Pe- nrose, and University of Arizona anesthesiology professor Stuart Ha- meroff, in a paper titled Quantum computing in microtubules: self- collapse as a possible mechanism for consciousness, proposed that con- sciousness can be explained as quantum computations orchestrated through groups of microtubules in the neurons of the brain. They ar- gued that the microtubules in a quantum coherent state could go through objective reduction or what in quantum physics is called the collapse of the wave function. In quantum theory, probability waves can be held in superposition, being in two places at once. When ob- servation occurs, the wave can collapse into an objective actuality. This is consistent with the present author’s position that the 2 wave 

collapses into its corresponding  particle as expressed in Section 4.4. 

Microtubules are self-assembling polymers (i.e. large molecules or macromolecules composed of repeating structural units) located in the cytoskeleton within neurons (and all cells). They are hollow cylinders composed of 13 linear tubulin chains (protofilaments) that align side-by-side, forming spiral patterns that are similar to a pine- cone or sunflower, with 8 spirals winding in one direction and 5 spir- als in the opposite direction (Figure 8). 

Occasionally double-microtubules form with 21 spirals, 13 spiraling in one direction and 8 in the other! Thus, the double microtubule has ‘addi- tively and geometrically’ moved up what one might term a ‘golden or Fibonacci octave’ so to speak. Again, the aesthetic principle is manifest in the microtubule structure itself. 

Early on in his book, Shadows of the mind, a
search for the missing science of consciousness, Pe-
nrose raised the question (perhaps somewhat rhe-
torically): «Why do Fibonacci numbers arise in mi-
crotubule structure?» (Penrose and Hameroff 1994, 362). Seventeen years later Penrose and Hameroff wrote that the «multiple winding patterns […] matching the Fibonacci series found widely in nature and possessing a helical symmetry, [are] suggestively sympathetic to large-scale quantum processes» (Penrose and Hameroff 2011, 226). 

As part of the cytoskeleton, the microtubules establish cell shape, direct growth, and organize cellular functions, «defining cell architecture like girders and beams in a building» (Penrose and Ha- meroff 2011, 226). But their lattice structure can be compared to computational systems. Penrose and Hameroff see them as biomole- cular quantum computers. This means there can be entanglement, superposition (being in two places at once), and immediate ‘nonlocal connection’ even when appearing ‘locally’ separated (see Section 4.3). 

And where quantum connections occur, the atoms can become quantum coherent. Quantum ‘in-phase states’ like this occur, for ex- ample, when the atoms of helium-4 are cooled to near Absolute Zero and become highly coherent, leading to superfluidity. Analogous phe- nomena also occur in the coherent light of lasers and in superconduc- tivity. The important point is that this creates a Bose-Einstein conden- sate where the atoms are said to «resonate in-phase within a com- mon Schrodinger wave function» (Merrick 2009, 196). Is there an underlying ‘golden in-phase resonance’ at work here? 

A serious question arose regarding the Penrose-Hameroff mi- crotubule/quantum computer hypothesis. Provocative research in 2003 began to demonstrate that quantum coherence occurs even in warm biological systems, including ‘bird brain’ navigation, DNA, pro- tein folding, biological water and microtubules. In fact Penrose and 

Hameroff have given this golden ratio driven resonance greater clari- ty, stating: «Moreover, geometrical resonances in microtubules, e.g. following helical pathways of Fibonacci geometry are suggested to enable topological quantum computing and error correction, avoid- ing decoherence perhaps effectively indefinitely as in a superconduc- tor» (Penrose and Hameroff 2011, 242). 

Clathrins, located at the tips of microtubules in the axon’s syn- aptic boutons, are buckyball shaped proteins (Figures 8 and 9) that selectively sort cargo at the cell membranes. As truncated icosahedra they have internal rectangles constructed in the ratio3:1 (see Fig- ure 2). During mitosis the clathrins bind directly to the microtubules (or microtubule-associated proteins called MAPS). But most impor- tantly, together with microtubules the clathrins regulate synaptic activity. The suggestion here is that their ‘golden in-phase resonance’ and attendant aesthetics may hold a central key to the mystery of consciousness itself! Hence, beauty, function and illumination may be intimately tied together. 

6. Conclusion 

The evidence presented here for the pervasiveness of the golden ratio throughout the sciences merely scratches the surface and one could easily cite more instances of its occurrence. The potential applica- tions for the golden ratio stretch from the macrocosmic down to the microcosmic scale, appearing to have scale independent relevance to everything in between. All of this scientific evidence for the role of the golden ratio throughout nature and the cosmos provides a very 

strong argument for the scientific foundation of aesthetics in the world. It should not be surprising that that which runs most perfectly is the most beautiful, and the most beautiful may be involved in illu- mination. 

References and footnotes

  1. Jump up
    ^
    Note: If the constraint on a and b each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. ϕ is defined as the positive solution. The negative solution can be written as 1

    5

    2

    {\displaystyle {\frac {1-{\sqrt {5}}}{2}}}

    . The sum of the two solutions is one, and the product of the two solutions is negative one.

  2. ^ Jump up to:
    a b c OEISA001622
  3. ^ Jump up to:
    a b c d e f g Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number. New York: Broadway Books. ISBN 0-7679-0815-5.
  4. ^ Jump up to:
    a b c Piotr Sadowski (1996). The knight on his quest: symbolic patterns of transition in Sir Gawain and the Green Knight. University of Delaware Press. p. 124. ISBN 978-0-87413-580-0.
  5. ^ Jump up to:
    a b Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997
  6. ^ Jump up to:
    a b Euclid, Elements, Book 6, Definition 3.
  7. Jump up
    ^
    Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) p. 37. “And the same applies in architecture, to the rectangles representing these and other ratios (e.g. the ‘golden cut’). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.”
  8. Jump up
    ^
    Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920
  9. Jump up
    ^
    William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport Publishers, 2003
  10. ^ Jump up to:
    a b c Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.
  11. Jump up
    ^
    Strogatz, Steven (September 24, 2012). “Me, Myself, and Math: Proportion Control”. The New York Times.
  12. ^ Jump up to:
    a b Weisstein, Eric W. “Golden Ratio Conjugate”. MathWorld.
  13. Jump up
    ^
    Markowsky, George (January 1992). “Misconceptions about the Golden Ratio” (PDF). The College Mathematics Journal. 23 (1).
  14. Jump up
    ^
    Mario Livio,The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number, p.6
  15. Jump up
    ^
    ῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν, translated in Richard Fitzpatrick (translator) (2007). Euclid’s Elements of Geometry. ISBN 978-0615179841., p. 156
  16. Jump up
    ^
    Euclid, Elements, Book 6, Proposition 30. Retrieved from http:/.aleph0.clarku.edu/~djoyce/java/elements/toc.html.
  17. Jump up
    ^
    Euclid, Elements, Book 2, Proposition 11; Book 4, Propositions 10–11; Book 13, Propositions 1–6, 8–11, 16–18.
  18. Jump up
    ^
    “The Golden Ratio”. The MacTutor History of Mathematics archive. Retrieved 2007-09-18.
  19. Jump up
    ^
    Weisstein, Eric W. “Golden Ratio”. MathWorld.
  20. Jump up
    ^
    Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp. 20–21. ISBN 1-4027-3522-7.
  21. Jump up
    ^
    Plato. “Timaeus”. Translated by Benjamin Jowett. The Internet Classics Archive. Retrieved 30 May 2006.
  22. ^ Jump up to:
    a b James Joseph Tattersall (2005). Elementary number theory in nine chapters (2nd ed.). Cambridge University Press. p. 28. ISBN 978-0-521-85014-8.
  23. Jump up
    ^
    Underwood Dudley (1999). Die Macht der Zahl: Was die Numerologie uns weismachen will. Springer. p. 245. ISBN 3-7643-5978-1.
  24. Jump up
    ^
    Cook, Theodore Andrea (1979) [1914]. The Curves of Life. New York: Dover Publications. ISBN 0-486-23701-X.
  25. Jump up
    ^
    Gardner, Martin (2001), The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems : Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics, W. W. Norton & Company, p. 88, ISBN 9780393020236.
  26. Jump up
    ^
    Jaric, Marko V. (2012), Introduction to the Mathematics of Quasicrystals, Elsevier, p. x, ISBN 9780323159470, “Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.”
  27. Jump up
    ^
    Van Mersbergen, Audrey M., “Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic”, Communication Quarterly, Vol. 46 No. 2, 1998, pp 194–213.
  28. Jump up
    ^
    Midhat J. Gazalé , Gnomon, Princeton University Press, 1999. ISBN 0-691-00514-1
  29. Jump up
    ^
    Keith J. Devlin The Math Instinct: Why You’re A Mathematical Genius (Along With Lobsters, Birds, Cats, And Dogs), p. 108. New York: Thunder’s Mouth Press, 2005, ISBN 1-56025-672-9
  30. Jump up
    ^
    Boussora, Kenza and Mazouz, Said, The Use of the Golden Section in the Great Mosque of Kairouan, Nexus Network Journal, vol. 6 no. 1 (Spring 2004), [1]
  31. Jump up
    ^
    Le Corbusier, The Modulor p. 25, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6
  32. Jump up
    ^
    Marcus Frings: The Golden Section in Architectural Theory, Nexus Network Journal vol. 4 no. 1 (Winter 2002), available online [2]
  33. Jump up
    ^
    Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. ISBN 0-419-22780-6: “Both the paintings and the architectural designs make use of the golden section”.
  34. Jump up
    ^
    Urwin, Simon. Analysing Architecture (2003) pp. 154–5, ISBN 0-415-30685-X
  35. Jump up
    ^
    Jason Elliot (2006). Mirrors of the Unseen: Journeys in Iran. Macmillan. pp. 277, 284. ISBN 978-0-312-30191-0.
  36. Jump up
    ^
    Patrice Foutakis, “Did the Greeks Build According to the Golden Ratio?”, Cambridge Archaeological Journal, vol. 24, n° 1, February 2014, pp. 71–86.
  37. Jump up
    ^
    Leonardo da Vinci’s Polyhedra, by George W. Hart[3]
  38. Jump up
    ^
    Livio, Mario. “The golden ratio and aesthetics”. Retrieved 2008-03-21.
  39. Jump up
    ^
    “Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like ‘which the ancient Greeks and others believed to have divine and mystical properties.’ Almost as compulsive is the urge to add a second factoid along the lines of ‘Leonardo Da Vinci believed that the human form displays the golden ratio.’ There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.” Keith Devlin (May 2007). “The Myth That Will Not Go Away”. Retrieved September 26, 2013.
  40. Jump up
    ^
    Donald E. Simanek. “Fibonacci Flim-Flam”. Archived from the original on February 1, 2010. Retrieved April 9, 2013.
  41. Jump up
    ^
    Salvador Dalí (2008). The Dali Dimension: Decoding the Mind of a Genius (DVD). Media 3.14-TVC-FGSD-IRL-AVRO.
  42. Jump up
    ^
    Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47, ISBN 1-883001-51-X
  43. Jump up
    ^
    Bouleau, Charles, The Painter’s Secret Geometry: A Study of Composition in Art (1963) pp. 247–8, Harcourt, Brace & World, ISBN 0-87817-259-9
  44. Jump up
    ^
    Olariu, Agata, Golden Section and the Art of Painting Available online
  45. Jump up
    ^
    Tosto, Pablo, La composición áurea en las artes plásticas – El número de oro, Librería Hachette, 1969, p. 134–144
  46. Jump up
    ^
    Jan Tschichold. The Form of the Book, pp.43 Fig 4. “Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well.”
  47. Jump up
    ^
    Jan Tschichold, The Form of the Book, Hartley & Marks (1991), ISBN 0-88179-116-4.
  48. Jump up
    ^
    Jones, Ronald (1971). “The golden section: A most remarkable measure”. The Structurist. 11: 44–52. “Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?”
  49. Jump up
    ^
    Art Johnson (1999). Famous problems and their mathematicians. Libraries Unlimited. p. 45. ISBN 978-1-56308-446-1. “The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.”
  50. Jump up
    ^
    Alexey Stakhov; Scott Olsen; Scott Anthony Olsen (2009). The mathematics of harmony: from Euclid to contemporary mathematics and computer science. World Scientific. p. 21. ISBN 978-981-277-582-5. “A credit card has a form of the golden rectangle.”
  51. Jump up
    ^
    Simon Cox (2004). Cracking the Da Vinci code: the unauthorized guide to the facts behind Dan Brown’s bestselling novel. Barnes & Noble Books. ISBN 978-0-7607-5931-8. “The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.”
  52. Jump up
    ^
    “THE NEW RAPIDE S : Design”. “The ‘Golden Ratio’ sits at the heart of every Aston Martin.”
  53. Jump up
    ^
    Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.
  54. Jump up
    ^
    Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (New York: Routledge, 2003) pp 83, ISBN 0-415-30010-X
  55. Jump up
    ^
    Roy Howat (1983). Debussy in Proportion: A Musical Analysis. Cambridge University Press. ISBN 0-521-31145-4.
  56. Jump up
    ^
    Simon Trezise (1994). Debussy: La Mer. Cambridge University Press. p. 53. ISBN 0-521-44656-2.
  57. Jump up
    ^
    “Pearl Masters Premium”. Pearl Corporation. Archived from the original on December 19, 2007. Retrieved December 2, 2007.
  58. Jump up
    ^
    An 833 Cents Scale: An experiment on harmony“, Huygens-Fokker.org. Accessed December 1, 2012.
  59. Jump up
    ^
    Richard Padovan (1999). Proportion. Taylor & Francis. pp. 305–306. ISBN 978-0-419-22780-9.
  60. Jump up
    ^
    Padovan, Richard (2002). “Proportion: Science, Philosophy, Architecture”. Nexus Network Journal. 4 (1): 113–122. doi:10.1007/s00004-001-0008-7.
  61. Jump up
    ^
    Zeising, Adolf (1854). Neue Lehre van den Proportionen des meschlischen Körpers. preface.
  62. Jump up
    ^
    “Golden ratio discovered in a quantum world”. Eurekalert.org. 2010-01-07. Retrieved 2011-10-31.
  63. Jump up
    ^
    J.C. Perez (1991), “Chaos DNA and Neuro-computers: A Golden Link”, in Speculations in Science and Technology vol. 14 no. 4, ISSN 0155-7785.
  64. Jump up
    ^
    Yamagishi, Michel E.B., and Shimabukuro, Alex I. (2007), “Nucleotide Frequencies in Human Genome and Fibonacci Numbers”, in Bulletin of Mathematical Biology, ISSN 0092-8240 (print), ISSN 1522-9602 (online). PDF full text
  65. Jump up
    ^
    Perez, J.-C. (September 2010). “Codon populations in single-stranded whole human genome DNA are fractal and fine-tuned by the Golden Ratio 1.618”. Interdisciplinary Sciences: Computational Life Science. 2 (3): 228–240. doi:10.1007/s12539-010-0022-0. PMID 20658335. PDF full text
  66. Jump up
    ^
    Pommersheim, James E., Tim K. Marks, and Erica L. Flapan, eds. 2010. “Number Theory: A Lively Introduction with Proofs, Applications, and Stories”. John Wiley and Sons: 82.
  67. Jump up
    ^
    The golden ratio and aesthetics, by Mario Livio.
  68. Jump up
    ^
    Max. Hailperin; Barbara K. Kaiser; Karl W. Knight (1998). Concrete Abstractions: An Introduction to Computer Science Using Scheme. Brooks/Cole Pub. Co. ISBN 0-534-95211-9.
  69. Jump up
    ^
    Brian Roselle, “Golden Mean Series”
  70. Jump up
    ^
    “A Disco Ball in Space”. NASA. 2001-10-09. Retrieved 2007-04-16.
  71. Jump up
    ^
    Chris and Penny. “Quandaries and Queries”. Math Central. Retrieved 23 October 2011.
  72. Jump up
    ^
    American Mathematical Monthly, pp. 49–50, 1954.
  73. ^ Jump up to:
    a b c Roger Herz-Fischler (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5.
  74. Jump up
    ^
    Koca, Mehmet; Koca, Nazife Ozdes; Koç, Ramazan (2010), “Catalan solids derived from three-dimensional-root systems and quaternions”, Journal of Mathematical Physics, 51: 043501, arXiv:0908.3272, Bibcode:2010JMP….51d3501K, doi:10.1063/1.3356985.
  75. Jump up
    ^
    Fibonacci Numbers and Nature – Part 2 : Why is the Golden section the “best” arrangement?, from Dr. Ron Knott’s Fibonacci Numbers and the Golden Section, retrieved 2012-11-29.
  76. Jump up
    ^
    Weisstein, Eric W. “Pisot Number”. MathWorld.
  77. Jump up
    ^
    Horocycles exinscrits : une propriété hyperbolique remarquable, cabri.net, retrieved 2009-07-21.
  78. Jump up
    ^
    Yee, Alexander J. (17 August 2015). “Golden Ratio”. numberword.org. Independent computations done by Ron Watkins and Dustin Kirkland.
  79. Jump up
    ^
    Radio, Astraea Web (2006). The Best of Astraea: 17 Articles on Science, History and Philosophy. Astrea Web Radio. ISBN 1-4259-7040-0.
  80. Jump up
    ^
    Midhat Gazale, Gnomon: From Pharaohs to Fractals, Princeton Univ. Press, 1999
  81. ^ Jump up to:
    a b c Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000
  82. ^ Jump up to:
    a b c “The Great Pyramid, The Great Discovery, and The Great Coincidence”. Archived from the original on 2014-01-02. Retrieved 2007-11-25.
  83. Jump up
    ^
    Lancelot Hogben, Mathematics for the Million, London: Allen & Unwin, 1942, p. 63., as cited by Dick Teresi, Lost Discoveries: The Ancient Roots of Modern Science—from the Babylonians to the Maya, New York: Simon & Schuster, 2003, p.56
  84. Jump up
    ^
    Burton, David M. (1999). The history of mathematics: an introduction (4 ed.). WCB McGraw-Hill. p. 56. ISBN 0-07-009468-3.
  85. Jump up
    ^
    Bell, Eric Temple (1940). The Development of Mathematics. New York: Dover. p. 40.
  86. Jump up
    ^
    Rice, Michael, Egypt’s Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C pp. 24 Routledge, 2003, ISBN 0-415-26876-1
  87. Jump up
    ^
    S. Giedon, 1957, The Beginnings of Architecture, The A.W. Mellon Lectures in the Fine Arts, 457, as cited in Rice, Michael, Egypt’s Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C pp.24 Routledge, 2003
  88. Jump up
    ^
    Markowsky, George (January 1992). “Misconceptions about the Golden Ratio” (PDF). College Mathematics Journal. Mathematical Association of America. 23 (1): 2–19. doi:10.2307/2686193. JSTOR 2686193.
  89. Jump up
    ^
    Taylor, The Great Pyramid: Why Was It Built and Who Built It?, 1859
  90. Jump up
    ^
    Matila Ghyka The Geometry of Art and Life, New York: Dover, 1977
  91. Jump up
    ^
    Man, John, Gutenberg: How One Man Remade the World with Word (2002) pp. 166–167, Wiley, ISBN 0-471-21823-5. “The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called ‘golden section’, which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.”
  92. ^ Jump up to:
    a b Pheasant, Stephen (1998). Bodyspace. London: Taylor & Francis. ISBN 0-7484-0067-2.
  93. ^ Jump up to:
    a b van Laack, Walter (2001). A Better History Of Our World: Volume 1 The Universe. Aachen: van Laach GmbH.
  94. Jump up
    ^
    Ivan Moscovich, Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles, New York: Sterling, 2004
  95. Jump up
    ^
    Peterson, Ivars. “Sea shell spirals”. Science News.
  96. Jump up
    ^
    For instance, Osler writes that “38.2 percent and 61.8 percent retracements of recent rises or declines are common,” in Osler, Carol (2000). “Support for Resistance: Technical Analysis and Intraday Exchange Rates” (PDF). Federal Reserve Bank of New York Economic Policy Review. 6 (2): 53–68.
  97. Jump up
    ^
    Roy Batchelor and Richard Ramyar, “Magic numbers in the Dow,” 25th International Symposium on Forecasting, 2005, p. 13, 31. “Not since the ‘big is beautiful’ days have giants looked better“, Tom Stevenson, The Daily Telegraph, Apr. 10, 2006, and “Technical failure”, The Economist, Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar’s research.

Further reading

  • Doczi, György (2005) [1981]. The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture. Boston: Shambhala Publications. ISBN 1-59030-259-1.
  • Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications. ISBN 0-486-22254-3.
  • Joseph, George G. (2000) [1991]. The Crest of the Peacock: The Non-European Roots of Mathematics (New ed.). Princeton, NJ: Princeton University Press. ISBN 0-691-00659-8.
  • Livio, Mario (2002) [2002]. The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number (Hardback ed.). NYC: Broadway (Random House). ISBN 0-7679-0815-5.
  • Sahlqvist, Leif (2008). Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and Design (3rd Rev. ed.). Charleston, SC: BookSurge. ISBN 1-4196-2157-2.
  • Schneider, Michael S. (1994). A Beginner’s Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperCollins. ISBN 0-06-016939-7.
  • Scimone, Aldo (1997). La Sezione Aurea. Storia culturale di un leitmotiv della Matematica. Palermo: Sigma Edizioni. ISBN 978-88-7231-025-0.
  • Stakhov, A. P. (2009). The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. Singapore: World Scientific Publishing. ISBN 978-981-277-582-5.
  • Walser, Hans (2001) [Der Goldene Schnitt 1993]. The Golden Section. Peter Hilton trans. Washington, DC: The Mathematical Association of America. ISBN 0-88385-534-8.

External lin

15. DNA molecules

Even the microscopic realm 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.

Thanks to Calvin Dvorsky for helping with the The famous Fibonacci sequence has captivated mathematicians, artists, designers, and scientists for centuries. Also known as the Golden Ratio, its ubiquity and astounding functionality in nature suggests its importance as a fundamental characteristic of the Universe.

We’ve talked about the Fibonacci series and the Golden ratio before, but it’s worth a quick review. The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on forever. Each number is the sum of the two numbers that precede it. It’s a simple pattern, but it appears to be a kind of built-in numbering system to the cosmos. Here are 15 astounding examples of phi in nature.

pastedGraphic_9.png

The Fibonacci Series: When Math Turns Golden

The Fibonacci Series, a set of numbers that increases rapidly, began as a medieval math joke about…

Read on io9. com

Leonardo Fibonacci came up with the sequence when calculating the ideal expansion pairs of rabbits over the course of one year. Today, its emergent patterns and ratios (phi = 1.61803…) can be seen from the microscale to the macroscale, and right through to biological systems and inanimate objects. While the Golden Ratio doesn’t account for every structure or pattern in the universe, it’s certainly a major player. Here are some examples.

1. Flower petals

pastedGraphic_10.png

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 (pictured at left), 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) allowing for the best possible exposure to sunlight and other factors.

2. Seed heads

The 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.

pastedGraphic_11.png

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. Interestingly, a highly irrational number is required to optimize filling (namely one that will not be well represented by a fraction). Phi fits the bill rather nicely.

3. Pinecones

article!

Sources and images: Top: Loskutnikov/Shutterstock; Buttercup: motorolka/shutterstock, ThinkQuest, Shell, Galaxy: FabulousFibonacci, American Museum of Natural History and here, honey bee, Hurricane: MNN, Faces: Goldennumber and here, DNA.

 

Happiness Medicine & Holistic Medicine Posts

Categories

Translate:

Follow me on Twitter

Translate »
error: Content is protected !!