To what extent is there a biological basis for the Golden Ratio?Introduction:

Generally, the game, "he (or she) loves me, he loves me not," is played with a wild daisy, and is therefore, otherwise known as the Daisy Oracle game. The petals are plucked one-by-one off the flower as the well-known couplet is recited, leaving the last petal and line of the poem to determine the answer to the daunting question. Unfortunately, simply by starting the Daisy Oracle game, players are faced with the inevitable “he loves me not,” answer. This is due to the fact that most wild daisies contain thirty-four petals, a number part of the Fibonacci sequence, producing the infamously heartbreaking answer. Whilst this may seem an omen of bad luck, believe it of not, there is a mathematical explanation.

Abstract:

The Golden Ratio is a mathematical curiosity that has been said to occur in numerous biological systems. However, it has been debated whether the Golden Ratio’s presence signifies a breakthrough ‘universal’ ratio or simply an interesting phenomenon taken too far. The scientific evidence and explanations, possible applications, as well as the psychology behind the Golden Ratio warrant exploration.

Mathematical/Historical Explanation:

Around 300 B.C., Euclid, a Greek mathematician, challenged himself and his fellow geometers to divide any line segment into two unequal parts, where the ratio of the larger section to the smaller section is the same as the complete line to the larger section. He completed this task himself and coined the term “extreme and mean ratio” to describe what is now more commonly known as the Golden Ratio. 1

Euclid, Thomas L. Heath, and Dana Densmore, Euclid's Elements: All Thirteen Books 1

Complete in One Volume, ed. by Thomas L. Heath (Santa Fe, N.M: Green Lion Press, 2002.

�1

Phi

This number has many additional names such as Phi, the Divine Proportion, the Golden Section, and the most irrational number, but is most bluntly rounded to 1.618. This number can be obtained algebraically with the aforementioned line segment.

Let the length of the shorter section equal 1 and the length of the longer section equal x. The length of the entire line segment is therefore, (x+1). The ratio of the longer section to the shorter section is therefore x/1, and the ratio of the entire line to the longer section is (x+1)/x. In order to meet the criteria of the Golden Ratio, these proportions must be equal, so that x/1=(x+1)/x. This equation can be rewritten into the form x^2-x-1=0, for which the positive solution is x=(1+√5)/2, which is equal to the Golden Ratio. Many years later, Phi became the centre of another mathematical dilemma.

The Rabbit Problem (Fibonacci Numbers):

Leonardo of Pisa, better known as Fibonacci, was an Italian mathematician whose answer to a problem revolutionised the way we think about patterns. In his 1202 2

book, Liber Abaci, Fibonacci detailed an illustrious hypothetical population dynamics problem.

Suppose a pair of newly born rabbits, one male and one female, are placed in a field surrounded on all sides by a wall. The rabbits are fully grown after a month. The pair then mates, and at the end of the second month, another pair of rabbits is born (one male and one female). After another month, these rabbits have reached maturity and can mate. The original pair mates too, and now there are five pairs of rabbits (two not-grown and three grown). If this pattern continues as described, how many rabbits will there be after one year?

At the end of the fourth month, the three pairs of mature rabbits mate. That leaves us with eight pairs of rabbits in total. At

Thomas, Rachel, “The Fibonacci sequence: A brief introduction.” plus.maths.org, https://2

plus.maths.org/content/fibonacci-sequence-brief-introduction. [accessed 25 Jun. 2017].

�2

this point, we can look at the sequence and determine the pattern: 1, 2, 3, 5, 8 pairs of rabbits. Each number in the sequence, deservingly dubbed the Fibonacci sequence, is defined by the sum of the two preceding ones. Therefore, to determine the number of rabbit pairs at the end of five months, all we have to do is add together 5 and 8, the two previous numbers of rabbit pairs logged. If we continue to do this until the end of the year…1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 we can see that there are 233 pairs of rabbits, making the final answer to the challenging rabbit problem not so challenging, with 466 rabbits in total.

As one can probably guess, this situation is clearly theoretical as it deals with perfect reproductive circumstances; thus, Fibonacci had to declare some assumptions: no rabbits die, each mature female reproduces at the end of every month, and every time a female reproduces, she gives birth to exactly one male and one female rabbit. In reality, a rabbit litter is around six, so in solving the acclaimed rabbit 3

population dilemma, Fibonacci really just discovered a new type of problem… 184,597,433,860 little bunnies from just one rabbit after seven years.4

Fibonacci numbers are also prevalent in several flowering plants. Whilst this is not always true, and scientists are not sure as to why this occurs, it is interesting that such a large number of flowers have a Fibonacci number of petals. For example, lilies and irises contain three petals, primroses, buttercups, and wild roses have five petals, delphiniums have eight petals, corn marigolds contain thirteen, and black-eyed Susan's have twenty-one petals. Finally, the wild daisy most often has the “he loves me not” number of, or thirty-four petals.

Knott, R. fibandphi (AT) ronknott.com. “Fibonacci Numbers and Nature - Part 2 Why is the 3

Golden section the "best" arrangement?” The Fibonacci Numbers and Golden Ratio in Nature - 2, www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html#section2.2. [Accessed 23 July 2017].

Lewis, Hazel, “The mathematics of Rabbit Island.” mathscareers.org.uk <http://4

www.mathscareers.org.uk/article/the-mathematics-of-rabbit-island/. [Accessed 14 Sept. 2017].

�3

Fibonacci’s numbers are astonishing in that they produce one of the most famous number sequences. In fact, the Fibonacci sequence was the first recursive sequence (a number sequence determined by using preceding terms to define succeeding terms) known in Europe. Fibonacci’s numbers are the approximations, in terms of whole numbers, of the Golden Ratio. If one were to look at the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and divide the adjacent Fibonacci numbers, the ratio would get closer and closer to 1.618. However, this process can be tedious, so in order to solve similar problems with Fibonacci numbers, the sequence can be expressed mathematically with the formula, Fn+2=Fn+1 + Fn. In this formula, n+2 is the unknown number, n+1 is the number just before, and n is the number preceding the latter. With this formula in mind, one can declare that Fn+1/Fn converges on the Golden Ratio.

The Myth of Phi- Nautilus Shell:

Perhaps the most iconic symbol of the Golden Ratio is a nautilus shell. With its beautifully designed spiral, it would seem to be nature’s quintessential display. For many years, this was thought to be the case, that is- until Clement Falbo proved otherwise in 2005.5

It had long been believed that the nautilus shell was a Golden Spiral, a configuration based upon the Golden Ratio

and derived from a Golden Rectangle. In order to create this, one should begin with two identical squares (one arbitrary unit) adjacent to one another. Next, another square (two arbitrary units) should be drawn, touching both of the already existing squares. A three-arbitrary unit square should then be drawn, touching both one of the original squares and the two-arbitrary unit square. Then, another square (five arbitrary units), should be added, touching the one-arbitrary unit original squares, as well as the three-arbitrary unit

Falbo. Clement. “The Golden Ratio- A Contrary Viewpoint,” Vol. 36, No.2, March 2005, 5

The College Mathematics Journal, http://web.sonoma.edu/Math/faculty/falbo/cmj123-134. [Accessed 21 Jun. 2017].

�4

Fibonacci Numbers (Fn)

Successive Fibonacci Number (Fn+1)

(Fn+1)/(Fn)

1 1 1.000000000

1 2 2.000000000

2 3 1.500000000

3 5 1.666666667

5 8 1.600000000

8 13 1.625000000

13 21 1.615384615

21 34 1.619047619

34 55 1.617647059

55 89 1.618181818

89 144 1.617977528

144 233 1.618055556

square. Continuing this process with squares of increasing Fibonacci units will result in the creation of the Golden Rectangle. In order to form the Golden Spiral, the corners of the squares should be connected in a spiral-shape. The spiral increases outward by a growth rate of 1.618 for each quarter-turn. Although this spiral does approximate a Golden Spiral, it would more appropriately be named a Fibonacci spiral, as a Golden Spiral, as found in nature should not end discretely, but continue infinitely.

The Nautilus is classified into the phylum Mollusca, a group also including snails, octopuses, and the giant squid. It is further characterised into the class Cephalopoda, a class typically categorised as the most intelligent and most mobile of all mollusks. The earliest cephalopods found in the fossil record are, in fact, nautiloids; however, these forms, instead of possessing the well-known spiral, were orthoconic, or had a straight shell. Over time, the nautilus evolved to have the recognisable spiral thought to involve the Golden Ratio. Falbo conducted research on Nautilus pompilius, or the chambered nautilus, and found that the actual ratios of the successive quarter turns of the shells were between 1.24mm and 1.43mm, significantly different from 1.618. Despite this, he was able to conclude that the nautilus does contain a logarithmic spiral, but not a Golden Spiral. A Golden Spiral is a logarithmic spiral that grows outward by a factor of the Golden Ratio every 90 degrees. Whilst the non-existence of the Golden Ratio in the nautilus shell was discouraging, investigations into other occurrences of Phi have led to more inspiring hypotheses.

Why Phi?:

Perhaps the growth patterns of petals, seeds, or leaves can best justify the biological benefit of the golden ratio. Consider the seed head of a sunflower- circular in shape and completely covered by seemingly endless seed spirals. Each new seed’s placement is relative to the location of the previous seed.

If a new seed grows at an angle of 180 degrees from the previous seed, or 0.5 turns, the resulting seed head would be two straight arms of seeds and much empty space.6

Knott, R. fibandphi (AT) ronknott.com. “Fibonacci Numbers and Nature - Part 2 Why is the 6

Golden section the "best" arrangement?”

�5

At a 0.6 turn between seeds, five arms of seeds would emerge; again, this does not optimise the available space.7

It seems that if the ratio between seeds is any rational number, the resulting seed head would simply be a fixed number of seed arms, separated by unused space. The next logical step is to try irrational numbers. Slightly better than the rational 0.5 and 0.6, pi and e still leave much empty space in the seed head. However, the seed head that emanates from utilising an approximation of Phi, 0.618*360=222.5, also recognised as 137.5 degrees (360-222.5), entails a maximised number of seeds with little to no unused space.8

A deviation from Phi, even if only by one degree, will result in a significant increase in empty seed head space. However, as noted by this model, an angle smaller than the Golden Ratio is preferable due to minimised empty space.9

The Golden Spiral Seed arrangement is beneficial because it maximises the number of seeds which can be packed into the flower’s seed head, which is important, evolutionarily, in terms of producing a large number of offspring with a similar advantageous trait.

Additionally, phyllotaxis, or the study of the ordered position of leaves on the stems of plants has convincing scientific evidence proving the existence of the

Knott, R. fibandphi (AT) ronknott.com. “Fibonacci Numbers and Nature - Part 2 Why is the 7

Golden section the "best" arrangement?”

Ibid..8

Emeny, William. “The Golden Angle.” Go Figure, 24 Apr. 2015, gofiguremath.org/natures-9

favorite-math/the-golden-ratio/the-golden-angle/. [Accessed 19 Nov. 2017].

�6

Golden Ratio.

The use of the Golden Spiral can also be found in the growth of new shoots, out of the axil, or the point/angle where a new branch or leaf grows out from the main stem of the plant.  For each new shoot, the angle at which it grows is relative to the one above and below. This was hypothesised to be (360 x (1-0.618)), or 137.52 degrees or the Golden Angle. This is referred to as the divergence of a plant. In order to 10

test this, King et al. constructed a mathematical model of needle leaves on a fir that grow toward the light. They were able to compare with a similar empirical study done by Pearcy and Yang in 1998, where the understory plant Adenocaulon bicolor was studied under the Redwood forest canopy. “Utilising the three-dimensional canopy architecture model YPLANT, they were able to transform their empirical data, obtained from 10 selected plants, into numerical data on plant geometry, photon flux density (PFD), and light capture efficiency.” Additionally, they were able to use this 11

model to study “hypothetical plants with (slightly) different geometric patterns, by changing the…parameters of their model.” King et al. found their results in 12

complete concordance with the previous empirical study done in 1998. They found that “…within (their) model, the golden angle is the divergence angle for maximum light capture.” An explanation for this could be that optimal light exposure, and 13

therefore, optimal light capture, act as a selective pressure. Maximum leaf exposure ot sunlight will increase the rate of photosynthesis, and therefore, the production of ATP to aid in growth. Thus, the biological fitness of the plant will be increased to its maximum as a result.14

In another study, Takuya Okabe created models of multiple plant species and their divergences. However, instead of the more common explanation for the Golden Ratio’s appearance in phyllotaxis (maximisation of light exposure), Okabe suggests that it all has to do with energy efficiency. The Golden Angle is the ‘optimal solution

Pearcy, R. W., and W. Yang. “The functional morphology of light capture and carbon gain in 10

the Redwood forest understory plant Adenocaulon bicolor Hook.” Functional Ecology, vol. 12, no. 4, 1998, pp. 543–552., doi:10.1046/j.1365-2435.1998.00234.x.

Pearcy, R. W., and W. Yang. “The functional morphology of light capture and carbon gain in 11

the Redwood forest understory plant Adenocaulon bicolor Hook.”

Ibid.12

King, S., Beck, F. and Luttge, U., “On the mystery of the golden angle in phyllotaxis,” 13

Plant, Cell & Environment, Vol. 27 (2004), 685–695. doi:10.1111/j.1365-3040.2004.01185.x, http://onlinelibrary.wiley.com/store/10.1111/j.1365-3040.2004.01185.x/asset/j.1365-3040.2004.01185.x.pdf;jsessionid=51CB8052758179C48BB8AC0933831B6B.f03t03?v=1&t=j8k1n97v&s=ea0cb4b527cf528dbd2ddf19e2f7597872eae2d8. [Accessed 16 May 2017].

King, S., Beck, F. and Luttge, U., ‘On the mystery of the golden angle in phyllotaxis,”14

�7

to minimise the energy cost of phyllotaxis transition…to within an accuracy of 1%,’ a degree of confidence usually only for medical studies.15

Although numerous studies have confirmed that the Golden Ratio appears in phyllotaxis, as one can see from the aforementioned studies, a solid explanation has not been proven, yet. Perhaps it is not simply one theory that defines the Golden Ratio’s presence, but a combination of justifications.

Phi in Anatomy:

Studies with plants suggest that there is a benefit for Phi over other ratios. It is therefore reasonable to presume that this may also be true for the presence of Phi in human anatomy and function. For this reason, it is important that the role of Phi be explored in depth to enhance and further develop medical diagnostic and treatment techniques.

Hearing Mechanism:16

The Golden Ratio can also be observed in the hearing mechanism of humans; specifically, logarithmic spirals that correspond to the Golden Ratio have been recognised in both the pinna (external ear) and cochlea (inner ear). As you may recall, humans hear sound 17

by the transferral of sound waves into electrical signals, which are then transported to the brain and perceived as noise. To begin, sound waves are collected by the equiangular spiral-shaped pinna. You may notice that if you bend your ears forward, essentially changing the shape of your pinna, sound volume increases, and if you bend your

Okabe, Takuya. “Biophysical optimality of the golden angle in phyllotaxis.” Scientific 15

Reports, vol. 5, no. 1, 2015, doi:10.1038/srep15358.

U.S. Department of Health & Human Services, “Cochlear Implants.” National Institute on 16

Deafness and Other Communication Disorders, https://www.nidcd.nih.gov/health/cochlear-implants. [Accessed 20 Mar. 2017].

Traynor, Robert. “Structure of the Ear and the Leaning Tower of Pisa.” Hearing 17

International, 8 July 2015, hearinghealthmatters.org/hearinginternational/2015/structure-of-the-ear-and-the-leaning-tower-of-pisa/. [Accessed 25. July. 2017].

�8

ears backwards, the volume decreases. Even though there is no change in sound 18

frequency, your perception of sound is altered as the pinna’s spiral is altered. Next, the sound waves travel through the ear canal, into the ear drum, and then to the middle ear to become fluid-membrane waves, which move past hair cells in the inner ear’s cochlea. As they pass the hair cells in the cochlea, the waves are converted into neural stimuli, which can then be processed by the brain as sound.19

The cochlea is a tube that enables humans to 20

differentiate sounds across a wide range of frequencies. One end of the tube is responsible for picking up higher frequencies, whilst the other is for picking up lower frequencies of sound.21

Batteau, D. W. “The role of the pinna in human localization.” Proceedings of the Royal 18

Society of London B: Biological Sciences, The Royal Society, 15 Aug. 1967, rspb.royalsocietypublishing.org/content/168/1011/158. [Accessed Apr. 4 2017].

Wright, C. G. Tonotopic Principal of the Cochlea. Southwestern Medical Centres, 19

s3.medel.com/images/triformance/tonotopic-principal-of-the-cochlea.jpg. [Accessed 6 Apr. 2017].

Drolet, Michael. JAC 2005 -- Acoustics, jac.michaeldrolet.net/acoustics/acoustics.htm. 20

[Accessed 6 Apr. 2017].

Manoussaki, Daphne, et al. “The influence of cochlear shape on low-frequency 21

hearing.”PNAS 2008 105 (16) 6162-6166; published ahead of print April 14, 2008, doi:10.1073/pnas.0710037105. [Accessed 6 Apr 2017].

�9

(a): Morganucodon, (b): Omithorhynchus, (c): Generalised for Eutriconodonts, Multituberculates, and Spalacotheroids, (d): Dryolestes, (e): Homo, (f): Didelphis

Research has shown that, over time in some mammals, such as humans and marsupials, cochlear shape and structure has evolved in a spiral-shape. This 22

phenotype has allowed for a greater range of sound frequencies to be heard and interpreted by the brain. Essentially, as the length of the cochlea increases, so does the range of sound that can be perceived due to a greater number of hair cells. It has been proposed by M. Pietsch et al. that a longer cochlea can best optimise space within the constraints of a mammalian skull by fitting into a coil shape, most efficiently, a Golden Ratio spiral, just like in the packing of a seed head in plants23

Despite this study suggesting the presence of the Golden Ratio in human cochleas being carried out at p=0.001, the typical certainty for medical studies, only 108 human cochleas were measured. There was also a mixture of males, females, right cochleas, and left cochleas, which are variables that should have been controlled in the investigation. The sample size was too small, especially when it is noted that 25 of the measurements had to be ‘adjusted manually’ by the scientists due to damage to the corrosion casts, their method used to measure the cochleas.24

Although that study showed some shortcomings, the notion that auditory structures follow the Golden Ratio is interesting in and of itself and most definitely necessitates further research. It has been theorised that, if the Golden Ratio’s presence was confirmed, it has the potential to help with medical advancements, such as in new research concerning cochlear implants. Cochlear implants bypass the transition of sound waves into fluid-membrane waves, by essentially acting as a hearing aid, changing sound waves directly into neural stimuli via electrode

Luo, Zhe-Xi, et al. “Fossil evidence on evolution of inner ear cochlea in Jurassic 22

mammals.” Proceedings of the Royal Society of London B: Biological Sciences, The Royal Society, 7 Jan. 2011, rspb.royalsocietypublishing.org/content/278/1702/28. [Accessed 3 May 2017].

Pietsch, M. et al. “Spiral Form of the Human Cochlea Results from Spatial Constraints.” 23

Scientific Reports 7 (2017): 7500. PMC. Web. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5548794/. [Accessed 21 Jun. 2017].

Pietsch, M. et al. “Spiral Form of the Human Cochlea Results from Spatial Constraints.”24

�10

implants attaching to the cochlea. It is the angle at which to place these 25 26

electrodes which is assisted by the Golden Ratio. Whilst there is some individual variation in the actual cochlear spirals in humans, it has been suggested that surgeons may use the Golden Ratio as a general guideline, or starting point in the placement of a cochlear implant. Nonetheless, further study is required to utilise this information in a clinical setting and we cannot really conclude that the Golden Ratio is actually present in auditory structures.

Heart:

It has also been proposed that the Golden Ratio is present in several aspects of the human heart.

The left ventricle contracts and pumps blood all around the body, thus making it an important part of the circulatory system. In a study led by Michael Y. Henein, the left ventricles of 30 Chinese, along with 30 Swedish patients were measured via echocardiogram. The Chinese population tended to have smaller measurements than the Swedish population; however, a comparison between the vertical (~8mm) and transverse (~5mm) measurements yielded a Golden Ratio in both populations. 27

Furthermore, significant deviations from the Golden Ratio in these measurements were found amongst patients with mild or severe heart failure. A larger deviation from the Golden Ratio in the vertical to transverse ratio was correlated with severe heart failure, with a ratio around 1.4; patients in this category exhibited 50% survival rate in three years follow-up. However, 60 patients is not a large enough sample 28

size to make a valid conclusion. Additionally, while 8mm/5mm does approximate the Golden Ratio, its actual value is 1.6, a significant difference away in terms of clinical applications.

In a study of healthy patients, Yetkin et al. utilised electrocardiography to measure the duration of both end-diastolic phase and end-systolic phase. Diastole is a 29

Isaacson, Brandon, WebMD, 1 June 2016, http://www.webmd.com/healthy-aging/25

understanding-cochlear-implants#1. [Accessed 1 Jul. 2017].

U.S. Department of Health & Human Services, “Cochlear Implants.” National Institute on 26

Deafness and Other Communication Disorders, https://www.nidcd.nih.gov/health/cochlear-implants. [Accessed 20 Mar. 2017].

Henein, Michael Y. et al., “The human heart: Application of the golden ratio and angle.” 27

International Journal of Cardiology, Vol. 150, Issue 3, 239 - 242. https://www.ncbi.nlm.nih.gov/labs/articles/21703707/. [Accessed 5 Sept. 2017].

Henein, Michael Y. et al., “The human heart: Application of the golden ratio and angle.”28

Yetkin, Gulay, et al. “Golden Ratio is beating in our heart.” International Journal of 29

Cardiology, vol. 168, no. 5, 2013, pp. 4926–4927., doi:10.1016/j.ijcard.2013.07.090.

�11

period of relaxation and filling, and systole is a period of contraction. The end-diastolic/end-systolic ratio was found to be 1.611, a close approximation of the Golden Ratio. This suggests that the beating of the heart in healthy humans aligns itself with the Golden Ratio. However, there were only 162 patients in the study, 30

not enough to make a valid conclusion about the presence of the Golden Ratio. Furthermore, the scientists, themselves, suggest that our hearts, ‘according to a uniformly agreed spiritual concept, are well-known to harbour our souls’ This implies that the scientists may view the Golden Ratio as a mystical or religious phenomenon, rather than a seminal scientific breakthrough.

In a separate study, Yetkin et al. monitored the blood pressure, an indication of heart function, of 462 patients using an ambulatory blood pressure monitor. They found that, at night, the systolic to diastolic blood pressure ratio of patients approximated the Golden Ratio. They postulated that the absence of the Golden Ratio during the day was due to an activated sympathetic nerves system’s influence on the heart, caused by daytime environmental stimuli and physical activity. In contrast, at night, 31

with patients in a more relaxed state, the parasympathetic nervous system tends to control bodily function, causing the systolic/diastolic ratio to gravitate toward the Golden Ratio. This may suggest the body’s natural gravitation towards a ratio 32

which appears to promote the least stress. However, one might question the validity of this study as, throughout the review, justification for the Golden Ratio was attributed to ‘divine energy.’33

Despite the findings from various experiments claiming the appearance of the Golden ratio in the cardiovascular system, there really is not a concrete biological explanation as to why this occurs. Throughout their paper, Yetkin et al. focussed on a ‘divine aesthetic’ in the cardiovascular system, rather than a practical function for the Golden Ratio. Perhaps the scientists were looking to find the Golden Ratio in the heart. This initial bias of the researchers may compromise a good methodology. However, these studies of human heart anatomy and function suggest a correlation between good health and the presence of the Golden Ratio, nonetheless. The implications of this interesting phenomenon have yet to be determined in a clinical

Yetkin, Gulay, et al. “Golden Ratio is beating in our heart.”30

Yalta, Kenan & Ozturk, Selcuk & Yetkin, Ertan, “Golden Ratio and the heart: A review of 31

divine aesthetics.” International Journal of Cardiology, Vol. 214, 10.1016/j.ijcard.2016.03.166., https://www.researchgate.net/publication/299421226_Golden_Ratio_and_the_heart_A_review_of_divine_aesthetics. [Accessed 5 Sept. 2017].

Yalta, Kenan & Ozturk, Selcuk & Yetkin, Ertan, “Golden Ratio and the heart: A review of 32

divine aesthetics.”

Ibid.33

�12

setting; therefore, further investigations should be undertaken in effort to incorporate the Golden Ratio in the development of cardiac therapies.

Facial Proportions:

Studies of human facial structure have also indicated that the presence of the Golden Ratio could be a marker of good health. Measurements have been taken 34

to describe a ratio between the width and length of a human face. When this proportion exceeded the Golden Ratio, subjects were declared to have long-face syndrome. Patients with this syndrome typically suffer from symptoms such as problems breathing through the nose, due to having narrow sinus cavities which inhibit airflow. This causes patients to breathe through their mouths, resulting in sleep apnea, crooked teeth, and snoring. Contrastingly, individuals with short-face syndrome, with width-to-length facial proportions significantly less than the Golden Ratio, tended to suffer from symptoms such as abnormal jaw development. This could potentially restrict blood flow to the brain, resulting in painful headaches. Subjects whose facial proportions approximated the Golden Ratio, in comparison, did not tend to suffer from the breathing issues or headaches, as seen with the other patients. Whilst these two syndromes are often subjective, Jefferson offers a ‘skeletal key’ in order to quantify and classify their diagnoses.35

Despite this seemingly helpful tool, the health benefits of Golden Ratio facial proportions seem to be thrown in for emphasis, whilst the real focus of the article concerns the aesthetics of human faces via Jefferson’s ‘skeletal key’. Jefferson’s 36

own suggestion of ‘breaking away our self-imposed image as mere “tooth” doctors and elevate our profession to that of “real” doctors’ gives the impression that he is simply looking for fame rather than a medical breakthrough. The poem embedded in the article, written by Jefferson himself, makes a reader question his motives and in no way ‘elevates (his) profession to that of a “real” doctor.’37

In addition to the aforementioned Golden Ratio facial proportion theories, several others have been suggested in a human face. Such examples are: the distance between pupils to the distance between eyebrows, the width of the nose to the distance between nostrils, and the length of the mouth to the width of the nose. It

Jefferson, Yosh, “Skeletal Types: key to unraveling the mystery of facial beauty and its 34

biological significance,” Journal of General Orthodontics., Vol. 7, June 1996.

Jefferson, Yosh, ‘Skeletal Types: key to unraveling the mystery of facial beauty and its 35

biological significance’.

Ibid.36

Ibid.37

�13

has been suggested that the more of these facial Golden Ratios a person has, the more attractive they appear to potential mates; thus, these individuals will then consequently have a higher chance to reproduce and pass their favourable characteristics on, due to natural selection. Despite the popular saying, “beauty is 38

in the eye of the beholder,” attraction to individuals with Golden Ratio facial proportions has been suggested to be an innate characteristic of mankind, based on the simple principle of the survival of the fittest. However, it was very difficult to find studies testing this theory.

According to a study out of the University of California a Golden Ratio face was not the most attractive to 126 college students. In the study, human faces were 39

digitally manipulated at different width to length ratios, and participants were asked to chose the one they found most attractive. Based on the Thurstonian attractiveness scale, the researchers predicted that a width to length ratio of 0.36 would be the most attractive. The researchers 40

found that the most attractive facial ratio, 0.46, to be significantly different than a Golden Ratio face at p=0.05 after conducting a t-test.41

Interestingly, a commonality throughout all of the studies on facial attractiveness and the Golden Ratio was that the faces tested were only women. Perhaps either the Golden Ratio is only applicable to females, or we just live in a world where females are objectified for their appearances.

Persaud-Sharma D and O’Leary JP. Fibonacci Series, Golden Proportions, and the Human 38

Biology. Austin J Surg. 2015;2(5): 1066.: http://digitalcommons.fiu.edu/cgi/viewcontent.cgi?article=1026&context=com_facpub

Pallett, Pamela M., et al. “New “Golden” Ratios for Facial Beauty.” Vision research, U.S. 39

National Library of Medicine, 25 Jan. 2010, www.ncbi.nlm.nih.gov/pmc/articles/PMC2814183/.

Thurstone, L. L. “The method of paired comparisons for social values.” The Journal of 40

Abnormal and Social Psychology, vol. 21, no. 4, 1972, pp. 384–400., doi:10.1037/h0065439.

Pallett, Pamela M., et al. “New “Golden” Ratios for Facial Beauty.”41

�14

Limb Proportions:

It has also been postulated that the Golden Ratio makes an appearance in various human limb proportions.

Wang et al. conducted some exciting research concerning the presence of the Golden Ratio in several human arm proportions. Their studies focussed on arm 42

segments created at intervals where the arm moves (wrist, elbow, shoulder); thus, creating functional partitions as opposed to anatomical partitions based simply on bone lengths, which had been previously disproved of containing the Golden Ratio. 43

The functional segments used in this study included the hand (measured from the tip of the middle finger to the wrist), the forearm (from the wrist to the elbow), and the upper arm (from the elbow to the shoulder). Based upon these measurements of thirty healthy adults, the scientists were able to conclude the presence of a Golden Spiral, dividing the arm into these three functional segments. Imagine an arm, elongated at the side, and the aforementioned partitions marked off. If a spiral is created, starting at the shoulder, curling towards the tip of the middle finger, then inwards to the elbow, and then finally curving to the wrist, the Golden Ratio is most easily observed. The arm also acts as a line, as described in the introduction of the Golden Ratio, where each segment and its consecutive one is also accordant with the Ratio.44

The results of this study have promising implications in the future for medical and bionic research, prosthetics, and plastic surgery. Additionally, the absence of the Golden Ratio in arm proportions could be influential in the diagnosis of patients with arm deformity or disfunction. However, there was only one study on the topic, 45

which had 30 human subjects, not enough to make a confident conclusion for or against the Golden Ratio in human limb proportions. Nonetheless, the researchers conducted a t-test, where the mean ratio sampled was said to not be statistically different from the Golden Ratio. Despite this, the scientists conducted their statistical test at p=0.05, when, for human investigations, studies should be carried out to a confidence limit of p=0.01, or 1%.

Wang, Nan, et al. “A Special Golden Curve in Human Upper Limbs’ Length Proportion: A 42

Functional Partition Which Is Different from Anatomy.” BioMed Research International, vol. 2017, 2017, pp. 1–6., doi:10.1155/2017/4158561.

Wang, Nan, et al. “A Special Golden Curve in Human Upper Limbs’ Length Proportion: A 43

Functional Partition Which Is Different from Anatomy.”

Ibid.44

Ibid.45

�15

Lungs:

It has also been suggested that the Golden Ratio can also be found in the lungs. Prior to Goldberger and his team’s research in 1986, it was well established that the two bronchi are asymmetrical, and thus, the branching into the bronchioles followed an irregular pattern. Their investigations not only confirmed the presence of this lung 4647

structure, but additionally proved the existence of the Golden Ratio. Each bronchus branches into two unequally sized bronchioles, with the ratio of these two tubes approximating the Golden Ratio. This pattern continues throughout the bronchiole tree, with each subset of bronchiole pairs demonstrating a relative proportion roughly equal to Phi. This organisation of branching in the bronchiole tree supports the fact that breathing is not a one-step process. Instead, it is more like a system of discrete steps as air moves through the differently sized bronchiole branches.

More recently, this knowledge has been used in the development of biologically variable ventilators that cater to the fact that breathing is not binary. Conventional 48

ventilation caused smaller bronchiole paths to collapse, due to the steady, monotonous respiration. The variable ventilation followed a fractal pattern that corresponded with the natural branching found within the patients’ lungs, that is the Golden Ratio. This variable ventilation proved to be much more effective in the 49

treatment of patients with acute lung injury and undergoing surgery. The natural 50

process evoked improvements in gas exchange, due to the oxygen’s ability to access a larger surface area of alveoli. The variable ventilation not only allowed more efficient oxygenation, but produced patient respiratory results comparable to healthy, non-ventilated individuals.

Goldberger, A. L., et al. “Bronchial asymmetry and Fibonacci scaling.” Experientia, vol. 41, 46

no. 12, 1985, pp. 1537–1538., doi:10.1007/bf01964794. [Accessed 27 Jan. 2017].

bronchoscopyatlas.ch, http://bronchoscopyatlas.ch/atlas/var/albums/Anatomic-variations/47

Anatomic-variations-of-the-tracheobronchial-tree/Asymmetry_mainstem_bronchi.jpg?m=1398440160. [Accessed 12 Nov. 2017].

Huhle, Robert, et al. “Variable ventilation from bench to bedside.” Critical Care, vol. 20, no. 48

1, 2016, doi:10.1186/s13054-016-1216-6. [Accessed 27 Jan. 2017].

Proctor, Shawn, “The Lung Redefined”, Respiratory Care & Sleep Medicine News, Jobs, 49

CE, and Events, 2008, http://respiratory-care-sleep-medicine.advanceweb.com/Article/The-Lung-Redefined.aspx. [Accessed 27 Jan. 2017].

Kowalski, Stephen et al. “Biologically Variable Ventilation in Patients with Acute Lung Injury: A Pilot Study.” Canadian Journal of Anaesthesia 60.5 (2013): 502–503. PMC. Web. 27 Jan. 2017.: https://www.ncbi.nlm.nih.gov/pmc/PMC3629278/

�16

However, investigating the presence of the Golden Ratio in bronchi and bronchiole branching was only attempted in one study. So, despite encouraging findings, this study had some problems. Several different types of mammals were studied: two adult male humans, two male beagles, a female Long-Evans rat, and a female Syrian hamster, so not enough of each type of species was used to confirm a result for any of them. Also, they used two different methods and got slightly different results from each one; however, the standard deviations of each overlap, which makes the results of the experiment not significant in most cases. Finally, a biological explanation was unable to be given as to why the Golden Ratio is the best possible ratio besides “natural selection”.

Psychology of Phi- Nutrition and Lifestyle:

Although no studies have been conducted, it has been suggested by a Tai Chi expert that breathing techniques based upon the Golden Ratio can promote good health. 51

This can be accomplished by a five-second inhalation followed by an eight-second exhalation. Five and eight are two consecutive Fibonacci numbers, and thus their relative proportion approximates the Golden Ratio. The longer exhalation period purportedly helps ensure that the lungs are emptied fully, thereby ridding the body of carbon dioxide and increasing the amount of oxygen able to enter upon the following inhalation. It has been proposed that by improving oxygen content in the bloodstream, this Golden Ratio breathing method can lead to pain relief, improved digestion, increased lung capacity, and better posture. After a bit of experience, 52

one could change to an eight-second inhalation and a thirteen-second exhalation to further reap the benefits of this breathing technique based upon Phi.53

In Joseph Mullen’s book, The Da Vinci Fitness Code, he promotes the use of the Golden Ratio to maximise one’s fitness routine. Specifically, he suggests that by 54

incorporating sequential numbers of the Fibonacci sequence into the determination of rest periods, reps for endurance, the number of sets, speed-of-movement, and resistance increases, one can reach his or her highest fitness potential.

Davies, Michael. Jiangan-- the Chinese Health Wand. Singing Dragon, 2011.51

“10 Benefits of Golden Ratio Breathing.” Wellnessmindbodyblog.wordpress.com, 12 Feb. 52

2016, wellnessmindbodyblog.wordpress.com/2016/02/07/18-benefits-of-golden-ratio-breathing/.

Davies, Michael, Jiangan- The Chinese Health Wand.53

Mullen, Joseph. The Da Vinci fitness code: maximum fitness--Minimum time. Fitness 54

Therapy Pub., 2006.

�17

Recently, there have been numerous accounts of a ‘Golden Ratio Lifestyle’. The most popular reference is a book titled The Golden Ratio Lifestyle Diet: Upgrade Your Life and Tap Your Genetic Potential for Ultimate Health, Beauty and Longevity. The guide promotes a “21-Day…Habit and Lifestyle Transformation,” and 21 is indeed a Fibonacci number. Within these 21 days, the book highlights the usage of 55

Fibonacci numbered days to begin, affirm, and enhance healthy lifestyle changes, as this helps to establish a new life-improving habit. What the book does suggest is a change from the traditional 40% carbohydrates, 30% fats, and 30% protein diet normally recommended to the average person. The two authors, Matthew Cross 56

and Robert Friedman M.D., advise a macronutrient ratio of no more than 40% carbohydrates as well as 60% of total combined protein and fat, for a healthier lifestyle. This 40:60 ratio is approximately the Golden Ratio; however, not enough research has been completed to confirm that macronutrient intake guidelines according to the Golden Ratio is a beneficial diet.

Interestingly, another startup company has also used the Golden Ratio in nutrition; however, the ratio between macros are slightly different. The Phi Bar specifically utilises the Golden Ratio in the creation of the recipes of their bars. Their Office Bar, for example, contains 25 grams of fat, of which 4.5 is saturated, 13 grams of carbohydrates, of which 5 is sugar, and 8 grams of protein. The ratios between fat 57

and carbohydrates, between carbohydrates and protein, as well as between protein and sugar, approximate the Golden Ratio. The Office Bar was intended to be eaten at an inactive phase of the day, and the higher fat content is believed to be the preference of muscles at rest. The carbohydrate and sugar contents are kept low in order to prevent a spike in insulin levels, and to minimise their storage as fat cells. 58

The amount of macronutrients are all equal to or come close to Fibonacci numbers, which, as you may recall, when divided, approximate the Golden Ratio; thus, the name Phi Bars.

The aforementioned lifestyle usages of the Golden Ratio have generated optimistic responses of success by those who have tried them; however, a lack of tested data has dissuaded others about the validity of these practices. Many Golden Ratio life-stylers report feelings of “balance” and “harmony”, but whether this can be scientifically proven remains to be seen.

Friedman, Robert, and Matthew Cross. The golden ratio lifestyle diet: upgrade your life & 55

tap your genetic potential for ultimate health, beauty & longevity. Hoshin Media, 2012.

Friedman, Robert, and Matthew Cross. The golden ratio lifestyle diet: upgrade your life & 56

tap your genetic potential for ultimate health, beauty & longevity

“Office Bar.” Phi Nutrition, eatphi.co/collections/frontpage/products/food-for-office. 57

[Accessed 20 Aug. 2017].

“Office Bar.” Phi Nutrition58

�18

Conclusion:

The Golden Ratio’s appearance in nature is not always obvious, as is its strategic placement in advertisements. Many companies use the Ratio, whether it be in the form of the Golden Rectangle, Golden Spiral, or Fibonacci numbers in order to draw customers in through an aesthetically pleasing logo, when, in fact, there is not any concrete research to support this theory.

Perhaps in the quest for optimal health and fitness, people are searching for a quick and neat solution. A fitness routine or diet based upon a mathematical concept may be appealing and appear promising; however, users are, in the words of Keith Devlin, a professor of mathematics at Stanford University, “victims to their natural desire to find meaning in the pattern of the universe, without the math skills to tell them that the patterns they see are illusory.”59

Another popular misconception is that a Golden Rectangle, a rectangle with the length and width of consecutive Fibonacci numbers, is more aesthetically pleasing than any other rectangle. However, a study from the Hass School of Business at Berkeley found that people favour rectangles with a ratio between 1.414 and 1.732, a range containing the Golden Ratio. However, the Golden Rectangle did not emerge as the most preferred.60

The Golden Ratio is also allegedly used as a marketing tactic by numerous corporations. A day out in London proved this to be true. I came across a vender selling merchandise with the Golden Spiral, Fibonacci Numbers, and Golden Rectangle. Most interestingly, he detailed that he did not really understand why the Golden Ratio allegedly sells, but purely uses it because of hearsay.

Another common misconception is that the Greeks used Phi in the design of the Parthenon. Whilst many swear that the Parthenon fits perfectly into a Golden Rectangle, Christine Flon denies the presence with the comment, “…it would be

Devlin, Keith. “The Golden Ratio and Fibonacci Numbers: Fact versus Fiction.” 8 Oct. 59

2012, Stanford University, Stanford University, www.youtube.com/watch?v=4oyyXC5IzEE. [Accessed 23 Feb. 2017].

Raghubir, Priya, and Eric Greenleaf. “Ratios in Proportion: What Should be the Shape of 60

the Package.” Studylib.net, studylib.net/doc/7346583/ratios-in-proportion--what-should-be-the-shape-of-the-pac… [Accessed 15 Nov. 2017].

�19

wrong to generalise… (as) architectural activity was an empirical practice in which experience and intuition, that is to say ‘mastery’, played a large part,’. Basically, there is no real evidence to 61

suggest that architects used the Golden Ratio, or even knew a thing about it.

Similarly to the Ancient Greek Parthenon, many have attempted to prove the existence of the Golden Ratio in the work of numerous artists, including: Da Vinci, Botticelli, Michelangelo, and Rafael. Golden Spiral 62

and Golden Rectangle overlays are added to the works of these artists claiming to confirm they employed the Golden Ratio; however, like the Parthenon, there is no credible evidence.

Interestingly, in many Golden Ratio articles, Da Vinci’s Vitruvian Man is the prominent image; however, the naval/height ratio in the drawing is 0.604, which is considerably smaller than 0.618. There is no mention of the Golden Ratio in the picture’s caption; thus, we are left to conclude that he had no intention to incorporate Phi into his illustration.63

In addition to its supposed appearance in art, architecture, and music, many creationist theories also claim their origins lie in the Golden Ratio. Peculiarly, the Golden Ratio may be a message from the Heavens that can only be worshipped and described as divine; however, 1.618 is more likely to be found amongst pi and e than the answer to the mystery of life.

Fundamental questions to propose at this point would be: do humans have a natural tendency to find and accept supposed ‘patterns’ that enhance their lives? Is the Golden Ratio simply a gimmick? Mathematical lecturer Ian Stewart emphasises the influence of illusions on everyday life. In his lecture, he describes this as an “illusory correlation”, or the human tendency to find patterns where they do not exist and manipulate them to fit their preconceived notions. Perhaps the role of the Golden 64

Ratio in biology is simply one of these so called manipulations and does not bear

Falbo. Clement. “The Golden Ratio- A Contrary Viewpoint,”61

Ibid62

Simanek, Donald E., ‘Fibonacci Flim-Flam’, lhup.edu, <https://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm> [accessed 8 Mar. 2017].

Stewart, Ian.64

�20

any true significance. However, until it is proven that there is not any biological connection with the Golden Ratio, more research should be done to explore possible implications.

Mathematically, the Golden Ratio has stood the test of time. Biologically, there are naysayers as well as Golden Ratio extremists. As for me, I am somewhere in between the two and believe that more evidence is required before we discard the Golden Ratio as a biological hoax or fully accept it as a medical miracle. In the end, nothing on Earth can truly possess the Golden Ratio, as Phi is an irrational number, and it is irrational to think otherwise.

�21

Bibliography:

Bates, Claire. “The Da Vinci Code of fertility: How the 'golden ratio' made popular by Dan Brown could reveal the most fertile wombs.” Daily Mail Online, Associated Newspapers, 15 Aug. 2012, www.dailymail.co.uk/health/article-2188598/How-golden-ratio-popular-Dan-Brown-reveal-fertile-wombs.html. [Accessed 31 Oct. 2016].

Batteau, D. W. “The role of the pinna in human localization.” Proceedings of the Royal Society of London B: Biological Sciences, The Royal Society, 15 Aug. 1967, rspb.royalsocietypublishing.org/content/168/1011/158. [Accessed 4 Apr. 2017].

Bentley, Brendan. “Why the golden proportion really is golden.” Australian Mathematics Teacher, vol. 72. no. 1, 20216, p.10+. PowerSearch, go.galegroup.com/ps/i.do?p=GPS&sw=w&u=pwpls_remote&v=2.1&id=GALE%7CA452052773&it=r&asid=d23e284061fcf61a565f5a79a5963e1e. ]Accessed 31 Oct. 2016].

Brownlee, John. “The Golden Ratio: Design's Biggest Myth.” Co.Design, Co.Design, 2 May 2017, www.fastcodesign.com/3044877/the-golden-ratio-designs-biggest-myth. [Accessed 17 Sept. 2017].

Carlson, Stephan C., “Golden ratio,” Encyclopaedia Britannica, https://www.britannica.com/topic/golden-ratio. [accessed 21 August, 2017].

Davies, Michael. Jiangan-- the Chinese Health Wand. Singing Dragon, 2011.

Devlin, Kate. “Human heart follows 'golden ratio' for beauty.” The Telegraph, 14 Dec. 2009, www.telegraph.co.uk/news/health/news/6790529/Human-heart-follows-the-golden-ratio-for-beauty.html. [Accessed 21 Jan. 2017].

Devlin, Keith. “The Golden Ratio and Fibonacci Numbers: Fact versus Fiction.” 8 Oct. 2012, Stanford University, Stanford University, www.youtube.com/watch?v=4oyyXC5IzEE. [Accessed 23 Feb. 2017].

Drolet, Michael. JAC 2005 -- Acoustics, jac.michaeldrolet.net/acoustics/acoustics.htm. [Accessed 6 Apr. 2017].

Dunning, B. "The Golden Ratio." Skeptoid Podcast. Skeptoid Media, 28 Aug 2012, http://skeptoid.com/episodes/4325. [Accessed 27 Feb. 2017].

Emeny, William. “The Golden Angle.” Go Figure, 24 Apr. 2015, gofiguremath.org/natures-favorite-math/the-golden-ratio/the-golden-angle/. [Accessed 19 Nov. 2017].

Euclid, Thomas L. Heath, and Dana Densmore, Euclid's Elements: All Thirteen Books Complete in One Volume, ed. by Thomas L. Heath (Santa Fe, N.M: Green Lion Press, 2002.

Falbo. Clement. “The Golden Ratio- A Contrary Viewpoint,” Vol. 36, No.2, March 2005, The College Mathematics Journal, http://web.sonoma.edu/Math/faculty/falbo/cmj123-134. [Accessed 21 Jun. 2017].

�22

French, Karen L. The hidden geometry of life: the science and spirituality of nature. Watkins Publishing / New York, 2012.

Friedman, Robert, and Matthew Cross. The golden ratio lifestyle diet: upgrade your life & tap your genetic potential for ultimate health, beauty & longevity. Hoshin Media, 2012.

Gardiner, John. “Fibonacci, quasicrystals and the beauty of flowers.” Plant Signaling & Behavior, vol. 7, no. 12, 2012, pp. 1721–1723., doi:10.4161/psb.22417.

Goldberger, A. L., et al. “Bronchial asymmetry and Fibonacci scaling.” Experientia, vol. 41, no. 12, 1985, pp. 1537–1538., doi:10.1007/bf01964794. [Accessed 27 Jan. 2017].

Gravett, Emily, Macmillan Children’s, “The Rabbit Problem.” 2009, http://educ.queensu.ca/sites/webpublish.queensu.ca.educwww/files/files/Community/COC/Melodies/Mister%20Fibonacci%20-%20Notes.pdf. [Accessed 24 Feb. 2017].

Henein, Michael Y. et al., “The human heart: Application of the golden ratio and angle.” International Journal of Cardiology, Vol. 150, Issue 3, 239 - 242. https://www.ncbi.nlm.nih.gov/labs/articles/21703707/. [Accessed 5 Sept. 2017].

Huhle, Robert, et al. “Variable ventilation from bench to bedside.” Critical Care, vol. 20, no. 1, 2016, doi:10.1186/s13054-016-1216-6. [Accessed 27 Jan. 2017].

Huntley, H.E. The divine proportion a study in mathematical beauty. Dover Publications, Inc., 2015.

Isaacson, Brandon, WebMD, 1 June 2016, http://www.webmd.com/healthy-aging/understanding-cochlear-implants#1. [Accessed 1 Jul. 2017].

Jefferson, Yosh, “Skeletal Types: key to unraveling the mystery of facial beauty and its biological significance,” Journal of General Orthodontics., Vol. 7, June 1996.

Jone, Pei-Ni, et al. “Right Ventricular to Left Ventricular Diameter Ratio at End-Systole in Evaluating Outcomes in Children with Pulmonary Hypertension.” Journal of the American Society of Echocardiography : official publication of the American Society of Echocardiography,

King, S., Beck, F. and Luttge, U., “On the mystery of the golden angle in phyllotaxis,” Plant, Cell & Environment, Vol. 27 (2004), 685–695. doi:10.1111/j.1365-3040.2004.01185.x, http://onlinelibrary.wiley.com/store/10.1111/j.1365-3040.2004.01185.x/asset/j.1365-3040.2004.01185.x.pdf;jsessionid=51CB8052758179C48BB8AC0933831B6B.f03t03?v=1&t=j8k1n97v&s=ea0cb4b527cf528dbd2ddf19e2f7597872eae2d8. [Accessed 16 May 2017].

Knott, R. fibandphi (AT) ronknott.com. “Fibonacci Numbers and Nature - Part 2 Why is the Golden section the "best" arrangement?” The Fibonacci Numbers and Golden Ratio in Nature - 2, www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html#section2.2. [Accessed 23 July 2017].

Kowalski, Stephen, et al. “Biologically variable ventilation in patients with acute lung injury: a pilot study.” Canadian Journal of Anesthesia/Journal canadien danesthésie, vol. 60, no. 5, June 2013, pp. 502–503., doi:10.1007/s12630-013-9899-5. [Accessed 12 Jan. 2017].

�23

Lewis, Hazel, “The mathematics of Rabbit Island.” mathscareers.org.uk <http://www.mathscareers.org.uk/article/the-mathematics-of-rabbit-island/. [Accessed 14 Sept. 2017].

Livio, Mario. “The golden number: nature seems to have a sense of proportion,” Natural History, Mar. 2003, p.64+. PowerSearch, go.galegroup.com/ps/i.do?p=GPS&sw=w&u=pwpls_remote&v=2.1&id=GALE%7CA98254967&it=r&asid=cebfa23d29a4679203b16481260e108f. [Accessed 31 Oct. 2016].

Livio, Mario. “The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. (Nonfiction).” Kirkus Review, 1 Sept. 2002, p. 1284+. General OneFile, go.galegroup.com/ps/i.do?p=GPS&sw=w&u=pwpls_remote&v=2.1&id=GALE%7CA92683038&it=r&asid=f9ea37fbfae0f446d3fc19f5e5f14c25. [Accessed 31 Oct. 2016].

Luo, Zhe-Xi, et al. “Fossil evidence on evolution of inner ear cochlea in Jurassic mammals.” Proceedings of the Royal Society of London B: Biological Sciences, The Royal Society, 7 Jan. 2011, rspb.royalsocietypublishing.org/content/278/1702/28. [Accessed 3 May 2017].

Manoussaki, Daphne, et al. “The influence of cochlear shape on low-frequency hearing.”PNAS 2008 105 (16) 6162-6166; published ahead of print April 14, 2008, doi:10.1073/pnas.0710037105. [Accessed 6 Apr 2017].

Moscarelli, Marco, and Ruggero De Paulis. “The golden perfection of the aortic valve.” International Journal of Cardiology, vol. 205, Dec. 2016, pp. 165–166., doi:10.1016/j.ijcard.2015.12.018. [Accessed 20 Mar. 2017].

Mullen, Joseph. The Da Vinci fitness code: maximum fitness--Minimum time. Fitness Therapy Pub., 2006.

Murali, Sruthy, Golden Ratio in Human Anatomy, (2012) Department of Mathematics Government College, https://www.researchgate.net/publication/234054763_GOLDEN_RATIO_IN_HUMAN_ANATOMY. [Accessed 31 Oct. 2016].

Nelson, S.M., E.E. Telfer, and R.A. Anderson, “The Ageing Ovary and Uterus: New Biological Insights.” Human Reproduction Update 19.1 (2013): 67–83. PMC. www.ncbi.nlm.nih.gov/pmc/articles/PMC3508627/. [accessed 3 Jun. 2017].

O’Connor, J., and Robertson, E., “The Golden Ratio.” history.mcs.st-andrews.ac.uk, http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html. [accessed 20 Feb. 2017].

Okabe, Takuya. “Biophysical optimality of the golden angle in phyllotaxis.” Scientific Reports, vol. 5, no. 1, 2015, doi:10.1038/srep15358.

Oxlund, Birgitte S, et al. “Collagen concentration and biomechanical properties of samples from the lower uterine cervix in relation to age and parity in non-Pregnant women.” Reproductive Biology and Endocrinology, BioMed Central, 6 July 2010, rbej.biomedcentral.com/articles/10.1186/1477-7827-8-82.

�24

Ozturk, Selcuk, et al. “Golden ratio: A subtle regulator in our body and cardiovascular system?” International Journal of Cardiology, vol. 223, Aug. 2016, pp. 143–145., doi:10.1016/j.ijcard.2016.08.147. [Accessed 20 Mar. 2017].

Pallett, Pamela M., et al. “New “Golden” Ratios for Facial Beauty.” Vision research, U.S. National Library of Medicine, 25 Jan. 2010, www.ncbi.nlm.nih.gov/pmc/articles/PMC2814183/.

Pearcy, R. W., and W. Yang. “The functional morphology of light capture and carbon gain in the Redwood forest understory plant Adenocaulon bicolor Hook.” Functional Ecology, vol. 12, no. 4, 1998, pp. 543–552., doi:10.1046/j.1365-2435.1998.00234.x.

Perez, J.C.. “Golden Ratio and Number in DNA- Codex Biogenesi.” golden-ratio-in-dna.blogspot.ae, http://golden-ratio-in-dna.blogspot.co.uk. [accessed 21 Oct. 2017].

Persaud-Sharma D and O’Leary JP. “Fibonacci Series, Golden Proportions, and the Human Biology.” Austin J Surg. 2015;2(5): 1066. http://digitalcommons.fiu.edu/cgi/viewcontent.cgi?article=1026&context=com_facpub. [Accessed 23 May 2017].

Pietsch, M. et al. “Spiral Form of the Human Cochlea Results from Spatial Constraints.” Scientific Reports 7 (2017): 7500. PMC. Web. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5548794/. [Accessed 21 Jun. 2017].

Proctor, Shawn, “The Lung Redefined”, Respiratory Care & Sleep Medicine News, Jobs, CE, and Events, 2008, http://respiratory-care-sleep-medicine.advanceweb.com/Article/The-Lung-Redefined.aspx. [Accessed 27 Jan. 2017].

Raghubir, Priya, and Eric Greenleaf. “Ratios in Proportion: What Should be the Shape of the Package.” Studylib.net, studylib.net/doc/7346583/ratios-in-proportion--what-should-be-the-shape-of-the-pac… [Accessed 15 Nov. 2017].

Reinhardt, Didier, et al. “Regulation of phyllotaxis by polar auxin transport.” Nature, vol. 426, no. 6964, 2003, pp. 255–260., doi:10.1038/nature02081.

Scales, Helen, and Aaron John Gregory. Spirals in time: the secret life and curious afterlife of seashells. St. Martins Press, 2015.

Schuster, Stefan, et al. “Use of Fibonacci numbers in lipidomics – Enumerating various classes of fatty acids.” Scientific Reports, Nature Publishing Group, 2017, ncbi.nlm.nih.gov/pmc/articles/PMC5223158/. [Accessed 13 Oct. 2017].

Simanek, Donald E., ‘Fibonacci Flim-Flam’, lhup.edu, <https://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm> [accessed 8 Mar. 2017].

Stewart, Ian. “The Mathematics of Visual Illusions.” Oxford Mathematics Christmas Public Lecture. Oxford Mathematics Christmas Public Lecture, 15 Dec. 2016, Oxford, University of Oxford Mathematical Institute.

Thomas, Rachel, “The Fibonacci sequence: A brief introduction.” plus.maths.org, https://plus.maths.org/content/fibonacci-sequence-brief-introduction. [accessed 25 Jun. 2017].

�25

Thurstone, L. L. “The method of paired comparisons for social values.” The Journal of Abnormal and Social Psychology, vol. 21, no. 4, 1972, pp. 384–400., doi:10.1037/h0065439.

Traynor, Robert. “Structure of the Ear and the Leaning Tower of Pisa.” Hearing International, 8 July 2015, hearinghealthmatters.org/hearinginternational/2015/structure-of-the-ear-and-the-leaning-tower-of-pisa/. [Accessed 25. July. 2017].

Ulmer, Hanno, et al. “George Clooney, the cauliflower, the cardiologist, and phi, the golden ratio.” The BMJ, British Medical Journal Publishing Group, 14 Dec. 2009, www.bmj.com/content/339/bmj.b4745. [Accessed 12 Apr. 2017].

Vendetti, Jann. “The Cephalopoda,” UCMP, 2006, UC Berkeley, http://www.ucmp.berkeley.edu/taxa/inverts/mollusca/cephalopoda.php. [Accessed 16 May 2017].

Verguts, J., et al. “Normative data for uterine size according to age and gravidity and possible role of the classical golden ratio.” Ultrasound in Obstetrics & Gynecology, vol. 42, no. 6, 2013, pp. 713–717., doi:10.1002/uog.12538.

Waichulis, Anthony, “A Spurious Affair: A Primer on Pictorial Composition (Part IV)”, http://anthonywaichulis.com/fools-gold-a-primer-on-pictorial-composition-part-v/. [Accessed 29 May 2017].

Wang, Nan, et al. “A Special Golden Curve in Human Upper Limbs’ Length Proportion: A Functional Partition Which Is Different from Anatomy.” BioMed Research International, vol. 2017, 2017, pp. 1–6., doi:10.1155/2017/4158561.

Wright, C. G. Tonotopic Principal of the Cochlea. Southwestern Medical Centres, s3.medel.com/images/triformance/tonotopic-principal-of-the-cochlea.jpg. [Accessed 6 Apr. 2017].

Yahya, Harun, “How does the ear’s Golden Ratio feature help hearing?” http://harunyahyainternationalmedia.com/en/Daily-Comments/32985/how-does-the-ears-golden, [Accessed 12. May. 2017].

Yalta, Kenan & Ozturk, Selcuk & Yetkin, Ertan, “Golden Ratio and the heart: A review of divine aesthetics.” International Journal of Cardiology, Vol. 214, 10.1016/j.ijcard.2016.03.166., https://www.researchgate.net/publication/299421226_Golden_Ratio_and_the_heart_A_review_of_divine_aesthetics. [Accessed 5 Sept. 2017].

Yetkin, Ertan, et al. “Left ventricular diameters as a reflection of "extreme and mean ratio".” International Journal of Cardiology, vol. 198, 27 June 2015, pp. 85–86., doi:http://dx.doi.org/10.1016/j.ijcard.2015.06.164.

Yetkin, Gulay, et al. “Golden Ratio is beating in our heart.” International Journal of Cardiology, vol. 168, no. 5, 2013, pp. 4926–4927., doi:10.1016/j.ijcard.2013.07.090.

bronchoscopyatlas.ch, http://bronchoscopyatlas.ch/atlas/var/albums/Anatomic-variations/Anatomic-variations-of-the-tracheobronchial-tree/Asymmetry_mainstem_bronchi.jpg?m=1398440160. [Accessed 12 Nov. 2017].

�26

National Oceanic and Atmospheric Administration U.S. Department of Commerce, “What is a nautilus?” https://oceanservice.noaa.gov/facts/nautilus.html. [Accessed 13 Oct. 2017].

“Office Bar.” Phi Nutrition, eatphi.co/collections/frontpage/products/food-for-office. [Accessed 20 Aug. 2017].

“The DaVinci Fitness Code?” Families.com, 12 Apr. 2006, www.families.com/blog/the-davinci-fitness-code. [Accessed 25 Aug. 2017].

“The Da Vinci Fitness Code.” Hitinpractice.motionforum.net, hitinpractice.motionforum.net/t65-the-da-vinci-fitness-code. [Accessed 25 Aug. 2017].

University of Miami Leonard M. Miller School of Medicine, Cochlear Implants, 1.bp.blogspot.com/-nc_DohyAXsk/Uj-xfM2aP_I/AAAAAAAAAZU/B-St6pOdchA/s1600/COCHLEAR-IMPLANT.jpg. [Accessed 24 Jan. 2017].

U.S. Department of Health & Human Services, “Cochlear Implants.” National Institute on Deafness and Other Communication Disorders, https://www.nidcd.nih.gov/health/cochlear-implants. [Accessed 20 Mar. 2017].

“10 Benefits of Golden Ratio Breathing.” Wellnessmindbodyblog.wordpress.com, 12 Feb. 2016, wellnessmindbodyblog.wordpress.com/2016/02/07/18-benefits-of-golden-ratio-breathing/. [Accessed 24 Feb. 2017].

�27