Review Article

Austin J Surg. 2015;2(5): 1066.

# Fibonacci Series, Golden Proportions, and the Human Biology

Dharam Persaud-Sharma* and James P O’Leary

Florida International University, Herbert Wertheim College of Medicine

***Corresponding author: **Dharam Persaud-Sharma,
Florida International University, Herbert Wertheim College of Medicine, Miami, FL, 33199, USA

**Received: **April 01, 2015; **Accepted: **June 25, 2015; **Published: **July 02, 2015

## Abstract

Pythagoras, Plato and Euclid’s paved the way for Classical Geometry. The idea of shapes that can be mathematically defined by equations led to the creation of great structures of modern and ancient civilizations, and milestones in mathematics and science. However, classical geometry fails to explain the complexity of non-linear shapes replete in nature such as the curvature of a flower or the wings of a Butterfly. Such non-linearity can be explained by fractal geometry which creates shapes that emulate those found in nature with remarkable accuracy. Such phenomenon begs the question of architectural origin for biological existence within the universe. While the concept of a unifying equation of life has yet to be discovered, the Fibonacci sequence may establish an origin for such a development. The observation of the Fibonacci sequence is existent in almost all aspects of life ranging from the leaves of a fern tree, architecture, and even paintings, makes it highly unlikely to be a stochastic phenomenon. Despite its wide-spread occurrence and existence, the Fibonacci series and the Rule of Golden Proportions has not been widely documented in the human body. This paper serves to review the observed documentation of the Fibonacci sequence in the human body.

**Keywords:** Fibonacci; Surgery; Medicine; Anatomy; Golden proportions;Math; Biology

## Introduction

Classical geometry includes the traditional shapes of triangles, squares, and rectangles that have been well established by the brilliance of Pythagoras, Plato, and Euclid. These are fundamental shapes that have forged the great structures of ancient and modern day civilization. However, classical geometry fails to translate into complex non-linear forms that are observed in nature. To explain such non-linear shapes, the idea of fractal geometry is proposed which states that such fractal shapes/images possess self-similarity and can be of non-integer or non-whole number dimensions. Whereas classical geometric shapes are defined by equations and mostly whole number (integer) values, the shapes of fractal geometry can be created by iterations of independent functions. Observing the patterns created by repeating ‘fractal images’ closely emulate those observed in nature, thus questioning whether this truly is the mechanism of origination of life on the planet. Two important properties of fractals include self-similarity and non-integer or non-whole number values. The idea of ‘self-identity’ is that its basic pattern or fractal is the same at all dimensions and that its repetition can theoretically continue to infinity. Examples of such repeating patterns at microscopic and macroscopic scales can be seen in all aspects of nature from the florescence of a sunflower, the snow-capped peaks of the Himalayan Mountains, to the bones of the human body.

The observation of self-similarity belonging to fractal geometry found in multiple aspects of nature, leads to the question of whether such an occurrence is merely stochastic or whether it has a functional purpose. While a singular unifying equation to define the creation and design of life has yet to be determined, an un-coincidental phenomenon known as the Fibonacci sequence and the rule of ‘Golden Proportions’ may serve as a starting point for uncovering the methods of a universal architect, should one exist.

## Fibonacci and rule of golden proportions

**Fibonacci sequences:** The Fibonacci sequence was first
recognized by the Indian Mathematician, Pingala (300-200 B.C.E.)
in his published book called the Chandasastra, in which he studied
grammar and the combination of long and short sounding vowels
[1,2]. This was originally known as matrameru, although it is now
known as the Gopala-Hemachandra Number in the East, and the
Fibonacci sequence in the West [1,2]. Leonardo Pisano developed
the groundwork for what is now known as “Fibonacci sequences” to
the Western world during his studies of the Hindu-Arab numerical
system. He published the groundwork of Fibonacci sequences in
his book called Liber Abaci (1202) in Italian, which translates into
English as the “Book of Calculations”. However, it should be noted
that the actual term “Fibonacci Sequences” was a tributary to
Leonardo Pisano, by French Mathematician, Edouard Lucas in 1877
[3]. The Fibonacci sequence itself is simple to follow. It proposes that
for the integer sequence starting with 0 or 1, the sequential number is
the sum of the two preceding numbers as in Figure 1.

**Figure 1:**Ratios of Fibonacci Numbers for the sequences starting with 1. Ratios approximate Fibonacci value of Phi (f) 1.618.

Figure 1:Ratios of Fibonacci Numbers for the sequences starting with 1. Ratios approximate Fibonacci value of Phi (f) 1.618.

As shown in Figure 1, Fibonacci sequences can either begin with
0 or 1. Although the sequence, originally proposed in Liber Abaci by
Leonardo (follows the sequence in A) begins with 1, this sequence
essentially reduces to the recurrence relationship of: F_{N} = F_{n-1} + F_{n-2}

**Figure 2:**Golden Rectangle and Golden Spiral.

Figure 2:Golden Rectangle and Golden Spiral.

Given that the starting numbers of the sequence can be either:
A) F_{1} = 1 F_{2} = 1 or B) F_{1} = 0 F_{2} = 1.In this sequence it can be seen
that each additional number after the given conditions of 1 or 0, is a
direct result of the sum of the two preceding numbers. For example,
in series (A), 1+ 1 = 2, where 2 is the third number in the sequence.
To find the third number (F_{4}), simply add F_{3} = 2 and F_{2} = 1, which
equals 3 (F_{4}).

**Golden proportion, rectangle, and spiral:** Another observation
that can be made from the series of the Fibonacci numbers includes
the rule of golden proportions. In essence, this is an observation that
the ratio of any two sequential Fibonacci numbers approximates to
the value of 1.618, which is most commonly represented by the Greek
Letter Phi (φ). The larger the consecutive numbers in the sequence,
the more accurate the approximation of 1.618. A summary of these
results illustrating this concept can be seen in Table 1.

**Table 1:**Ratios of Fibonacci Numbers for the sequences starting with 1. Ratios approximate Fibonacci value of Phi (f) 1.618.

F_{N + 1}

F_{N}

(F_{N + 1})/F_{N}

21

2

32

1.5000

53

1.6666

85

1.6000

138

1.6250

2113

1.6154

3421

1.6190

5534

1.6176

8955

1.6182

144

89

1.6179(Nearest approximation of f)

Table 1:Ratios of Fibonacci Numbers for the sequences starting with 1. Ratios approximate Fibonacci value of Phi (f) 1.618.

Furthering this observation, shapes can be created based upon length measurements of the Fibonacci numbers in sequence. For example, creating a rectangle with the values of any of the two successive Fibonacci numbers form what is known as the “Golden Rectangle”. These rectangles can be divided into squares that are equally sided and are of smaller values from the Fibonacci sequence Figure 2.

Rectangles created using consecutive numbers from the Fibonacci sequence can be divided into equally sided squares of such numbers as well. The length of a single side can be divided into smaller values of the Fibonacci sequence. Connecting the diagonals of each of the squares creates a spiral, known as the Golden Spiral.

This image compares the spiral of the human ear, the shell of a nautilus, and the golden spiral. Both the human ear and the shell of a nautilus approximate the dimensions of the golden spiral.

Connecting the corners of each of the squares by an arc reveals a spiral pattern. This spiral pattern is known as the ‘golden spiral’. This pattern is seen in several different forms in nature including, but not limited to the human ear and the shell of a nautilus as seen in Figure 3.

**Figure 3:**Golden Spiral in Nature.

Figure 3:Golden Spiral in Nature.

## Human anatomy and Fibonacci

Proportion of human bones: In the human body, there are instances of the Fibonacci Phi (φ) although it has not been widely discussed. In 1973, Dr. William Littler proposed that by making a clenched fist, Fibonacci’s spiral can be approximated in Figure 4 [4].

**Figure 4:**The human hand superimposed on the Golden Rectangle, with approximations to the Golden Spiral [4].

Figure 4:The human hand superimposed on the Golden Rectangle, with approximations to the Golden Spiral [4].

In this paper, Litter proposed that such geometry is observed based upon ratio of the lengths of the phalanges, and that the flexor and extensor movement of the primary fingers approximate the golden spiral. Such a spiral would have to be created based upon the relationship between the metacarpophalangeal and interphalanges of the digits [4]. However, due a lack of statistical and well documented empirical data, accurate representations of the golden spiral could not be readily ascertained. In 1998, Gupta et al., confirmed that the motion of the phalanges do approximate the pattern of the golden spiral, excluding for the fifth digit because of abduction of the digit during extension [5]. Through empirical evidence obtained through radiographic analysis of the hands of 197 individuals, Hamilton and Dunsmuirin 2002 concluded that the ratios between the phalange lengths of the digits do not approximate the golden ratio value of (φ) 1.618 [6]. However, Hutchison and Hutchison (2010) showed that the data collected by Hamilton et al. (2002) overlooked the relationship that Littler actually proposed. Hamilton and Dunsmuir (2010) confirmed that the phalangeal length ratio data obtained from their subjects compared to those that were almost arbitrarily listed by Littler in 1973, were in fact comparable and approximated the Fibonacci value of (φ) of 1.618. What Hamilton et al. failed to notice in their own data was that the values they obtained from their research approximated a Lucas series, which essentially underscores a Fibonacci sequence in that the sum of the first two lengths equaled the third length [7].

Hamilton and Dunsmuir’s findings highlighted that the index, second, and third digits follow a series of 1x, 1.3x, 2.3x [7]. Hamilton et al. provided length measurement data for the fourth digit (little finger) that follows the initial values of a Fibonacci sequence of 0,1,1,2, represented by y, y, and 2y in Figure 5 [7]. Hence, the data collected by Hamilton et al., supported Littler’s initial proposal of a clenched fist approximating the dimensions of the golden spiral. Additionally, Hamilton’s data showed that the ratios for the length of the phalanges follows the additive rule of a Lucas series for the index, second and third digits, while approximating Fibonacci values for the fourth digit (little finger).

**Figure 5:**Phalangeal lengths as reported by Hamilton et al. [6]. For the index, second and third digits: ratios of the DIP-tip/PIP-DIP/MCP-PIP distances were 1:1.3:2.3, and 1:1:2 for the little finger. DIP: Distal interphalangeal; PIP: Proximal Interphalangeal; MCP: Metacarpophalangeal Joint.

Figure 5:Phalangeal lengths as reported by Hamilton et al. [6]. For the index, second and third digits: ratios of the DIP-tip/PIP-DIP/MCP-PIP distances were 1:1.3:2.3, and 1:1:2 for the little finger. DIP: Distal interphalangeal; PIP: Proximal Interphalangeal; MCP: Metacarpophalangeal Joint.

The aesthetics of a smile and beauty: The rule of golden proportions has been proposed in an attempt to define anatomical beauty. It is commonly accepted that facial beauty is correlates with anatomical symmetry [8]. One of the main features of the human face is the mouth and teeth. Thus, professionals within the field of dentistry have attempted to quantitatively characterize the parameters of an aesthetically appealing smile. The most logical starting point is that of the rule of golden proportions. In 1973, Lombardi was the first to officially propose the existence of having proportionate teeth, but dismissed the idea of using the rule of golden proportions to create aesthetic teeth [9]. He did suggest an underlying repeated ratio of the maxillary anterior teeth for aesthetics, but stated the inappropriate use of the rule of golden proportions for such purposes. In 1978, Levin was the first to observe that: 1) the width of the maxillary central incisor is in golden proportion to the width of the lateral incisor. 2) The width of the maxillary lateral incisor is in golden proportion to the width of the canine [10]. Based upon this observation, Levin later developed a tooth caliper to gauge whether an individual’s teeth are in golden proportion to one another, and diagnostic grid to verify if the teeth are appropriately spaced. Further research into the relationship between the golden proportion, dental arrangement, and an aesthetic smile was developed by Rickets in 1982. He implemented the use of the golden proportions in the treatment of patients [11,12]. While Levin’s observation of the golden ratio existing in dentition is undeniable, its application to developing an aesthetically appealing smile has recently been questioned and often dismissed [9,13-15].

In 2007, Dio et al. attempted to empirically define the basis for determining/judging beauty [16]. Dio et al., utilized fMRI imaging of the brain to formulate an association between brain stimulation from the interpretation of visual models with different physical proportionsusing a model constructed according to golden proportions as an independent variable. The results of the study indicate that defining beauty is a joint process of cortical neurons, where Dio et al. identified that objective beauty stimulates the insula, whereas subjective beauty based upon the judgment of the individual perceiving the experience shows functional excitation of the Amygdala. Dio et al. echo the sentiments referenced by Gombrich [17] and Ramachandran [18], in that the criteria for making overall assessments of beauty seem to be related to our biological heritage, although it cannot be explained by a conscious explanation [16]. In our opinion, while the existence of the golden ratio continues to be revealed in several different instances of the human anatomy, attempting to use its proportions as a measure of beauty is not something that should be standardized. It is influenced by many different factors including cultural influences. This is something that may have been overlooked by Dio et al. study, in that in the demographic information about the 14 participants in the study was not included. The idea of ‘beauty’ itself is a subjective term and not one that can be universally held constant. The diversity of biology in its essence is what constitutes its beauty, while the observation of the golden proportion in nature highlights an architectural design, proportions, and symmetry.

*Phyllotaxis, coronary arteries, and the human heart beat:*
Although the blood vessels of the human body and a plant have
different physiological components, they are very similar in their
basic function and visual appearance Figure 6. The vascular network
in its basic structure is nearly identical to shapes that can be created by
fractal geometry, as seen in Figure 6C. Mathematically, the properties
of these structures are created by the self-similarity of the fractal. This
means that the whole image is the result of multiple smaller copies
of itself that share the same statistical properties at different scales of
measurement.

**Figure 6:**Comparison between the silhouettes of a tree (A), the blood vessels of a heart (B), and a fractal image (C).

Figure 6:Comparison between the silhouettes of a tree (A), the blood vessels of a heart (B), and a fractal image (C).

Of importance here is that in 1977, Mitchison revealed that the phyllotaxis of plants follows the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21… as seen in Figure 1. Likewise in 2011, Ashrafian and Atasiounoted that the number of branches from vessels in the heart follows the Fibonacci sequences Table 2 [19]. Ashrafian and Atanasiou, supported their observation by correlating studies which showed that atherosclerotic lesions in coronary arteries follow a Fibonacci distribution [20].

**Table 2:**Table summarizing the number of branching from blood vessels of the Heart showing that branching numbers follow a Fibonacci Series [19].

Number of branches

2LCA

RCA

3LAD

LCx

RCA

5LAD

LCx

RI

AM

PDA

8LAD

Ldiag1

LCx

OM1

Sep1

PDA

Rdiag1

AM

13LAD

PLV1

PRV1

Sep3

Sep4

OM2

LDiag2

Sep2

OM3

PLV2

PLV3

PRV2

PRV3

Sequence branding of the coronary arterial tree. LCA: Left Coronary Artery; RCA: Right Coronary Artery; LAD: Left Anterior Descending; LCX: Left Circumflex; OM: Obtuse Marginal; AM: Acute Marginal; PDA: Posterior Descenting Artery; PRV: Posterior Right Ventricular (1) = lateral; (2) = intermediate; (3) = medial, Sep: Septal; RI: Ramus Intermedius (authors interpretation).

Table 2:Table summarizing the number of branching from blood vessels of the Heart showing that branching numbers follow a Fibonacci Series [19].

In 2013, Yetkin et al. showed that the golden ratio of Phi (φ) 1.618, exists within the cardiac cycle of the human heart beat. It is known that the time periods for the systolic and diastolic phases vary with the method of measurement. The three most commonly used techniques for determining systolic and diastolic phases of the cardiac cycleinclude: electrographical, echocardiogaphical, and phonocardiographical techniques. Yetkin et al. defined systole as the time between the ‘R wave’ and the end of the ‘T wave’ using an electrocardiogram [21]. The data from their research included an assessment of 162 healthy subjects, and concluded that the ratio of the Diastolic time interval/Systolic time interval was 1.611, while the R-R interval/Diastolic time interval was 1.618 which is the value of Phi (φ) [21].

**Fibonacci, Fractals, and the Human Genome:** The supposition
that the structure of DNA and its organization pattern is a fractal
was first proposed by Benoit Mandelbot in 1982 [22]. Mandelbot,
for the most part, is responsible for first coining the term “fractal”
and providing mathematical explanation non-euclidean geometries
observed in nature. In the early 1990’s, based upon the supposition
made by Mandelbot, it was proven that the Human genome contains
fractal behavior - Fibonacci series, and the golden proportion. For the past 25 years, this work has mainly been led by French independentresearcher
Jean-Claude Perez. In 1991, Perez proposed that the DNA
gene-coding region sequences were strongly related to the Golden
Ratio and Fibonacci/Lucas integer numbers [23]. His work was
bolstered by Yamagishi et al who found consistency of a Fibonacci
series level of organization across the whole human genome [24]. In
2010, Perez provided evidence that the genetic code in Table 3 not
only maps codons to amino acids, but also serves as a global errordetection
scheme [25].

**Table 3:**The 64 codon populations of the whole human genome for the 3 codon reading frames of single stranded DNA (2843411612 codons). In this table, the 3 values in each cell are: the codon label, the codon’s total population, the “Codon Frequency Ratio” (CFR). CFR is computed as: codon population x 64 / 2.843.411.612. (Where 2.843.411.612 is the whole genome cumulated codons). Then, if CFR < 1, the codon is rare, If CFR>1, the codon is frequent.

SECOND NUCLEOTIDE

FIRST NUCLEOTIDET

C

A

G

THIRD NUCLEOTIDE

T

TTT 109591342

2.4667TCT 62964964

1.4172TAT 58718182

1.3216TGT 57468177

1.2935T

TTC 56120623

1.2632TCC 43850042

0.9870TAC 32272009

0.7264TGC 40949883

0.9217C

TTA 59263408

1.3339TCA 55697529

1.2536TAA 59167883

1.3318TGA 55709222

1.2539A

TTG 54004116

1.2133TCG 6263386

0.1410TAG 36718434

0.8265TGG 52453369

1.1806G

C

CTT 36828780

1.2791CTT 50494519

1.1365CAT 52236743

1.1758CGT 7137644

0.1607T

CTC 47838959

1.0768CCC 37290873

0.8393CAC 42634617

0.9596CGC 6737724

0.1517C

CTA 36671812

0.8254CCA 52352507

1.1784CAA 53776608

1.2104CGA 6251611

0.1407A

CTG 57598215

1.2964CCG 7815619

1.1759CAG 57544367

1.2952CGG 7815677

0.1759G

A

ATT 71001746

1.5981ACT 45731927

1.0293AAT 70880610

1.5954AGT 45794017

1.0307T

ATC 37952376

0.8542ACC 33024323

0.7433AAC 41380831

0.9314AGC 39724813

0.8941C

ATA 58649060

1.3201ACA 57234565

0.7433AAA 109143641

2.4566AGA 62837294

1.4144A

ATG 52222957

1.1754ACG 7117535

0.1602AAG 56701727

1.2763AGG 50430220

1.1351G

G

GTT 415577671

0.9354GCT 39746348

0.8946GAT 37990593

0.8551GGT 33071650

0.7444T

GTC 26866216

0.6047GCC 33788267

0.7605GAC 26820898

0.6037GGC 33774099

0.7602C

GTA 32292235

0.7268GCA 40907730

0.9208GAA 56018645

1.2609GGA 43853584

0.9871A

GTG 42755364

0.9623GCG 6744112

0.1518GAG 47821818

1.0764GGG 37333942

0.8203G

Table 3:The 64 codon populations of the whole human genome for the 3 codon reading frames of single stranded DNA (2843411612 codons). In this table, the 3 values in each cell are: the codon label, the codon’s total population, the “Codon Frequency Ratio” (CFR). CFR is computed as: codon population x 64 / 2.843.411.612. (Where 2.843.411.612 is the whole genome cumulated codons). Then, if CFR < 1, the codon is rare, If CFR>1, the codon is frequent.

In the simplest explanation of his research, Perez analyzed the entirety of the human genome yet only evaluated a single strand considering that 1 strand is a mere complement of the other. As outlined by Perez in 2010, the cumulative number of each of the 64 genetic codons for the three different codon reading frames was obtained [25]. Then analysis on 24 human chromosomes, using the same method, was performed and the results were totaled for the 3 codon reading frames and the 24 chromosomes. The 64 codon population was arranged according the Genetic Code as seen in Table 3. The table was then separated according to the 6 binary splits of a Dragon Curve. A dragon curve is an irregularly shaped curve with a self-similar pattern, hence a fractal image as shown in Figure 7.

**Figure 7:**Dragon curve after 7 folds (a), and after 11 folds (b) [27].

Figure 7:Dragon curve after 7 folds (a), and after 11 folds (b) [27].

Conceptually, a Dragon Curve can be created by folding of a piece of paper into a long strip. Then as the strip of paper is unfolded, the adjacent segments are rotated and formed at right angles [26,27]. Computer modeling of this concept is shown in Figure 7. Perez’s research in 2010makes several conclusions regarding the fractal nature of DNA. First, the universal genetic code serves as a macrolevel structural matrix [25]. This matrix serves as a regulatory mechanism, controlling and balancing the codon population within the whole human genome. By modeling of the arrangement of the nitrogenous bases, Adenine (A), Guanine (G), Cytosine (C), and Thymine (T), using both the Universal Genetic Code Table Figure 3 and the Dragon-Fractal paradigm, symmetry in the data was revealed. Such symmetry showed two attractors towards values of “1” and that of Phi (φ) 1.618 [25]. According to Chaos theory of dynamic systems, attractors can be thought of ‘magnetic’ points to which the initial state of a complex system evolves towards in its final conditions. Attractors are important in mathematical modeling and systems because they serve as the link between Chaos theory and Fractal geometry [28]. Perez concludes that the ratios between 3-base pair codons sorted by A or G in the second base positions and those by T and C in the second base position tend to an attractor value of “1” [25]. The ratios between codons C or G and T or A in the second base position tend to a cluster value of “3(φ)/2” [25]. Furthermore, the distance that separates these two attractors of “1” and “3(φ)/2”, is ½(φ). In a 2009 follow-up study by Perez, he analyzed the codon populations of 20 various species like Eukaryotes, bacteria and viruses. The results showed that 3 parameters (1, 2, and Phi (φ)) define codon populations within their genomes to a precision of 99% and often 99.999% [29]. For the human and chimpanzee genomes, codon frequencies are 99.99% correlated [29]. This exemplifies the widespread occurrence of Phi (φ) and Fibonacci series not only within the human genome but in other species as well.

## Conclusion

Fractal geometry lays the foundation to understanding the complexity of the shapes in nature. In the exploration of the origins of life through mathematics, the occurrence of the Golden Ratio, Fibonacci Series and the underlying Lucas series are observed in several aspects of life on planet earth and within the cosmos. Although widely identified in non-biological fields such as architecture and art, it has not been well explored in the human biology. Recent work has begun to explore the understanding of such phenomenon documented at several different scales and systems in the human anatomy and physiology ranging from orthopedics, dentistry, the spiral of the human ear, the cardiovascular system and the human genome. The observance of such a seemingly universal concept begs the question of the origin of life; however, more research needs to be performed to explore its physiological role in biology. Understanding its functional role may be a keystone to making quantum advances in several fields such as artificial intelligence, biomedical engineering designs, and human regeneration, amongst others.

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