3. The mathematics of golden ratio: Fibonacci series

In view that understanding the golden ratio mathematics is imminent, especially when it is reflected in the Fibonacci Series [2]. Another perspective is the how the golden ration interrelates with the square root of 5. The two perspectives are clarified in the next paragraphs. Accordingly, this section will discuss the mathematical properties of golden ratio (Phi), especially in Fibonacci series, while noting that phi with small letter p is equal to 1.618033988749895, whereas Phi with capital P equals 0.618033988749895.

Fibonacci series, named after Leonardo Fibonacci, is a simple series that when starting with 0 and 1, each new number in the series is simply the sum of the two before it. The ratio of each successive pair of numbers in the series approximates phi (1.618). In fact, after the 40th number in the series, the ratio is accurate to 15 decimal places. Furthermore, the value of golden ratio (phi) is reciprocal to the value of golden ratio (Phi), noting that the ratios of the successive numbers in the Fibonacci series quickly converge on golden ratio (Phi), the ratio is accurate to 15 decimal places:

$$\mathbf{f\_n} = P \text{hů}^n / 5^{\text{>}}$$

For example, the 40th number in the Fibonacci series is 102,334,155, which can be computed as (Table 1):

4. Vitruvian man

Table 1. Fibonacci series converge on Phi.

creates the greatest harmony in the symmetrical relations.

Proportions used to model, paint, and sculpt a human body are essential, as [18] Luca Pacioli, a contemporary of Da Vinci, indicates that "without mathematics there is no art," proportions are an integral part of design and beauty of nature, to achieve beauty, balance, and harmony, thereby presenting visual parity to the audience. According to [9], such use of proportion

n Fn Fn+1 phi Phi 0 1 infinity 0 1 2 2 0.5 3 5 1.666666667 0.6

 8 13 1.62500000000000 0.615384615384615 21 34 1.61904761904762 0.617647058823529 55 89 1.61818181818182 0.617977528089888 144 233 1.61805555555556 0.618025751072961 377 610 1.61803713527851 0.618032786885246 987 1597 1.61803444782168 0.618033813400125 2584 4181 1.61803405572755 0.618033963166707 6765 10,946 1.61803399852180 0.618033985017358 17,711 28,657 1.61803399017560 0.618033988205325 46,368 75,025 1.61803398895790 0.618033988670443 121,393 196,418 1.61803398878024 0.618033988738303 317,811 514,229 1.61803398875432 0.618033988748204 832,040 1,346,269 1.61803398875054 0.618033988749648 2,178,309 3,524,578 1.61803398874999 0.618033988749859 5,702,887 9,227,465 1.61803398874991 0.618033988749890 14,930,352 24,157,817 1.61803398874990 0.618033988749894 39,088,169 63,245,986 1.61803398874990 0.618033988749895 102,334,155 165,580,141 1.61803398874989 0.618033988749895 267,914,296 433,494,437 1.61803398874989 0.618033988749895 701,408,733 1,134,903,170 1.61803398874989 0.618033988749895 1,836,311,903 2,971,215,073 1.61803398874989 0.618033988749895 4,807,526,976 7,778,742,049 1.61803398874989 0.618033988749895 12,586,269,025 20,365,011,074 1.61803398874989 0.618033988749895 32,951,280,099 53,316,291,173 1.61803398874989 0.618033988749895

A Human Body Mathematical Model Biometric Using Golden Ratio: A New Algorithm

http://dx.doi.org/10.5772/intechopen.76113

117

$$\mathbf{f\_{40}} = \text{Phi}^{40}/\text{5}^{\circ\_{\hat{2}}} = 102,334,155$$

Concluding this section, where the mathematical perspective of the Golden Ratio was illustrated through the Fibonacci Series and the relationship of the successive elements of the series that converge to the golden ratio.

A Human Body Mathematical Model Biometric Using Golden Ratio: A New Algorithm http://dx.doi.org/10.5772/intechopen.76113 117


Table 1. Fibonacci series converge on Phi.
