4. Vitruvian man

3. The mathematics of golden ratio: Fibonacci series

Figure 3. A golden rectangle with longer side a and shorter segment b.

capital P equals 0.618033988749895.

that converge to the golden ratio.

15 decimal places:

116 Machine Learning and Biometrics

(Table 1):

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

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

fn <sup>¼</sup> Phi<sup>n</sup>=5½

For example, the 40th number in the Fibonacci series is 102,334,155, which can be computed as

f40 <sup>¼</sup> Phi<sup>40</sup>=5½ <sup>¼</sup> <sup>102</sup>; <sup>334</sup>; <sup>155</sup>

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 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 creates the greatest harmony in the symmetrical relations.

The use of proportions when drawing a human body is not new, in fact, Leonardo da Vinci suggested a 15 rules of proportions which will be discussed in this section. Also, Marcus Vitruvius Pollio (born c. 80–70 BC, died after c. 15 BC), named by Taylor in an article in The New Zealand Herald called Marcus Vitruvius Pollio "the world's first known engineer" [12]. Marcus Vitruvius Pollio wrote his work On Architecture in 25 BC. In his book, he stated that human body and a perfect building are similar [1]. Furthermore, the use of proportion in buildings, paintings, and sculpting is not a new idea so is the use of human body proportions in buildings.

Leonardo da Vinci developed 15 proportion rules used to model a human in his work named VITRUVIAN MAN. The famous proportions were written underneath the illustration shown in Figure 4. The rules are further explained in Figure 5 and in the list of proportions shown at the end of this section.

The first rule is that the length of a man is equal to the width of a man with both arms extended. Hence, a man can be inscribed in a square, as seen in Figure 4. A man can be inscribed in circle where the extended arms while raised to the level of the head and both feet are on the circumstance of the circle, as seen in Figure 4.

The second rule is for the proportion of the face which is 1/10 of the man height. The third rule is for the head proportion which is 1/8 of man height. The fourth and fifth rules, from the chest to the hair line is 1/7-man height, and from chest to the head is 1/6-man height. The sixth rule pertains to the shoulders width, which is ¼ of man height. The seventh rule measures the distance from breast to the top of the head and the proportion is ¼ of the man height, noting

that measurements from the rule 6 and 7 correlate. Rules 8–10 all relate to forearm, upper arm, and hand which are ¼, 1/8, and 1/10 height of a man, respectively. It is worthwhile to note that the aforementioned three rules on measurement of forearm, shoulders width correlate with rule 7. Rule 11 is the measure of half the man, from the root of the penis to the top of the head or the root of the penis to the sole of the feet is ½ of man height. Rules 12–14 pertain to foot leg and thigh. The foot is 1/6 of man height while the leg and thigh are both equal to ¼ of man height. The last rule explains proportions used in the face which divides the face to threethirds: one-third from hair line to eyebrows, second third from eyebrows to below nose, and from below the nose to end of chin. The last rule, pertaining to face proportions will be thoroughly discussed in Section 6 of this chapter. The original quote from Leonardo da Vinci

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

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

119

In short, [5], the following table summarized the previous quote as illustrated in Figure 5.

2. Face: from the hairline to the bottom of the chin is 1/10 of the height of a man. 3. Head: from below the chin to the top of the head is 1/8 of the height of a man.

4. From above the chest to the top of the head is 1/6 of the height of a man.

6. Shoulders: the maximum width of the shoulders is ¼ of the height of a man.

1. The length of the outspread arms is equal to the height of a man.

Figure 5. Vitruvian man with proportions suggested by Leonardo da Vinci.

5. From above the chest to the hairline is 1/7 of the height of a man.

7. From the breasts to the top of the head is ¼ of the height of a man.

work is in the research [16, 17].

Figure 4. Vitruvian man by Leonardo da Vinci, Galleria dell' Accademia, Venice (1485–1490).

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Figure 5. Vitruvian man with proportions suggested by Leonardo da Vinci.

The use of proportions when drawing a human body is not new, in fact, Leonardo da Vinci suggested a 15 rules of proportions which will be discussed in this section. Also, Marcus Vitruvius Pollio (born c. 80–70 BC, died after c. 15 BC), named by Taylor in an article in The New Zealand Herald called Marcus Vitruvius Pollio "the world's first known engineer" [12]. Marcus Vitruvius Pollio wrote his work On Architecture in 25 BC. In his book, he stated that human body and a perfect building are similar [1]. Furthermore, the use of proportion in buildings, paintings, and sculpting is not a new idea so is the use of human body proportions in

Leonardo da Vinci developed 15 proportion rules used to model a human in his work named VITRUVIAN MAN. The famous proportions were written underneath the illustration shown in Figure 4. The rules are further explained in Figure 5 and in the list of proportions shown at

The first rule is that the length of a man is equal to the width of a man with both arms extended. Hence, a man can be inscribed in a square, as seen in Figure 4. A man can be inscribed in circle where the extended arms while raised to the level of the head and both feet

The second rule is for the proportion of the face which is 1/10 of the man height. The third rule is for the head proportion which is 1/8 of man height. The fourth and fifth rules, from the chest to the hair line is 1/7-man height, and from chest to the head is 1/6-man height. The sixth rule pertains to the shoulders width, which is ¼ of man height. The seventh rule measures the distance from breast to the top of the head and the proportion is ¼ of the man height, noting

buildings.

the end of this section.

118 Machine Learning and Biometrics

are on the circumstance of the circle, as seen in Figure 4.

Figure 4. Vitruvian man by Leonardo da Vinci, Galleria dell' Accademia, Venice (1485–1490).

that measurements from the rule 6 and 7 correlate. Rules 8–10 all relate to forearm, upper arm, and hand which are ¼, 1/8, and 1/10 height of a man, respectively. It is worthwhile to note that the aforementioned three rules on measurement of forearm, shoulders width correlate with rule 7. Rule 11 is the measure of half the man, from the root of the penis to the top of the head or the root of the penis to the sole of the feet is ½ of man height. Rules 12–14 pertain to foot leg and thigh. The foot is 1/6 of man height while the leg and thigh are both equal to ¼ of man height. The last rule explains proportions used in the face which divides the face to threethirds: one-third from hair line to eyebrows, second third from eyebrows to below nose, and from below the nose to end of chin. The last rule, pertaining to face proportions will be thoroughly discussed in Section 6 of this chapter. The original quote from Leonardo da Vinci work is in the research [16, 17].

In short, [5], the following table summarized the previous quote as illustrated in Figure 5.


In closing, this section attempted to shed light on the significance of using the divine proportions in design, beauty, and parity. Famous and profound painters, designers over the centuries have been influenced to use proportion when painting and sculpting human body; in this context, 15 proportional rules used by Leonardo da Vinci in drawing the Vitruvian Man was explicitly discussed, noting that proportion use was extended to be used in architecture and by architects like Marcus Vitruvius Pollio, which will be further discussed next. As such, proportion use in art is equally profound and archaic.
