5. Bauentwurfslehre by Ernst Neufert

Bauentwurfslehre is a German word means Architects' Data a book authored by Ernst Neufert a German engineer. Architects' Data is reference book for spatial requirements. The author developed the book based on the previous work of Leonardo da Vinci with golden ratio proportions. In this section, the work of Bauentwurfslehre by Ernst Neufert is first explained. Then 29 proportional rules are derived to further the work of the algorithm that is being developed to provide 35 measures that helps building a human model.

circle is drawn with the navel as center and the radius from navel to the outer side of the arm; (6) a fifth circle is drawn with the center is mid-chest center, with radius outer arms; (7) a sixth circle is centered in the throat and the radius from throat to the line extended from shoulder to the ground; (8) a seventh circle is sketched with center in mid thighs and the radius is the outer side of the thigh; (9) an eighth circle is sketched and centered in the is mid-chest center, with radius inside arms.; (10) a ninth circle is sketched and centered in the navel and the radius is the sides of the middle section; (11) a tenth circle is drawn and centered in the point between the eyes and the radius is the sides of the face (temples); (12) the eleventh circle is drawn and centered to touch the seventh circle while reducing the radius to the outer side of the legs, below the knee, and finally (13) the twelfth circle is sketched similar to circle 11 while touching

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To explain the golden ratio from within the 29 measurements little m and big M, one must start with explaining the base case M16. The measurement M16 is measured from bottom of foot to

foot end (see Figure 7). M17 is derived according to the golden ratio rule seen in (1):

the second circle.

Figure 6. Bauentwurfslehre by Ernst Neufert (1936) [14].

The Bauentwurfslehre shown in Figure 6 is the composed of 29 proportional measurements, named m (in small letter) and M (capital letter) used on a human model. Also, there are 12 circles to derive the proportions. Both the circles and the proportions will be explained in the next paragraphs.

The 12 circles are numbered to explain each and differentiate one from the other. To build the model in Figure 6, the following steps are done: (1) a square is sketched with height of the man as the height of the square as well as the width; (2) two lines are sketched from opposite corners to locate the center of the first circle (the biggest circle); (3) a second circle is drawn with center is the navel and the radius is extended from navel to perpendicular line drawn from hand rest to ground; (4) a third circle is sketched with the navel as the center, and the radius is distance from navel to perpendicular line extended from elbow to ground; (5) a fourth

Figure 6. Bauentwurfslehre by Ernst Neufert (1936) [14].

8. Forearm: The distance from the elbow to the tip of the hand is ¼ of the height of a man.

9. Upper arm: the distance from the elbow to the armpit is 1/8 of the height of a man.

10. Hand: the length of the hand is 1/10 of the height of a man.

13. Leg: from below the foot to below the knee is ¼ of the height of a man.

14. Thigh: from below the knee to the root of the penis is ¼ of the height of a man.

15. Face: the distances from the below the chin to the nose and the eyebrows and the hairline

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,

Bauentwurfslehre is a German word means Architects' Data a book authored by Ernst Neufert a German engineer. Architects' Data is reference book for spatial requirements. The author developed the book based on the previous work of Leonardo da Vinci with golden ratio proportions. In this section, the work of Bauentwurfslehre by Ernst Neufert is first explained. Then 29 proportional rules are derived to further the work of the algorithm that is being developed to

The Bauentwurfslehre shown in Figure 6 is the composed of 29 proportional measurements, named m (in small letter) and M (capital letter) used on a human model. Also, there are 12 circles to derive the proportions. Both the circles and the proportions will be explained in the

The 12 circles are numbered to explain each and differentiate one from the other. To build the model in Figure 6, the following steps are done: (1) a square is sketched with height of the man as the height of the square as well as the width; (2) two lines are sketched from opposite corners to locate the center of the first circle (the biggest circle); (3) a second circle is drawn with center is the navel and the radius is extended from navel to perpendicular line drawn from hand rest to ground; (4) a third circle is sketched with the navel as the center, and the radius is distance from navel to perpendicular line extended from elbow to ground; (5) a fourth

11. The root of the penis is at 1/2 the height of a man.

are equal to the ears and to 1/3 of the face.

proportion use in art is equally profound and archaic.

provide 35 measures that helps building a human model.

5. Bauentwurfslehre by Ernst Neufert

next paragraphs.

12. Foot: is 1/6 of the height of a man.

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circle is drawn with the navel as center and the radius from navel to the outer side of the arm; (6) a fifth circle is drawn with the center is mid-chest center, with radius outer arms; (7) a sixth circle is centered in the throat and the radius from throat to the line extended from shoulder to the ground; (8) a seventh circle is sketched with center in mid thighs and the radius is the outer side of the thigh; (9) an eighth circle is sketched and centered in the is mid-chest center, with radius inside arms.; (10) a ninth circle is sketched and centered in the navel and the radius is the sides of the middle section; (11) a tenth circle is drawn and centered in the point between the eyes and the radius is the sides of the face (temples); (12) the eleventh circle is drawn and centered to touch the seventh circle while reducing the radius to the outer side of the legs, below the knee, and finally (13) the twelfth circle is sketched similar to circle 11 while touching the second circle.

To explain the golden ratio from within the 29 measurements little m and big M, one must start with explaining the base case M16. The measurement M16 is measured from bottom of foot to foot end (see Figure 7). M17 is derived according to the golden ratio rule seen in (1):

$$\frac{M17 + M16}{M17} = \frac{M17}{M16} = \varphi \tag{1}$$

M11 ¼ M20 þ M21: (8)

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

M4 ¼ M9 þ M11 (9)

M1 ¼ M3 þ M4 (11)

M2 ¼ M6 þ M5 (13)

M6 ¼ M15 þ M14 (16)

<sup>M</sup><sup>3</sup> <sup>¼</sup> <sup>φ</sup> (10)

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<sup>M</sup><sup>2</sup> <sup>¼</sup> <sup>φ</sup> (12)

<sup>φ</sup> (14)

<sup>φ</sup> <sup>þ</sup> <sup>1</sup> (15)

<sup>φ</sup> (17)

<sup>φ</sup> <sup>þ</sup> <sup>1</sup> (18)

The upper limit of M4 is tangent to circle 8 and the center of circle 9 and crosses the navel. M4 is the addition of M9 and M11 (see Figure 6). Also, M4 is driven according to rules explained in

<sup>M</sup><sup>4</sup> <sup>¼</sup> <sup>M</sup><sup>4</sup>

Once you reach the navel, one can easily calculate M2 According to (12). The M27 is from top of the head to hair line tip. The M27 is used to derive M26 according to rules explained in Figure 3. The lower limit of M27 is tangent to circle 10, while the lower limit of M26 is tangent to circle 6 and crosses the center of circle 10 (see Figure 6). Again, to derive M26 apply:

<sup>M</sup><sup>1</sup> <sup>¼</sup> <sup>M</sup><sup>1</sup>

According to (13), the M2 is the addition of M6 and M5. Hence, both M6 and M5 can be driven from M2 according to (14) and (15). The upper limit of M2 is tangent for circles 1 and 2. The M2

<sup>M</sup><sup>5</sup> <sup>¼</sup> <sup>M</sup><sup>2</sup>

<sup>M</sup><sup>6</sup> <sup>¼</sup> <sup>M</sup><sup>2</sup>

The M6 is from the base of the throat to the top of the head. Since M6 is the total of M14 and

<sup>M</sup><sup>14</sup> <sup>¼</sup> <sup>M</sup><sup>6</sup>

<sup>m</sup><sup>15</sup> <sup>¼</sup> <sup>M</sup><sup>6</sup>

The M5 is from the navel to the throat, the upper limit of M5 is same level as center of circle 6. Since M5 is the total of M13 and M12 according to (16). Hence, M12, and M13 can be driven

M15 according to (16). Hence, M14, and M15 can be driven according to (17) and (18):

M4 þ M3

M1 þ M2

lower limit crosses center of circle 9 which is the navel (see Figure 6).

Figure 3 and as seen in (9):

according to (20) and (21):

Furthermore, M7 is the total of both M16 and M17 [see (2)]. In addition, M7 is used to derive M8 according to golden ratio rule as illustrated in Eq. (3). Note that the upper limit of M8 is below the knee and it is the tangent of both circles 12 and 2 (see Figure 6). Also, note that M3 is the total of M7 and M8 as seen in Eq. (4):

$$\mathbf{M}\mathbf{\mathcal{T}} = \mathbf{M}16 + \mathbf{M}17 \tag{2}$$

$$\frac{M8 + M7}{M8} = \frac{M8}{M7} = \varphi \tag{3}$$

$$\mathbf{M}\mathbf{3} = \mathbf{M}\mathbf{7} + \mathbf{M}\mathbf{8} \tag{4}$$

Again, M18 is the measure of the knee, below knee to above the knee. The M18 is used to derive M19 according to rule in Eq. (5). The upper limit of M19 is the center of circle 7 and tangent to circle 3 also, if a line drawn with the hand extended on the sides. In addition, note that M9 is the total of M18 and M19 see (6):

$$\frac{M19 + M18}{M19} = \frac{M19}{M18} = \varphi \tag{5}$$

$$\mathbf{M}\mathbf{9} = \mathbf{M}\mathbf{1}\mathbf{8} + \mathbf{M}\mathbf{1}\mathbf{9} \tag{6}$$

The M9 is used to derive M11 = M9 � 1.61803398874989. The M20 is the measure of length of the hand (tip of figure to rest). The M20 is used to derive M21 according to golden ratio rule in (7). The upper limit of M20 and the lower limit of M21 is tangent to circles 7 and 4 (see Figure 6). The upper limit of M21 is tangent to circle 8 and the center of circle 9 and crosses the navel:

$$\frac{M21 + M20}{M21} = \frac{M21}{M20} = \varphi \tag{7}$$

The upper limit of M11 is tangent to circle 8 and the center of circle 9 and crosses the navel. And M11 is the addition of M20 and M21 (see Figure 6). M11 is driven from M9 according to rule in Figure 3:

Figure 7. Limit of M16.

$$\mathbf{M}11 = \mathbf{M}20 + \mathbf{M}21.\tag{8}$$

The upper limit of M4 is tangent to circle 8 and the center of circle 9 and crosses the navel. M4 is the addition of M9 and M11 (see Figure 6). Also, M4 is driven according to rules explained in Figure 3 and as seen in (9):

M17 þ M16

M8 þ M7

M19 þ M18

M21 þ M20

the total of M7 and M8 as seen in Eq. (4):

122 Machine Learning and Biometrics

that M9 is the total of M18 and M19 see (6):

Figure 3:

Figure 7. Limit of M16.

<sup>M</sup><sup>17</sup> <sup>¼</sup> <sup>M</sup><sup>17</sup>

Furthermore, M7 is the total of both M16 and M17 [see (2)]. In addition, M7 is used to derive M8 according to golden ratio rule as illustrated in Eq. (3). Note that the upper limit of M8 is below the knee and it is the tangent of both circles 12 and 2 (see Figure 6). Also, note that M3 is

<sup>M</sup><sup>8</sup> <sup>¼</sup> <sup>M</sup><sup>8</sup>

Again, M18 is the measure of the knee, below knee to above the knee. The M18 is used to derive M19 according to rule in Eq. (5). The upper limit of M19 is the center of circle 7 and tangent to circle 3 also, if a line drawn with the hand extended on the sides. In addition, note

<sup>M</sup><sup>19</sup> <sup>¼</sup> <sup>M</sup><sup>19</sup>

The M9 is used to derive M11 = M9 � 1.61803398874989. The M20 is the measure of length of the hand (tip of figure to rest). The M20 is used to derive M21 according to golden ratio rule in (7). The upper limit of M20 and the lower limit of M21 is tangent to circles 7 and 4 (see Figure 6). The upper limit of M21 is tangent to circle 8 and the center of circle 9 and crosses the navel:

<sup>M</sup><sup>21</sup> <sup>¼</sup> <sup>M</sup><sup>21</sup>

The upper limit of M11 is tangent to circle 8 and the center of circle 9 and crosses the navel. And M11 is the addition of M20 and M21 (see Figure 6). M11 is driven from M9 according to rule in

<sup>M</sup><sup>16</sup> <sup>¼</sup> <sup>φ</sup> (1)

<sup>M</sup><sup>7</sup> <sup>¼</sup> <sup>φ</sup> (3)

<sup>M</sup><sup>18</sup> <sup>¼</sup> <sup>φ</sup> (5)

<sup>M</sup><sup>20</sup> <sup>¼</sup> <sup>φ</sup> (7)

M7 ¼ M16 þ M17 (2)

M3 ¼ M7 þ M8 (4)

M9 ¼ M18 þ M19 (6)

$$\mathbf{M4} = \mathbf{M9} + \mathbf{M11} \tag{9}$$

$$\frac{M4 + M3}{M4} = \frac{M4}{M3} = \wp \tag{10}$$

$$\mathbf{M1} = \mathbf{M3} + \mathbf{M4} \tag{11}$$

Once you reach the navel, one can easily calculate M2 According to (12). The M27 is from top of the head to hair line tip. The M27 is used to derive M26 according to rules explained in Figure 3. The lower limit of M27 is tangent to circle 10, while the lower limit of M26 is tangent to circle 6 and crosses the center of circle 10 (see Figure 6). Again, to derive M26 apply:

$$\frac{M1 + M2}{M1} = \frac{M1}{M2} = \varphi \tag{12}$$

According to (13), the M2 is the addition of M6 and M5. Hence, both M6 and M5 can be driven from M2 according to (14) and (15). The upper limit of M2 is tangent for circles 1 and 2. The M2 lower limit crosses center of circle 9 which is the navel (see Figure 6).

$$\mathbf{M2} = \mathbf{M6} + \mathbf{M5} \tag{13}$$

$$M\mathfrak{F} = \frac{M\mathfrak{2}}{\varphi} \tag{14}$$

$$M\mathfrak{G} = \frac{M\mathfrak{2}}{\mathfrak{q} + 1} \tag{15}$$

The M6 is from the base of the throat to the top of the head. Since M6 is the total of M14 and M15 according to (16). Hence, M14, and M15 can be driven according to (17) and (18):

$$\mathbf{M}\mathbf{6} = \mathbf{M}1\mathbf{5} + \mathbf{M}1\mathbf{4} \tag{16}$$

$$M14 = \frac{M6}{\varphi} \tag{17}$$

$$m15 = \frac{M6}{\wp + 1} \tag{18}$$

The M5 is from the navel to the throat, the upper limit of M5 is same level as center of circle 6. Since M5 is the total of M13 and M12 according to (16). Hence, M12, and M13 can be driven according to (20) and (21):

$$\mathbf{M}\mathbf{5} = \mathbf{M}\mathbf{1}\mathbf{3} + \mathbf{M}\mathbf{1}\mathbf{2} \tag{19}$$

$$M12 = \frac{M5}{\varphi} \tag{20}$$

M24 ¼ M29 þ M28 (31)

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

<sup>φ</sup> (32)

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<sup>φ</sup> <sup>þ</sup> <sup>1</sup> (33)

<sup>M</sup><sup>29</sup> <sup>¼</sup> <sup>M</sup><sup>24</sup>

<sup>M</sup><sup>28</sup> <sup>¼</sup> <sup>M</sup><sup>24</sup>

This section explained 29 proportions and 12 circles used in Bauentwurfslehre by Ernst Neufert. The section first explained the work which is based on Leonardo da Vinci with golden ratio proportions. Then the research explained each circle's radius and center. Next, the research explained each proportion and how to calculate each ratio from M1 to M29. The findings of this section will be reflected in the algorithm suggested in Section 7 of this research. The next section will discuss the golden proportions used to model a human face and hands.

The golden proportion is repeated in the human face and is used by plastic surgeons to follow as guidelines. Human face from the bottom of the chin to the hairline in length to the edge of the eyebrow is 1:1.618 or 1: φ as illustrated in Figure 12, the red rectangle. Likewise, proportion is illustrated in the same figure using the green rectangle; the green rectangle is edges of the eyebrows and the height is from the tip of the nose to the top of the eyebrows. The third golden proportion is seen in the blue rectangle. The blue rectangle extends from the eyebrow to the bottom lip, and from the space between the eyes to the end of the eyebrow. Based on the previous one can calculate the width of the face to equal 1: length of the face. The length of the face = M26 + M25 + M29 and the length of the green box equals M25 and the length of the blue rectangle is 2 � M25. Once the length is calculated, then the width can be easily calculated

6. Face and hand proportions

Figure 8. Face and hand golden proportions.

(Figure 8).

$$M13 = \frac{M5}{q+1} \tag{21}$$

The M12 is from the navel to rib cage, the upper limit of M12 is same level as tangent to of circle 4. Since M12 is the total of M23 and M22 according to (22). Hence, M22, and M23 can be driven according to (23) and (24):

$$\mathbf{M12} = \mathbf{M23} + \mathbf{M22} \tag{22}$$

$$M22 = \frac{M12}{\varphi} \tag{23}$$

$$M23 = \frac{M12}{q+1} \tag{24}$$

The M15 is from eyebrows to top of the head, the upper limit of M12 is same level as tangent to of circle 6. Since M12 is the total of M27 and M26 according to (25). Hence, M26, and M27 can be driven according to (26) and (27):

$$\text{M15} = \text{M27} + \text{M26} \tag{25}$$

$$M26 = \frac{M15}{\varphi} \tag{26}$$

$$M27 = \frac{M15}{\wp + 1} \tag{27}$$

The M14 is from the throat to eyebrows. M14 is tangent to circles 3 and 5 while the upper limit is tangent to circle 6. Note that, M14 is the addition of M24 and M25, and is derived according (29) and (30):

$$\text{M14} = \text{M24} + \text{M25} \tag{28}$$

$$M24 = \frac{M14}{\varphi} \tag{29}$$

$$M25 = \frac{M14}{q+1} \tag{30}$$

The M24 upper limit will be tangent to circle 10 and the lower limit is tangent to circle 5. M24 is tangent to circles 3 and 5 while the upper limit is tangent to circle 6. Note that, M24 is the addition of M29 and M28, and is derived according (32) and (33):

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$$\text{M24} = \text{M29} + \text{M28} \tag{31}$$

$$M29 = \frac{M24}{\varphi} \tag{32}$$

$$M28 = \frac{M24}{\wp + 1} \tag{33}$$

This section explained 29 proportions and 12 circles used in Bauentwurfslehre by Ernst Neufert. The section first explained the work which is based on Leonardo da Vinci with golden ratio proportions. Then the research explained each circle's radius and center. Next, the research explained each proportion and how to calculate each ratio from M1 to M29. The findings of this section will be reflected in the algorithm suggested in Section 7 of this research. The next section will discuss the golden proportions used to model a human face and hands.
