Takashi Ida
Advanced Ceramics Research Center
Nagoya Institute of Technology, Japan
[ Japanese ]
The drawing scheme of “Vitruvian Man” by Leonardo da Vinci has been analyzed. It is concluded that the ratio of the radius of the circle to the side length of the square was intended to be 137 / 225 = 0.6088···, but not the golden ratio (1 / r) = (5^{1/2} − 1) / 2 = 0.6180··· .
The golden ratio is the proportion given by the numerical value: r = (1 + 5^{1/2}) / 2 = 1.6180··· , or (1 / r) = (5^{1/2} − 1) / 2 = 0.6180··· . It is often assumed that the ratio of the radius of the circle to the side length of the square (= the height of a man) in the “Vitruvian Man”, which is said to be drawn by Leonardo da Vinci, is given by the golden ratio. The author has evaluated the radius of the circle for a normalized image, obtained by slight modification, where the figure intended to be square by da Vinci was adjusted to be square by linearly mapping the image with Adobe Photoshop. The image after the normalization is shown in Fig. 1. The ratio was estimated at 0.606 ∼ 0.609, which is significantly smaller than the golden ratio 0.618. |
“Vitruvian man” by Leonardo da Vinci (normalized) |
The discrepancy in the “golden-ratio model” is more pronounced in Fig. 2 , where the square and circle drawn by da Vinci (red) and the circle calculated with the golden ratio (blue) are superimposed. The fingertip touches both the red circle and square drawn by da Vinci, when it is located at the hight of the top of the head. However, such a condition could never be satisfied with the circle calculated with the golden ratio (0.618). When the “golden-ratio circle” is in contact with the base line of the square, the upper part of the circle should necessarily be located closely to the upper vertices of the square. It looks impossible to achieve the situation that the fingertip touches both the square and the golden-ratio circle by ordinary motion of human shoulders. The radius of the circle drawn by da Vinci was not intended to be the golden ratio. |
Square and circle drawn by da Vinci (red) and a circle with the radus of golden ratio (blue) |
It is suggested on a web site [ Geometrical construction of the Vitruvian Man by Leonardo da Vinci ] that the circle goes through the top vertex of the 45°-rotated square, as shown in Fig. 3. In this case, the radius of the sphere should have the value (2^{1/2} + 1) / 4 = 0.604, which is certainly closer to the measured value 0.606 ∼ 0.609 than the golden ratio 0.618. However, no trace to support this assumption is found in the drawing. Even if the result of this model shows good fit to the observed data, it is still difficult to justify the assumption. By the way, the circle described by Vitruvius should satisfy the following conditons,
However, the best position of the navel will be ambiguous, and the circumference that touches fingertips and feet will depend on the angles of the arms and legs from the trunk of the body. Then the conditions (i) and (ii) cannot fully determine the position and radius of the circle. The circle drawn by da Vinci has following additional properties, which may not have been described by Vitruvius,
Since the allowable range for the radius of the circle becomes much more restricted by the conditions (iii) and (iv), it indicates that da Vinci really intended to introduce those additional conditions. |
Lines drawn by da Vinci (red) and a circle going through the top vertex of the 45°-rotated square (blue). |
Lines marking the proportions about human body are drawn in “Vitruvian Man”, and there is a kind of a scale below the figure. Those markers and scale are superimposed as red lines in Fig. 4. There are four points marking the edges of two lines, AA′ and BB′ in Fig. 4. Refer to the original figure Fig. 1, if necessary. |
Lines, segments, dots and scale drawn by da Vinci (red). |
In the texts on the lower section of “Vitruvian Man”, it is written:
The distances between line markers on the drawing are in good coincidence with the above description, as shown in Fig. 5. The divisons on the scale below the drawing correspond to 1/96 and 1/24 of the height of the man. |
Distances between segments based on the note in the lower section. |
The four edge points of lines AA′ and BB′ are clearly marked by da Vinci ( Fig. 4 ), while no mark is found for other edge points. It suggests that da Vinci attached special meaning to the length or the location of the edge points of lines AA′ and BB′. The distances of AA′ and BB′ from the top of the head (T) are respectively 1/6 and 1/4, both of which are consistent with the description in the text area. However, no description about the length or the locations of the edge points has been found. The measured length of the line BB′ marking the location of the “breasts” is close to 1/5 of the height of the man ( Fig. 6 ). It is likely that da Vinci assumed the value 1/5 as the width of the breasts, which is equal to the twice the length of a hand (1/10). On the other hand, the definition of the length of the line AA′, which is close to 1/7.5 = 2/15, still remains unclear. As described later, the points A and A′ can be assigned to the centers for the rotation of the arms. Then the length AA′ should be less than the width of the shoulders (1/4), but tolerable range remains, because not only the motion of arms but also shoulders affects the location of moved fingertips. It should be noted that the locations A, A′, B and B′ appear to be arranged so that lines AB and A′B′ crosses just at the top of the head (T), as can be seen in Fig. 6 . It is likely that da Vinci applied this condition to remove amguity and fix the positions of the points A and A′. The length of the line AA′ should be (1/5) × (1/6) / (1/4) = 2/15 to satisfy this assumption. |
Positioning of four special points. |
The fingertips (C and C′) for horizontally stretched arms are located at the same height as the points A and A′ ( Fig. 7 ). The location of a fingertip at the height of the top of the head (D) and the point C appear to be equally distant from the point A. Even if the point A deviates from the location of the shoulder joint, it can virtually be the center about rotation of an arm, when associative motion of the arm and shoulder is taken into account. We can imagine that da Vinci treated the point A as the center for the rotation of the arm, determined the rotation radius from the distance between A and C, and defined the point D as the crossing point of the rotated fingertip and the top line of the square. The circle going through the point D and touching the baseline of the square can uniquely be determined. |
Rotation of the arms (1). |
It is likely that da Vinci geometrically determined the radius of the circle, but the radius can also be evaluated by calculation. When the side length of the square (height of the man) is assumed to be one, the distance between A and C should be AC = 1/2 − (2/15)/2 = 13/30, as shown in Fig. 7, and then AD = AC = 13/30. The distance between points D and T (top of the head) is given by 1/15 + [(13/30)^{2} − (1/6)^{2}]^{1/2} = 7/15 . See ( Fig. 8). |
Rotation of the arms (2). |
Let R be the radius of the circle da Vinci intended to draw. Then the following equation should be satisfied, [R^{2} −(7/15)^{2}]^{1/2} + R = 1 , as shown in Fig. 9. The solution of the above equation is given by R = [1 + (7/15)^{2}] / 2 = 137/225 = 0.6088··· . |
Calculation of the radius of da Vinci's circle (R = 137/225). |
Figure 10 shows superimposed image of the square and circle drawn in “Vitruvian Man” by Leonardo da Vinci as red figures, and a blue circle with the radius of 137/225 (0.609). In conclusion, Leonardo da Vinci assumed the width of the breasts BB′ to be 1/5 of the height of the man, and defined two points A and A′ as the 2:1 internally dividing points of TB and TB′ for the top of the head (T) (Fig. 6) . The fingertip (C) initially located at the same height as the point A was rotated around A, and found the point D as the crossing point with the top line of the square (Fig. 7) . Finally, da Vinci has drawn the circle touching the point D and the baseline of the square (Fig. 8) . Consequently, Leonardo da Vinci intended to draw the circle with the radius of 137/225 of the side length of the square (Fig. 9) , and has certainly succeeded in drawing such a circle (Fig. 10). |
Square and circle in the “Vitruvian Man” (red) and the circle with the radius of 137/225 (blue). |
June 18, 2012