Da Vinci Man Drawing Circle
"Vitruvian Man" by Leonardo da Vinci and the Gold Ratio
Takashi Ida
Advanced Ceramics Research Center
Nagoya Establish of Applied science, Japan
[ Japanese ]
The drawing scheme of "Vitruvian Man" past Leonardo da Vinci has been analyzed. It is ended that the ratio of the radius of the circle to the side length of the square was intended to exist 137 / 225 = 0.6088···, but non the gilded ratio (ane / r) = (51/2 − 1) / 2 = 0.6180··· .
1. "Vitruvian Man" by Leonardo da Vinci and the golden ratio
| The golden ratio is the proportion given by the numerical value: r = (1 + 5ane/2) / two = 1.6180··· , or (one / r) = (five1/2 − 1) / 2 = 0.6180··· . Information technology is oftentimes assumed that the ratio of the radius of the circle to the side length of the square (= the acme of a human) in the "Vitruvian Man", which is said to exist fatigued by Leonardo da Vinci, is given by the golden ratio. The author has evaluated the radius of the circle for a normalized image, obtained past slight modification, where the figure intended to be foursquare by da Vinci was adjusted to be square by linearly mapping the image with Adobe Photoshop. The prototype afterward the normalization is shown in Fig. 1. The ratio was estimated at 0.606 ∼ 0.609, which is significantly smaller than the gold ratio 0.618. | "Vitruvian man" by Leonardo da Vinci (normalized) |
2. Golden-ratio model
| The discrepancy in the "golden-ratio model" is more pronounced in Fig. 2 , where the foursquare and circle drawn by da Vinci (red) and the circle calculated with the golden ratio (blue) are superimposed. The fingertip touches both the scarlet circumvolve and foursquare drawn by da Vinci, when it is located at the hight of the superlative of the head. Withal, such a condition could never exist satisfied with the circle calculated with the aureate ratio (0.618). When the "gilded-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. Information technology looks impossible to achieve the situation that the fingertip touches both the foursquare and the gilded-ratio circle by ordinary motion of human shoulders. The radius of the circumvolve drawn past da Vinci was non intended to exist the aureate ratio. | Square and circumvolve drawn by da Vinci (red) and a circle with the radus of gold ratio (blueish) |
3. Model based on 45°-rotation of square
| It is suggested on a web site [ Geometrical construction of the Vitruvian Man past Leonardo da Vinci ] that the circle goes through the top vertex of the 45°-rotated square, as shown in Fig. three. In this case, the radius of the sphere should take the value (21/2 + 1) / iv = 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 supposition is constitute in the cartoon. Fifty-fifty if the outcome of this model shows expert 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 torso of the body. Then the conditions (i) and (two) cannot fully determine the position and radius of the circle. The circle drawn by da Vinci has post-obit boosted properties, which may not have been described past Vitruvius,
Since the allowable range for the radius of the circle becomes much more restricted by the weather (iii) and (iv), information technology indicates that da Vinci really intended to introduce those additional weather condition. | Lines drawn by da Vinci (ruby) and a circle going through the pinnacle vertex of the 45°-rotated foursquare (blue). |
4. Lines and points fatigued past da Vinci
| Lines marking the proportions almost man body are fatigued in "Vitruvian Man", and there is a kind of a calibration below the figure. Those markers and scale are superimposed as reddish lines in Fig. 4. There are four points marker the edges of 2 lines, AA′ and BB′ in Fig. iv. Refer to the original figure Fig. i, if necessary. | Lines, segments, dots and calibration fatigued by da Vinci (red). |
v. Texts noted by da Vinci
| In the texts on the lower department of "Vitruvian Man", it is written:
The distances betwixt 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 summit of the man. | Distances betwixt segments based on the annotation in the lower section. |
six. Four points marked by da Vinci
| The four edge points of lines AA′ and BB′ are conspicuously marked by da Vinci ( Fig. 4 ), while no mark is institute 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 superlative of the head (T) are respectively 1/6 and 1/4, both of which are consequent with the description in the text surface area. Still, no description about the length or the locations of the border points has been found. The measured length of the line BB′ marking the location of the "breasts" is close to ane/5 of the height of the homo ( Fig. 6 ). It is probable 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 (i/10). On the other paw, the definition of the length of the line AA′, which is close to 1/vii.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. And so the length AA′ should be less than the width of the shoulders (1/iv), but tolerable range remains, because not just the motion of arms simply also shoulders affects the location of moved fingertips. It should be noted that the locations A, A′, B and B′ appear to be bundled so that lines AB and A′B′ crosses only at the acme of the head (T), equally can be seen in Fig. half-dozen . Information technology 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 exist (1/five) × (1/6) / (1/iv) = 2/xv to satisfy this assumption. | Positioning of iv special points. |
7. Rotation of arms (1)
| 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 as 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 motility of the arm and shoulder is taken into account. We can imagine that da Vinci treated the bespeak A every bit the center for the rotation of the arm, determined the rotation radius from the distance betwixt A and C, and defined the point D as the crossing point of the rotated fingertip and the peak line of the foursquare. The circle going through the point D and touching the baseline of the foursquare can uniquely be determined. | Rotation of the arms (1). |
8. Rotation of the arms (2)
| It is likely that da Vinci geometrically determined the radius of the circle, but the radius tin besides be evaluated by calculation. When the side length of the square (meridian of the man) is assumed to be one, the distance between A and C should exist Air-conditioning = ane/2 − (2/fifteen)/two = thirteen/30, every bit shown in Fig. 7, and and so AD = AC = 13/30. The distance betwixt points D and T (superlative of the head) is given by ane/fifteen + [(13/30)2 − (ane/6)two]one/2 = 7/15 . Run across ( Fig. eight). | Rotation of the arms (two). |
9. Calculation of the radius of da Vinci's circle
| Let R exist the radius of the circle da Vinci intended to draw. And then the post-obit equation should be satisfied, [R 2 −(7/xv)ii]1/2 + R = 1 , every bit shown in Fig. 9. The solution of the above equation is given by R = [1 + (7/fifteen)two] / two = 137/225 = 0.6088··· . | Calculation of the radius of da Vinci'due south circle (R = 137/225). |
x. Conclusion
| Figure 10 shows superimposed epitome of the square and circle drawn in "Vitruvian Man" past Leonardo da Vinci as carmine figures, and a blueish circumvolve with the radius of 137/225 (0.609). In conclusion, Leonardo da Vinci assumed the width of the breasts BB′ to be i/5 of the height of the man, and defined ii points A and A′ as the ii:1 internally dividing points of TB and TB′ for the top of the caput (T) (Fig. half-dozen) . The fingertip (C) initially located at the same height as the point A was rotated effectually A, and institute the betoken D as the crossing indicate 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. nine) , and has certainly succeeded in drawing such a circle (Fig. x). | Square and circumvolve in the "Vitruvian Man" (red) and the circle with the radius of 137/225 (blueish). |
June xviii, 2012
woolfbeheriseed83.blogspot.com
Source: http://www.crl.nitech.ac.jp/~ida/education/VitruvianMan/index.html
Post a Comment for "Da Vinci Man Drawing Circle"