### Takashi Ida

### Advanced Ceramics Research Center, Nagoya Institute of Technology, Japan

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…

# 1. “Vitruvian Man” by Leonardo da Vinci and the golden ratio

The golden ratio is the proportion given by the numerical value: *r* = (5^{1/2} + 1) / 2 = 1.6180…, or (1 / *r*) = (5^{1/2} – 1) / 2 = 0.6180… .

It is sometimes assumed that the ratio of the radius of the circle to the side length of the square (= height of a man) in the “Vitruvian Man”, which is said to be drawn by Leonardo da Vinci c. 1490, is given by the golden ratio (“Da Vinci code” by Dan Brown, for example).

The author has obtained image data of “Vitruvian Man” through the internet, and retouched the image with Adobe Photoshop to make a normalized image, where the square drawn by Leonardo da Vinci appear square on the PC screen. The image after normalization is shown in Figure 1.

The ratio of the radius of the circle to the square in the normalized image is estimated at 0.606 – 0.609, which is significantly smaller than the golden ratio.

# 2. Golden-ratio model

The discrepancy in the “golden-ratio model” is more pronounced in Figure 2, where the square and circle drawn by Leonardo da Vinci (red lines) and the circle (blue line) with the radius of the golden ratio ((1 / *r*) = (5^{1/2} – 1) / 2 = 0.6180…) are superimposed to the normalized image.

Figure 2

Lines drawn by Leonardo da Vinci (red) and a circle with the radius of golden ratio (blue)

The fingertips touch both the red circle and square drawn by Leonardo da Vinci, when they are located at the height of the top of the head.

Such restrictions could never be satisfied with the circle with the radius 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 more closely to the upper vertices of the square drawn by Leonardo da Vinci. It looks impossible to realize the situation that the fingertips touch both the square and the golden-ratio circle by ordinary motion of human shoulders.

The relative locations of the fingertips and shoulders of a human body are not affected by the artificial modification about the image of a human body.

# 3. Model based on 45º-rotation of square

It was suggested on a web page (now deleted) entitled “Geometrical construction of the Vitruvian Man by Leonardo da Vinci” that the circle goes through the top vertex of the 45º-rotated square, as shown in Figure 3.

Figure 3

Lines drawn by Leonardo da Vinci (red) and a circle going through the top vertex of the 45º-rotated square (blue).

In this case, the radius of the circle should have a ratio (2^{1/2} + 1) / 4 = 0.604, which is certainly closer to the measured value in 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 results of this model show better fit to the observed data than the golden-ratio circle, it is still difficult to justify the assumption.

It is said that the circle described by Vitruvius should satisfy the following requirements,

- the center is placed on the navel,
- the circumference touches both fingertips and feet.

The best location of the navel in a drawing of a human body would be ambiguous, and the circumference that touches fingertips and feet would depend on the angles of the arms and legs from the trunk of the body. The requirements of Vitruvius do not fully determine the location and radius of the circle.

The circle drawn by Leonardo has following additional characters, which were not described by Vitruvius,

- the circumference touches the soles of the man standing upright.
- the circumference touches the fingertips at the height of the top of the head.

The allowable range for the radius of the circle drawn by Leonardo is much more restricted than that of Vitruvius. It is likely that Leonardo understood what Vitruvius assumed, and intended to add other restrictions by himself.

# 4. Lines and points drawn by Leonardo da Vinci

Lines drawn by Leonardo da Vinci can be found in the image of “Vitruvian Man”. It is likely that Leonardo intended to show proportions of a human body by drawing those lines.

A kind of scales is also drawn by Leonardo, below the drawing of a human body.

Figure 4 shows the lines the author found in “Vitruvian Man” as red lines superimposed on the normalized image.

There are many lines found in the image of “Vitruvian Man”, but only two lines are different from other lines. They are shown in Figure 4 as AA’ and BB’.

# 5. Texts written by Leonardo da Vinci

It is said that Italian texts were written by Leonardo da Vinci in a mirror writing manner below the drawing of the Vitruvian Man. They say:

- from above the chest to the top of the head is one-sixth of the height of a man
- from above the chest to the hairline is one-seventh of the height of a man
- the maximum width of the shoulders is a quarter of the height of a man
- rom the breasts to the top of the head is a quarter of the height of a man
- the distance from the elbow to the tip of the hand is a quarter of the height of a man
- the distance from the elbow to the armpit is one-eighth of the height of a man
- the length of the hand is one-tenth of the height of a man
- the root of the penis is at half the height of a man
- from below the foot to below the knee is a quarter of the height of a man
- from below the knee to the root of the penis is a quarter of the height of a man

The distances between line markers on the drawing are in good coincidence with the above description, as shown in Figure 5.

The divisions on the scale bellow the drawing correspond to 1/96 and 1/24 of the height of the man. It is likely that Leonardo used the scale to measure, design, or adjust dimensions of the drawing.

# 6. Four points marked by Leonardo da Vinci

Four edge points of lines AA′ and BB′ are clearly marked by Leonardo, while no mark is found for other edge points of lines. It suggests that he attached a 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 length of the line BB′ marking the “breasts” is close to 1/5 of the height of the man (Figure 5). It is likely that Leonardo assumed the value 1/5 as the width of the breasts, which is equal to the twice the length of a hand (1/10).

The length of the line AA′ is close to 1/7.5 = 2/15. As will be 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 fingertips moved upward.

It may be noted that the locations A, A′, B, and B′ appear to be arranged so that lines AB and A′B′ crosses at the top of the head (T), as shown in Figure 6. It is likely that Leonardo applied this restriction to remove ambiguity and determine the positions of the points A and A′.

The length of the line AA′ should be (1/5) × (1/6) / (1/4) = 2/15, when the points A and A’ are on the sides of triangle TBB’.

## 7. Rotation of arms (1)

As shown in Figure 7, the fingertips C and C’ of horizontally stretched arms are located at the same height as the points A and A’.

The location of a fingertip D at the height of the top of the head and the point C are equally distant from the point A (AC = AD).

The position A may look displaced from the location of the shoulder joint, but it can virtually be the center about rotation of an arm. The shoulder will associatively be moved, when a man moves an arm upward.

Leonardo assumed the points A and A’ to be the centers of the rotation of arms. We can imagine that the axis of a compass tool is fixed to the point A, and the radius of rotation of the tool is adjusted as the pen attached to the tool points C. The point D is unambiguously determined as the crossing point of a trace of rotation and the top line of the square.

## 8. Rotation of arms (2)

It is likely that Leonardo geometrically determined the circle drawn in “Vitruvian Man”, but the radius can also be evaluated by calculation.

The side length of the square drawn by Leonardo is assumed to be 1. The distance between A and C should be

AC = 1 / 2 – (2 / 15) / 2 = 13 / 30,

as can be found in Figure 7, and the distance between A and D should also be AD = 13 / 30, even if Leonardo did not want to show the trace of other circles in “Vitruvian Man” to the viewers.

The distance between points D and T (top of the head) is then given by

DT = 1 / 15 + [(13 / 30)^{2} – (1 / 6)^{2}]^{1/2} = 7 / 15 .

See Figure 8.

## 9. Calculation of the radius of the circle in the Vitruvian man

Let *R* be the radius of the circle Leonardo intended to draw. The following equation should by satisfied,

[R^{2} – (7 / 15)^{2}]^{1/2} + *R* = 1 ,

as shown in Figure 9.

The solution of the above equation is given by

*R* = [1 + (7 / 15)^{2}] / 2 = 137 / 225 = 0.6088 … .

Figure 9

Calculation of the radius of Leonardo da Vinci’s circle (*R* = 137 / 225)

# 10. Conclusion

Figure 10 show the square and circle drawn by Leonardo da Vinci as red lines, and a blue circle with the radius of 137 / 225 (0.609) that touches to the bottom line of the square.

Figure 10

Square and circle in the “Vitruvian Man” (red lines) and circle with the radius of 137/255 (blue line)

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 and TB’ for the top of the head T (Figure 6). The fingertip C initially located at the same height as the point A was rotated about A, and the point D was found as the crossing point of the trace of the rotated fingertip C with the top line of the square (Figure 7). Finally, Leonardo has drawn the circle passing through the point D and touching the baseline of the square (Figure 8).

It is concluded that Leonardo da Vinci drew the circle with the radius of 137/225 of the side length of the square (Figure 9 & 10).

# 11. The text above the drawing of Vitruvian man

The above conclusion was derived before the author of this page know the meanings of the text written above the drawing of Vitruvian man.

Now it is known that the text above the drawing of Vitruvian man says that Vitruvius, an architect, says the following things.

- 4 fingers make a palm
- 4 palms make a foot
- 6 palms make a cubit
- 4 cubits make a man
- 4 cubits make a footstep
- 24 palms make a man
- When the legs are opened as lowering head by 1 / 14 of the height of a man, and the arms are raised as the finger tips touch the line of the top of the head, the end of the open limbs are on a circle with the center located at the navel, and space between the legs makes an equilateral triangle.

It may be difficult to find an equilateral triangle between the legs of the Vitruvian man, but Leonardo understood that the legs are rotated about the hip joints (H and H’). The ends of the legs (F and F’) and hip joiints (H, H’) certainly make an equilateral triangle, as shown in Figure 11.

It seems that the text written above the drawing also supports the assumptions made in this study.

publicated June 18, 2012

updated May 12, 2024