Question
Asked 4th Mar, 2016

# Is the navel of the Vitruvian Man of Leonardo da Vinci in golden section?

This is widely believed. However, if we draw accurately the golden section the center of the circle is above the navel, and therefore the circle is greater than Leonardo´s drawing.
If the navel is 1/10 above the center of the human figure, then the circle will be smaller .
Leonardo said legs form a triangle, and feet up 1/14, but does not say how the feet are separated.
If this separation is the golden section, the circles match.
Therefore, the navel is not golden section but is involved in position.
Regards

## Most recent answer

Very interesting the contribution of Ugur Gulcugil. Indeed, the arithmetic solution 17/28 justifies both the geometric solution of Vesica Pisces and the Royal Cubit of 7 palms. However, none justifies the position of the navel using the only scale in the drawing (the cubit of 6 palms) without making calculations, which I think is what it is about.
Regards
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## All Answers (57)

No, he does not seem to obey the golden ratio rule, but he looks better this way, doesn't he? Also look at Michelangelo's David or Venus di Milo: not obeying the golden ratio, but excellent sculptures.
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Dear colleagues,
I agree with both, even in that "all that is beautiful in humans and nature have inherent proportions closely related to the Golden Section". I must add that all that is beautiful in humans and nature have inherent proportions closely related to numerous proportions and mathematical constants. Some are more obvious than the Golden Section. For example, binary divisions by symmetry axes.
Of course, the position of navel in Leonardo drawing is anecdotal regarding the anthropometric system of the "Ad Quadratum" Man. It's a great solution of the dilemma of the "Ad Circulum" Vitruvian Man. Others tried it with less success (beauty).
This navel is also an icon of the myth of the divine proportion in architecture and classical art. This ratio is very occasionally used in the architecture of the past, contrary to what most people think.
Regards
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Lesley Karlene Chandler
Macquarie University
Art historian Martin Kemp discusses this exact point of the schema of Vitruvius (followed by Da Vinci) in his latest book, Structural Intuitions: Seeing Shapes in Art and Science. Virginia Press, London. 2016.
Regards
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Dear Lesley,
Thank you for this reference. Although I see that the author does not mention the hypothesis of the golden ratio.
It is true that Leonardo, without saying so, used the golden ratio to place the navel. However he also used other proportions, so the navel is not golden section of height.
This theory spread Le Corbusier in his Modulor, and has been repeated by several authors. However, it is not correct.
The widespread use of the golden section in architecture and classic art is a myth.
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Lesley Karlene Chandler
Macquarie University
Dear Francisco,
My thinking is that the work of Martin Kemp supports your hypothesis - not contests it and I am in agreement with that position in relation to the use (or lack thereof) of the golden mean as arguable measurement tool in Vitruvian Man.
Regards
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Chris Impens
Ghent University
In da Vinci's drawing the navel is at exactly 3/5 of the body height. This can be verified by transferring the length of the hand (1/10) above the center of the square. Irrational proportions in the drawing would contradict the very ideas of Vitruvius that da Vinci wants to illustrate.
Dear Chris Impens,
If the navel is 3/5 from the body height, the circle comes out smaller than the one drawn by da Vinci. It would be the second case drawn (0.6).
I agree that the irrational proportions are contradictory to the Vitruvian treatise, yet they are not contradictory with da Vinci's legacy. They are also consistent with the system of anthropometric proportions in architecture.
Thanks for the link. I attached you another.
...and check your big points with GeoGebra. I use Autocad and with 3/5 I get a smaller circle than da Vinci.
Chris Impens
Ghent University
If we agree that the hand is 1/10 (as stated by da Vinci), then the navel is at height 1/2+1/10=6/10, as my overlay in white lines shows. I don't understand what you mean by "da Vinci's legacy", and I'm not aware of him using or mentioning the golden section anywhere. (Apart from illustrating Pacioli, but this is entirely mathematical.)
Chris Impens's white lines are not 1/10. If the center of the circle was at 6/10 its diameter would be 12/10, and the circle drawn by da Vinci is somewhat larger. Can anyone check this evidence?
Thank you
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Chris Impens
Ghent University
Here is the reasoning: AB=1/10 (length of hand, according to da Vinci, copying Vitruvius), hence BC=1/10 (equal sides of a square), hence NM=1/10 (equal opposite sides of parallelogram).
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Chris Impens
Ghent University
Or, cut-and-paste without any geometry: I rotated and translated the left hand (1/10), so that it points from the groin to the navel.
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I insist. If the center of the circle (the navel) is exactly 6/10 then it does not match the one drawn by da Vinci. In addition da Vinci says and draws that the feet rise 1/14, not that the center is 6/10.
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Chris Impens
Ghent University
There may be some distortion in the image; I took every precaution to eliminate such artefacts. Moreover: where is "my" geometry wrong?
Chris Impens
Ghent University
By the way: da Vinci's drawing is geometrically impossible, as I explain elsewhere, and he's cheating ever so slightly. So perhaps we are both right.
I also thought it was 6/10, and I also thought there could be distortion because the circle did not exactly match (as it should happen to you). Later I checked that the feet rise 1/14, and then the side of the triangle is the golden section. Exactly, or at least with much more approximation than the previous hypothesis of the golden section and 6/10.
As to the drawing is geometrically impossible, I would like to know the original source of da Vinci. Where did he explain it?
I raise another question: how to measure non-duodecimal values (1/10, 1/7, 1/14) with the anthropometric scale drawn by da Vinci?
Regards
Chris Impens
Ghent University
(1) I did my homework again, with greater precision. The circle with radius 3/5+1/160 fits to perfection, and is also compatible with Da Vinci's recipe to raise the arm until it meets the upper side of the square, which gives an irrational radius indistinguishable from the rational one. The tiny correction 1/160 can be easily obtained from the Homo ad Quadratum, since the distance from the hairline to the top is 1/40.
(2) It's very easy to verify "de visu" that the radius is NOT given by the golden ratio. Finding the real value is more difficult.
(3) For 1/10, 1/7, 1/14 and the like, for the geometrical impossibility and for more details to the above: http://ci47.blogspot.be/2017/05/da-vinci-homo-ad-quadratum.html and http://ci47.blogspot.be/2017/05/da-vinci-homo-ad-circulum.html
Dear Chris Impens,
I am very grateful to you for the replica and the new calculations. I also reviewed that image with AutoCad.
Conclusions on the question would be:
1.- The radius of the circle is neither in gold ratio (HIP.1 = 0.618) nor in 6/10 (HIP.2 = 0.6)
2.- Its value approaches 1/14 plus the height of a triangle whit side in gold ratio
HIP.3 = 1/14 + ((√5-1) / 2 · (√3 / 2)) = 0.6067
3.- It is great that this value coincides in a 99.92% with that obtained by raising the arms
HIP.4 = 0.6062
4.- To give a more accurate value an orthophoto is necessary. In my detail it is closer to 0.61
5.- The approximation 6/10 +1/160 = 97/160 = 0.60625 is very good, but it is decimal. It can not be measured with an anthropometric duodecimal scale. Another simpler fraction is 17/28 = 0.6071, but neither is duodecimal, but in base 7. Geometric procedures are necessary for its determination.
6.- According to my observations, in order to quantify all these measures in  Architecture by means of integers and low numbers, another scale in proportion √2 was used. Sides and diagonals of square allow approximations to many remarkable values. Among them:
(1 + √2) / 24 = 1/10 (error 0.59%)
(2 + √2 / 24 = 1/7 (error -0.42%)
7.- For this reason, I believe that da Vinci may have also used this resource to locate the navel of the Vitruvian man: The silver ratio.
HIP.5 = (1 + √2) / 4 = 0.6036
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Chris Impens
Ghent University
As to the silver navel: see newly added P.S.3 in http://ci47.blogspot.be/2017/05/da-vinci-homo-ad-circulum.html
Perhaps da Vinci used a sophisticated geometric process for the navel, with another Sacred Geometry that Vitruvius does not mention. The Homo bene figuratus of the Codex Huygens is considered preparatory work (so I think). Several regular polygons appear (3-4-5-6-8).
Regards
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For greater precision an orthoimaging with graphic scale would be necessary.
However, as an approximation to 1/14 plus the golden triangle, I find the hypothesis of the silver navel more beautiful than 97/160.
Leno Mascia
Loughborough University
From a ‘modern’ mathematics perspective Leonardo was seeking intuitively a practical solution to the problem of estimating the unknowns for the two simultaneous equations, respectively the position of the navel (radius of circle, R) and the altitude of the collarbone (k), to obtain the interception with the circle for the rotational movement of the arms centered at the collarbone. Using the general equation of the circle (x- h)2 + (y- k)2 = r2, where the co-ordinates of the two parameters are respectively, h = 0, r = R and k = R for the large circle centered at the navel, and h = 0 and r = H/2 (where H= height = width) for the circle describing the rotation of the arms. Setting x2 as the equal term one obtains the equation 2yk – 2yR – k2 + H2/4 = 0. Substituting the two empirically derived parameters, respectively H = 24 palms and k = 20 palms (i.e. 5H/6 - Leonardo's empirical parameter), the solution produces an H/R ratio equal to 1.637, which is very close to the directly measured values (range 1.64 - 1.65) while the small discrepancy can be attributed to a likely approximation inherent to the H/6 parameter.
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Thanks Leno Mascia for this new contribution!
We are analyzing it.
Great, we have another hypothesis 6 about the navel thanks to Leno Mascia. In this case the arms rotate on a only one center at the height of the collarbone (5/6 H), like the legs with the navel. The value obtained for R/H is 0.6111 and can be expressed as 11/18, that is, 44 exact inches.
The solution is convincing, but questions remain unanswered:
The result would be rational, but the solution would also be geometric. It seems contrary to the total commensuration represented by Vitruvian Man.
The small discrepancies are not due to approximations, since the geometric solution is somewhat larger than the circle drawn by da Vinci, and its center does not coincide with the navel.
Although Vitruvio mentions the inches for engineering works, da Vinci neither mentions them nor draws them.
Why does the man lift his legs 1/14? What da Vinci says is not justified.
The image presents certain deformation, surely and partly by photographic aberration. For more precision an orthoimage is necessary (hopefully with metric reference).
Until then I still prefer hypothesis 5 of the Silver Navel. I feel it more beautiful.
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Dr. Masayuki Ohtani´s Answer
Attached new hypothesis. I am processing it...
Comments are welcome.
The radius is 121/200. Grids is punctuated in the ratio of 2:3:5. It's the principle for symmetry. And Maybe Da Vinci wrote the answer. Solution is:
Leno Mascia
Loughborough University
The result corresponds to H/R = 1.652, which is very close to the measured values, varying from 1.64 to 1.65 and therefore considerably higher than 1.618 ( the Golden Section ). But I cannot see how the factor 8/10 was derived.
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Masayuki Ohtani
Kanazawa College of Art
Hi, Francisco
This is how to measure non-duodecimal values
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Masayuki Ohtani
Kanazawa College of Art
A thinking way to solve the math riddle raised by Vitruvius more than 2000 years ago
The last paragraph of the Chapter 1 of the Vitruvius’ Book III is:
“Therefore, if it is agreed that number was found out from the human fingers, and that there is a symmetrical correspondence between the members separately and the entire form of the body, in accordance with a certain part selected as standard, we can have nothing but respect for those who, in constructing temples of the immortal gods, have so arranged the members of the works that both the separate parts and the whole design may harmonize in their proportions and symmetry.” ( http://www.gutenberg.org/files/20239/20239-h/20239-h.htm )
“Symmetrical correspondence” can be understood as something like self-similarity between parts and whole in fractal geometry. Of course, the problem is not shape but proportion. And note that the word of “fingers” in this paragraph is the translation of the Latin word “articulis” that has also the meaning of “joints”. If we focus on the positions of joints to select a certain part as standard, we can grasp the scaling principle.
According to Vitruvius, since the open hand from the wrist joint to the tip of the middle finger is 1/10 and the forearm is 1/4, the length to the wrist joint from the elbow joint is 1/4-1/10=3/20 and the length from the elbow joint to the spine is 1/2-1/4=1/4. Therefore, the length from the tip of the middle finger to the spine is divided into three segments in the ratio of 2:3:5 at the points of the wrist joint and the elbow joint.
Then, let’s apply the ratio of 2:3:5 to the whole height of body so that the upper limb and whole body have a symmetrical correspondence i.e., same proportion. Division points of the height correspond to the intersection of outstretched arms and spine and the position of end of the spine (the root of penis). If the base of the square is at y=0 and the height=1, the division points are at y=8/10 and y=5/10, respectively.
Then, if the foot is 1/7 and the ratio of 2:3:5 is applied to the lower limb, the length from the knee joint to the position of spine (navel) is (1/7)x(5/2)=5/14 and the length from the ankle joint to the knee joint is (1/7)x(3/2)=3/14. Therefore, the length from the ankle joint to the navel is 5/14+3/14=4/7. The length from the sole of the foot to the ankle joint is obtained as the difference from the radius of 121/200 to be 47/1400. In this case, the length from the sole to the knee joint is 3/14+47/1400=1/4-3/1400 or 121/200-5/14=1/4-3/1400, almost equal to 1/4.
As already answered, the radius must be 121/200, if human body inscribed in a circle and a square has a symmetrical correspondence between parts and whole by the scaling principle of 2:3:5.
Celebrate the 500th anniversary of the death of Leonardo Da Vinci.
Leno Mascia
Loughborough University
Hi Masayuki,
Your notes make very interesting reading:
My views are that in studies of the proportions of the human body, Leonardo identifies the pubic bone and the mid-point of the collarbone as two reference anthropometric parameters replicated in his drawing “The Vitruvian Man”. However, although these two points do correspond to the two extremes of the 2:3:5 proportions derived from Vitruvius canon of human proportions, taking the base as the reference points, there is no marking in Leonardo’s drawing in correspondence with the middle number of the series, i.e. 3/10 or 7/10. Therefore, there are some doubts as to whether Leonardo has considered Vitruvius 2:3:5 ratio. My interpretation is that Leonardo derived the anthropometric parameters for the drawing directly from the notes on his “experimental” studies and has diverted the attention from Vitruvius canon of human proportions towards geometry based illustrations, such as the introduction of the equilateral triangle for the opening of the legs and “asymmetry management and rectification” of the whole.
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Masayuki Ohtani
Kanazawa College of Art
Hi, Leno
The Real Da Vinci Code: The reason why I guess that Leonardo had considered the ratio of 2:3:5
If we see experimental works for human body inscribed in a circle and a square in the Codex Huygens, we may be persuaded that Leonardo’s quest could not bear fruit without meditation and perspiration prior to inspiration. If we see the other Vitruvian Man drawn by Giacomo Andrea de Ferrara, we would guess that the Giacomo’s drawing decisively influenced Leonardo’s thinking way. And Leonardo should end up noticing that the original Vitruvius’ description on the foot is wrong or trick. Because, if the foot is 1/6, legs protrude beyond the circle.
“This calls for wisdom. Let the person who has insight calculate the number of the beast, for it is the number of a man. That number is 666.” (Revelation 13:18)
5/12+1/4=?
Leonardo noticed the difference between beast and normal human. In order to imply this, it seems likely that the three lines of the first paragraph of upper text part are written outside the circle and the scale is drawn and arranged outside the square, as shown in the attached Fig. The content of the first paragraph and the scale are based on the Vitruvius’ original description. Three lines suggests that he considered three segments to solve the math riddle.
On the other hand, the first line of second paragraph of upper text part is written to touch the circle with very small ca. 140 Latin letters and spaces per line, and the second and the third lines are written across the circle. In other words, these three lines are arranged to fit within the circle in a geometrical sense. Leonardo probably appeals that legs fit within the circle by his drawing way.
By the way, since the length from the sole to the navel is the same whether the legs are closed or opened, it is not always necessary to make an equilateral triangle. Leonardo would draw the open legs to show that the de-centering of the circle and the square is reasonable. But Leonardo dared to write “1/14”. Therefore, the repetition of number 14 is important, as professor Langdon says “repeating symbol is the simplest way to strengthen its meaning” in the Dan Brown’s novel The Da Vinci Code. It is remarkable that Leonardo has changed the proportion of the foot to be 1/7, instead of 1/6 in the Vitruvius' treatise. Note that 1/7 is 2/14 and about 0.14.
“Out of the temple came the seven angels with the seven plagues. “ (Revelation 15:6)
Since Leonardo has considered the ratio of 2:3:5, eventually he would adopt 1/7 as the proportion of the foot, instead of 1/6, in order to "have so arranged the members of the works that both the separate parts and the whole design may harmonize in their proportions and symmetry” within the circle and the square. Furthermore, Leonardo has specified the proportions of body parts that had not been mentioned in the Vitruvius' treatise. Except for the divisions of face, all the values enumerated by Leonardo can be calculated, based on the arrangements in the ratio of 2:3:5. It’s interesting that Leonardo dared to specify that the proportion of the length from above the chest to the hairline is 1/7.
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Fernando Blasco
Universidad Politécnica de Madrid
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Leno Mascia
Loughborough University
Thank you Fernando! The article is most informative. It also covers some of the points that I have made in a submitted manuscript, under the title "Observations on the geometry behind the human proportions of the Vitruvian Man by Leonardo da Vinci". I will make it available to followers in due course if it fails to go through.
Dear colleagues,
Thanks to the latest contributions we know that the question is not new and that it has already been answered: The navel is not in divine proportion.
So, where is it?
It has also been said that it is 6/10, 97/160, 11/18, 121/200, 137/225, ... each time more complicated (more ugly) fractions.
If we only use the duodecimal scale we can not use decimal or sexagesimal fractions without geometrical procedures (diagonal or Thales). The same goes for the seventh, eleventh, ...
For now it only turns 11/18, although the navel does not match.
The navel is close to 0.605. In any case, the image presents distortions / deformations in the circle and in the scale. Is this the original?
Also left unanswered what Leonardo wrote about the legs raised a fourteenth in a triangle...
I advance another rational value that I will defend soon: 29/48
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Masayuki Ohtani
Kanazawa College of Art
Appendix
The y coordinates of the lines at above the chest and breasts can be determined, based on the Vitruvius’ description. And both ends of these line segments are positioned as shown in attached Fig.
We will notice that the position of vertical line at left armpit is different from the right.
Masayuki Ohtani
Kanazawa College of Art
Hi, Francisco
If the radius is 121/200 and the height is lowered by 1/14, the space between the legs is exactly an isosceles triangle with the legs of 0.515 and the base of 0.570. In order to draw an equilateral triangle accurately in the space between the legs, the height must be lowered by 10000/176115.
Therefore, as already answered, I think that number 14 is meaningful as a code to imply the difference of leg between beast and normal human. If the height is lowered by 1/14 (larger than 1/16), somehow the ankle of the beast will fit within the circle (∵2/3-1/14≈0.595<0.605). But, in the case of a normal human, the space between the legs must be a triangle clearly.
Leonardo drew one foot from the side to emphasize the size of the foot, because he adopted 1/7 instead of 1/6.
Masayuki Ohtani
Kanazawa College of Art
But.... After obtaining the intersection of the circle x^2+(y-r)^2=r^2 and the line y=1/14, if applying the Euclid's Proposition 1 to the line segment from the intersection to the center of the square, an equilateral triangle appears. Although the base is not horizontal, the legs of Vitruvian Man are drawn along the two legs of the triangle.
Leonardo does not say that the triangle is inscribed in the circle, but points to its center (the navel).
Its value approaches 1/14 plus the height of a triangle whit side in gold ratio
HIP.3 = 1/14 + ((√5-1) / 2 · (√3 / 2)) = 0.6067
The question is how to express this value through duodecimal fractions, without geometric procedures.
Leno Mascia
Loughborough University
Leonardo’s magic triangle:
Usually it is assumed that he refers to the space between legs, with sides formed by the inside delineation and the vertex at the pubic bone. But it is also true that an equilateral triangle can be obtained by taking the vertex at the navel and the sides tangentially to the outer surface of the muscles of the legs. In this latter case the height of the equilateral triangle is equal to the base of the equilateral triangle for the inner side of the legs. The resulting isosceles triangle has the upper vertex at the navel and the base as the inner distance between the heel (on the right) and the toe on the left), which is the constitutive triangle for a heptagon inscribed in the circle, forming the two lateral upper vertices at -approximately- the points of contact of the fingers in the raised arms position (see Mascia’s article in Nexus Network Journal -2015, DOI 10.1OO7/s0004-015-0258-4). Although we find no reference to such a heptagonal inscription in Leonardo’s annotations (not enough space or too complicated) it is intriguing that he - apparently – used this procedure (a few years later during his association with Luca Piacioli) for the construction of the heptagon. He was accused of “lack of originality” (plagiarism?) by modern mathematicians (The Mathematics of Great Amateurs by J.L. Cullidge). Personally, - I am sure- Leonardo did not know that a certain Arab Mathematician (Abul Wafa) had already used the same procedure. It was bad enough having to cope with lessons on Euclid from Maestro Luca!
Masayuki Ohtani
Kanazawa College of Art
Leonardo may have drawn two equilateral triangles. If he could do extraction of square root, he would be able to calculate the coordinate to draw an equilateral triangle accurately.
The distance between left corners is 0.0327 as the length of line and 0.0333 as the length of arc. If the radius is 121/200, the length from the sole to the ankle joint is 47/1400=0.0336=ca.1/28=(1/14)/2.
Masayuki Ohtani
Kanazawa College of Art
By the way, instead of the ratio of 2:3:5, if using the ratio of 1/Φ:1:Φ, the radius is 3Φ/8=0.60676. I made a short movie using a stick figure as an art work last year and posted at https://youtu.be/599igc5iyyQ
As far as I read the Vitruvius treatise, he did not write the direct information about y coordinate of the upper limb. Without finding the principle at first, y coordinate of the upper limb can't be determined. Therefore, the radius also can't be determined with solid basis. Since Leonardo found the principle, he could determine the position of the knee and the root of penis with solid basis.
Since Vitruvius could not do extraction of square root, I am using 2:3:5 this year.
There can be many interpretations and triangles, but the only equilateral triangle at 1/14 of height that points to the navel has Φ as its side.
This triangle is not inscribed in the circle because it is prior to the determination of the center. To his height it is necessary to add 1/14.
Then for its mathematical calculation it would be necessary to operate with several irrational numbers:
HIP.3 = 1/14 + ((√5-1) / 2 · (√3 / 2)) = 0.6067
If the design is geometric (with ruler and compass) then we must give the numerical results approximate to integer dimensions.
If we divide the height into 100 (decimal system) none of these values ​​will be integer, and we will need several decimals to approximate.
If we divide the height into 96 (duodecimal system) we will have worse approximations and fractions with high numbers, which complicates any calculation.
If we divide into 60 (sexagesimal system) we will improve the approximations, but the calculations will be even more complicated.
Leonardo's ruler is only divided into 96.
So, what system justifies these values ​​with good approximations and using simple (and beautiful) numerical relationships?
Leno Mascia
Loughborough University
Going back to my line of reasoning, that evolves around the heptagonal inscription of the man in the circle, it should be noted that the sagitta (S) of the arc formed by the base of the equilateral triangle with its upper vertex at the root of the penis (centre of the square) is equal to ~ 0.101 R (calculated using the equation S = R (1- cos 26°), where R is the radius of the arc centred at the navel and the 26° value corresponds to half the vertex angle of the isosceles triangle at the navel. Since the measured height divided by the radius of the circle is ~1.64 the calculated S/H ratio is approximately equal to ~ 1/16. In Leonardo’s drawing the S/H ratio varies respectively from ~1/14 at the lower vertex of the triangle for the contact of right foot to a value varying between 1/16 and 1/18 for the left foot.
My guess is that Leonardo choses the 1/14 value because it is the closest corresponding simple fraction when using the foot as the measuring scale for the height (i.e. H = 7 ft) and because it would not fit into conventional mathematical systems (decimal, duodecimal or sexagesimal).
My view is that Leonardo did not use the factor H/14 as design parameter, but the value is consequential to the intended inclusion of the equilateral triangle for the space between the legs. A similar argument can be made for the heptagonal inscription: i.e. a mere consequence of the accidental juxtaposition of both the equilateral triangle and the navel as the centre of the circle.
It seems that Agrippa identified the heptagonal inscription at ~ 1520 (?), but did Leonardo think about this when he was producing the drawing?
Masayuki Ohtani
Kanazawa College of Art
If you attempt the direct attack by yourself to solve the Vitruvius' math riddle rather than superimposing polygons or rulers on Leonardo’s drawing, you will be able to infer Leonardo's intention that he drew to emphasize leg part. I think that Vitruvius devoted a very excellent screening test to Augustus, because it involves the process: (i) induction to find the principle, (ii) deductions based on the principle and (iii) check and correction based on the principle.
Book III starts from the description on breasts:
“Apollo at Delphi, through the oracular utterance of his priestess, pronounced Socrates the wisest of men. Of him it is related that he said with sagacity and great learning that the human breast should have been furnished with open windows, so that men might not keep their feelings concealed, but have them open to the view. Oh that nature, following his idea, had constructed them thus unfolded and obvious to the view!”
Leonardo wrote his thinking way:
“In every true science experience has penetrated through the senses and imposed silence on the tongues of the disputants. Its disciples pasture not on dreams but always on basic and known principles. They move forward step by step, making valid deductions all the way, as is evidenced in the elementary mathematical sciences: numbers and measures, called arithmetic and geometry.” (Lu.33)
Very interesting those words of Leonardo. The numbers are arithmetic and the measures are geometry. So, arithmetic would not be sufficient to measure, geometry would also be necessary. But does he mean any geometry, or only one?
The anthropometric measures are of duodecimal base (halves and thirds). Vitruvius introduces 1/10, so arithmetically it would mean using the sexagesinal base (ratio 2-3-5). In addition Leonardo adds the 1/7 and 1/14, with which there is no viable arithmetic solution.
Also enter the triangle (assumed to be equilateral) and the circle, that is, several geometric proportions.
However Vitruvius does not mention geometric proportion, except the diagonal of the square. The diagonal does not appear in the Vitruvian Homo of Leonardo, but in the Homo Vitruviano of many other authors.
Masayuki Ohtani
Kanazawa College of Art
The length from the root of the penis to above the chest is 1/3. The length from the top of the head to the intersection of the outstretched arms and the spine is 1/5. 1/9 can be found as the distance between nipples.
Today we define all measurements by means of a quantity (a number) and a reference, but Leonardo says that the measure is also geometry. However, he expresses them by fractions of the height of man that do not belong to a known arithmetic base (prime numbers 2,3,5 and 7).
What geometry allows to divide the unit like this so that it can be measured with whole and low numbers without making big mistakes?
Dear colleagues, thanks to your contributions we know that the navel is not in golden proportion but is close.
In my opinion it was located graphically with an equilateral golden triangle at a height of 1/14 >HIP.3 = 1/14 + ((√5-1) / 2 · (√3 / 2)) = 0.6067
Any of the real numbers that have been proposed (and futures) can be expressed in small fractions of some arithmetic basis, always approximately as we do with decimals. They will be high values but that anyone understands, the Vitruvius´s Ordinatio.
For that reason I believe that the navel approached some remarkable value in Architecture, specifically to the Number of Silver, with which very good harmonic approximations to 1/5, to 1/7, and practically to any real value are obtained.
2 Recommendations
Aineias Oikonomou
National Technical University of Athens
Dear colleagues, please refer to:
for a thorough analysis.
Takashi Ida concludes in 2012, that the ratio is definitely 0.609.
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Thanks Aineias Oikonomou. I will review it.
Regards
Yes, it could be 137/225, or it could be any of the other proposed values, whose average is 0.607.
The hypothesis closest to this average is one fourteenth plus the height of a golden triangle, as Leonardo himself states (except that the triangle is golden).
To calculate with a close arithmetic fraction, we could take some of the values already mentioned (11/18; 29/48; 97/160; 121/200 or 137/225), or some new ones such as 39/64 or 17/28.
However I still think that the resource of measuring with sides and diagonals of square was used, as I have seen in Anthropometric-Architecture. My favorite is the Silver Navel, but if you need more accuracy I also propose (√2 + 18)/32.
New proposals! @Vitor Murtinho exposes hypotheses already seen and new ones. He proposes the Vesica-Piscis. What do you think?
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The design in Vesica Piscis by @Vitor Murtinho is great! The inverse geometric layout (starting from the square) is simple and revealing. Look at the orientation of the arms and the location of the nipples. The value obtained is 0.6076, very close to our average.
Considering this coherent geometric design, it is necessary to explain how Leonardo quantified / measured this value in the drawing.
I think he could use fraction 17/28. This would explain why he uses 1/7 in the foot, and 1/14 to raise the feet. The navel would be located 3/28 above the center. It would also explain the small discrepancy in the diameter of the circumference detected by Murtinho.
Although it would remain the doubt of how it measured in base 7 with a duodecimal scale, without using geometric resources like Thales.
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Masayuki Ohtani
Kanazawa College of Art
Vitruvius wrote " It is true that posterity, having made progress in refinement and delicacy of feeling, and finding pleasure in more slender proportions, has established seven diameters of the thickness as the height of the Doric column" in Book IV Ch.1.
This means that the foot size should be improved to be 1/7, though Vitruvius wrote it was 1/6 in Book III.
We must think about the Vitruvius' intention. The problem is not the measurement but how to determine the position of navel from the description of human body in Book III.
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Look, read and calculate.
1. The geometry of the circumference of Leonardo's Vitruvian man is the Vesica Piscis indicated by Murtinho, with radius 0.6076.
2. If we divide the unit arithmetically into 28 then:
The legs are raised 2 (1/14 as Leonardo writes).
The navel is 17 high (0.6071).
The equilateral triangle is 15 high and Phi of side.
The base of the triangle cuts the circumference with a chord of 16.
3. If we use sides and diagonals of square all the measurements can be determined by the duodecimal scale drawn by Leonardo.
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Ugur Gulcugil
Erasmus University Rotterdam
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Very interesting the contribution of Ugur Gulcugil. Indeed, the arithmetic solution 17/28 justifies both the geometric solution of Vesica Pisces and the Royal Cubit of 7 palms. However, none justifies the position of the navel using the only scale in the drawing (the cubit of 6 palms) without making calculations, which I think is what it is about.
Regards
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