Vitruvian Man by Leonardo da Vinci

January 12, 2011

Post image for Vitruvian Man by Leonardo da Vinci

Vitruvian Man

Introduction

The Vitruvian Man is a world-renowned drawing created by Leonardo da Vinci around the year 1487. It is accompanied by notes based on the work of Vitruvius. The drawing, which is in pen and ink on paper, depicts a nude male figure in two superimposed positions with his arms and legs apart and simultaneously inscribed in a circle and square. The drawing and text are sometimes called the Canon of Proportions or, less often, Proportions of Man. It is stored in the Gallerie dell’Accademia in Venice, Italy, and, like most works on paper, is displayed only occasionally.

Leonardo da Vinci, Vitruvian Man, 1487, 34.4 × 25.5 cm (13.5 × 10.0 in)

A passage from Roman architect Vitruvius (Marcus Vitruvius Pollio), describing the perfect human form in geometrical terms, was the source of inspiration for numerous renaissance artists. Only one of these, the incomparable Leonardo da Vinci, was successful in correctly illustrating the proportions outlined in Vitruvius’ work De Architectura, and the result went on to become the most recognized drawings in the world, and came to represent the standard of human physical beauty. It was the version produced by Leonardo da Vinci, whose vast knowledge of both anatomy and geometry made him uniquely suited to the task.

The drawing is based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De Architectura. Vitruvius described the human figure as being the principal source of proportion among the Classical orders of architecture. Other artists had attempted to depict the concept, with less success. The drawing is traditionally named in honour of the architect.

The Vitruvian Man image exemplifies the blend of art and science during the Renaissance and provides the perfect example of Leonardo’s keen interest in proportion. In addition, this picture represents a cornerstone of Leonardo’s attempts to relate man to nature. Encyclopaedia Britannica online states, “Leonardo envisaged the great picture chart of the human body he had produced through his anatomical drawings and Vitruvian Man as a cosmografia del minor mondo (cosmography of the microcosm). He believed the workings of the human body to be an analogy for the workings of the universe.” It is also believed by some that Leonardo symbolized the material existence by the square and spiritual existence by the circle. [ Source: Wikipedia.org ]

Vitruvius, De Architectura: THE PLANNING OF TEMPLES, Book 3, Chapter I
1. The planning of temples depends upon symmetry: and the method of this architects must diligently apprehend. It arises from proportion (which in Greek is called analogia). Proportion consists in taking a fixed module, in each case, both for the parts of a building and for the whole, by which the method of symmetry is put to practice. For without symmetry and proportion no temple can have a regular plan; that is, it must have an exact proportion worked out after the fashion of the members of a finely-shaped human body.
2. For Nature has so planned the human body that the face from the chin to the top of the forehead and the roots of the hair is a tenth part; also the palm of the hand from the wrist to the top of the middle finger is as much; the head from the chin to the crown, an eighth part; from the top of the breast with the bottom of the neck to the roots of the hair, a sixth part; from the middle of the breast to the crown, a fourth part; a third part of the height of the face is from the bottom of the chin to the bottom of the nostrils; the nose from the bottom of the nostrils to the line between the brows, as much; from that line to the roots of the hair, the forehead is given as the third part. The foot is a sixth of the height of the body; the cubit a quarter, the breast also a quarter. The other limbs also have their own proportionate measurements. And by using these, ancient painters and famous sculptors have attained great and unbounded distinction.
3. In like fashion the members of temples ought to have dimensions of their several parts answering suitably to the general sum of their whole magnitude. Now the navel is naturally the exact centre of the body. For if a man lies on his back with hands and feet outspread, and the centre of a circle is placed on his navel, his figure and toes will be touched by the circumference. Also a square will be found described within the figure, in the same way as a round figure is produced. For if we measure from the sole of the foot to the top of the head, and apply the measure to the outstretched hands, the breadth will be found equal to the height, just like sites which are squared by rule.
4. Therefore if Nature has planned the human body so that the members correspond in their proportions to its complete configuration, the ancients seem to have had reason in determining that in the execution of their works they should observe an exact adjustment of the several members to the general pattern of the plan. Therefore, since in all their works they handed down orders, they did so especially in building temples, the excellences and the faults of which usually endure for ages. [Source: aiwaz.net]

Geometrical construction of the Vitruvian Man by Leonardo da Vinci

It is assumed that proportions of the circle and square reflect Golden Division. Here we present analysis that shows that this assumption is incorrect.

If a circle has radius = 1 unit, square side is equal to:

1.656 for Vitruvian Man
1.618 for Golden section construction
1.571 for the condition: circumference of the circle = perimeter of the square
1.772 for the condition: area of the circle = area of the square

Fig. 1 Comparison of true Golden Rectangle with Vitruvian Man drawing

Fig. 2 Circle and square based on Golden Section

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.


Fig. 2b  Squaring the circle.
Image on the right: Squaring the circle: the areas of this square and this circle are equal.
Image on the left: Circumference of the circle equals the perimeter of the square.

Fig. 2b Left shows a circle with Radius = 1 and a square with side = 1.571.
The Circumference of the Circle = 6.28… [ 2 x Pi = 6.28 ]
The square with side 1.571 has perimeter equal 6.28 [ 4 x 1.571 = 6.28 ].

Fig. 2b Right shows a circle with Radius = 1 and a square with side = 1.772.
The Area of the circle is 3.14 [ as determined by pi multiplied by the radius squared ].
The area of the square is also 3.14…  [1.772 x 1.772 ].

Vitruvian Man – methods of geometrical construction of the circle and the square

The simplest composition is based on a square, which is duplicated and rotated 45º to form an octagram.  The distance between the base line of the first square and the apex of the rotated one simply represents the diameter of the circle.

Fig. 3a The simplest way to describe
the geometrical construction of the Vitruvian Man.

Fig.3b Steps explaining the simplest
geometrical construction of the Vitruvian Man.


Another method of geometrical construction of the Vitruvian Man

Step 1: Draw a square and circle (radius R1) as shown on the Fig. 4

Fig. 4

Step 2: Move circle so point A overlaps with point B (see Fig. 5)

Fig. 5

Step 3: Locate center of the final circle (point O) by Dividing distance AB in a half.
Draw new circle with radius R2=OA (see Fig.6)

Fig. 6

The result matches perfectly Leonardo’s drawing:

Fig. 7 Superimposed image of Fig.6 and Leonardo’s drawing.

Related articles: Fibonacci Numbers in Nature & the Golden Ratio

Article Source: Vitruvian Man

PS Vitruvian-Man omission/oversight

By Anthony G. Cila

3D explanation of Vitruvian Man drawing

I greatly appreciated you article and graphics, well stated.  However, how is it you missed what was actually drawn/alluded to by Da Vinci?
 
While it is fashionable to assume he may have been attempting the “rounding the square”,  of missed relationships to the golden rule… such is a plausible distraction from what is actually depicted… at first “glance” (the error most commonly made… exposes the genius the master has relied upon to conceal a greater truth):
 
1) a well proportioned man (measurements alluded to based on “a foot” and its/his relationship to “a cubit”)
2) inside a square
3) inside a circle
 
…all are depicted as two dimensional figures (but the man is not a two dimensional object, neither should one assume it is a square or circle).
 
…it appears to be drawn “as if to scale” but oddly, not actually drawn to scale
(i.e. the man is 4 cubits tall… by Da Vinci’s own “standard”… a “foot and a half” each cubit is six “feet”
(oddly he didn’t use an inch to represent a foot in keeping with his scale scale, as multiple measurement systems were in play then)
…drawn on parchment approximately 13.5 inches by 10 inches the images are not drawn in inches, instead he free handed them
(perhaps he actually used centimeters/millimeters or fractions of inches?)
(digitally we have “pixel” counts of 137-139 for an imperfect circle and 224-225 of a near perfect square)
…the square and circle do not seem to be related mathematically (leastwise as erroneously expected… knowing full well Pi and Phi are irrational numbers)
(exact matches cannot made using irrational numbers, even if the “square’s sides are a multiple of Pi (or Phi) how could the square be “rounded” unless “rounding” is employed)
 
Thus, the square and circle are, like the 3D man, representing 3D objects (a cube and a sphere… both of which contain the same volume relative to their sizes… that is the comparison Da Vinci is concealing in plain view… along with what had to be rounding the irrational aspects of the equation)… that is how they are related mathematically… in which ever unit of measure one chooses to use…. where you choose to do your rounding directly influences the accuracy of the outcome, but there should be no mistake in the obvious once revealed.
 
Typically Da Vinci seems to be rounding (irrationally) sometimes at the tenth and other times elsewhere…
 
World-Mystery measurements/figures result in the following volume(s):
 
… with a cube: 2 x 2 x 2 = 8
…and an R2 (Fig 6) of 1.2071:  rounded 1.21
…1.21 (to the 3rd power) x Pi (rounded to 3.142) x (4/3) =
…1.77 (rounded 1.8) x Pi (3.142) x (4/3) = 7.54 (who wouldn’t round that up to “8” not only irrational numbers must be rounded to be used/fit)
…1.77 (rounded to “2”) x Pi x (4/3) = 8.34 (who wouldn’t round that down to 8, even compound rounding achieves the desired answer, his answer)
 
If his cube has a SIDE= 6 x 6 x 6 = 216 is the volume
If your R2 theory is correct his sphere Radius= 3.621 (but its off by less than a tenth) because,
His sphere requires a Radius=: 3.722 (to the 3rd power) x Pi (rounded to 3.142) x (4/3) = 216
 
…a 139 pixel circle/sphere radius and a 224 pixel square/cube side work quite well together requiring no rounding whatsoever (and achieve a far less than a % error)… this is no doubt what Da Vinci was actually doing despite the herculean efforts of the various math departments world wide in depth analysis (some measuring inside, on and outside the objects… ignoring his work is more carved onto the paper (delineating actual boundaries) and subsequently traced… its can be viewed at the museum in Venice (for closer inspection/measurement). 
 
Along those lines with his imperfect circle/sphere’s radius measured at 137-139 (in pixels) and his square/cube’s at 225 (pixels) an interesting anomaly presents itself and should be noted:
 
137 + 225 + Pi (rounded 3.142) = 365.142 days in a year (off by less than one tenth(?) just like his sphere and cube volumes). 
 
Clearly there is more going on in his drawing than the combined intellect of the last 600+ years (leastwise until I came along to solve the misinterpretation of the 3D objects depicted in the 2D rendering of the Vitruvian man), enjoy…
 
Its obvious Da Vinci was taking his liberty with rounding… at his convenience (lacking an IBM/TI calculator), wink, and depending on where he rounded he could easily have thought he hit “the golden rule” too.

{ 3 comments… read them below or add one }

Ron Expeth February 1, 2011 at 2:45 pm

The measure of a man is 4 parts vertically and 4 parts horizontally. If you look closely, you will see there are in fact 16 figures superimposed not just the 2 that most people see. If an artist were to draw those 16 figures separately that would be a new an interesting insight into this very special piece.

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Rev. Allorrah Be January 15, 2011 at 10:51 pm

I find this subject fascinating. I would like to be able to email this to everybody on my distribution list… as well as my own In Box. I do hope a copy will appear there, so I may save it in my Folders!

In the Light,
Allorrah I AM

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Ron O. Cook January 12, 2011 at 11:42 am

I am totally taken by this figure as one would know if they saw some of my paintings. I have done my versions of the same and continue to study the symbolic meanings of the human form. I know, it was one of the first logos that I embedded within my mentality. There are times when I feel so close to those great painters of the Renaissance that I lived way back then. There is deep meaning held within those lines.

Ron O.

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