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Vitruvian Man by Leonardo da Vinci

Return to the main article: Fibonacci Numbers in Nature & the Golden Ratio

Geometrical construction of the Vitruvian Man by Leonardo da Vinci

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.


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.

This 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 ]

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

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


Fig. 2 Circle and square based on Golden Section

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

 

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. 3 The simplest way to describe
the 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 Click to enlarge

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



 Fig. 5 Click to enlarge

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 Click to enlarge

 

The result matches perfectly Leonardo's drawing:

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

Return to the main article:
Fibonacci Numbers in Nature & the Golden Ratio


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Leonardo da Vinci, Vitruvian Man, Fibonacci numbers and the Golden Ratio in nature, architecture and art. Includes extensive resources