These drawings are based on the correlations of ideal human proportions
with geometry described by the ancient Roman architect Vitruvius in Book III
of his treatise De Architecture. Vitruvius described the human figure as being
the principal source of proportion among the Classical orders of
architecture. His drawing is named in honor of the architect.
A method to construct a golden rectangle. The square is outlined in red. The
resulting dimensions are in the golden ratio
Mathematician Mark Barr proposed using
the first letter in the name of Greek
sculptor Phidias, phi, to symbolize the
golden ratio. Usually, the lowercase
form (φ) is used. Sometimes, the uppercase
form (Φ) is used for the reciprocal of the
golden ratio,
A golden rectangle with longer side a and shorter side b, when placed
adjacent to a square with sides of length a, will produce a similar
golden rectangle with longer side a + b and shorter side a. This
illustrates the relationship. Two quantities a and b are said to be in the
golden ratio φ if:
Two quantities a and b are said to be in the golden ratio φ if:
.
One method for finding the value of φ is to start with the left fraction. Through
simplifying the fraction and substituting in b/a = 1/φ,
,
It is shown that,
Multiplying by φ gives
Which can be rearranged to?
.
Using the quadratic formula gives the only positive solution as,
.
Also referred to as the "Golden section" and the GOLD MEAN the
Golden mean is an ancient fine arts formula that mathematically
defines a rectangle of specific proportions, the golden section is a
line segment divided according to the golden ratio (approximately
1.6180339887): The total length a + b is to the length of the longer
segment a as the length of a is to the length of the shorter segment b
This image below 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. Encyclopedia 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 cosmography 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.
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
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 is Leonardo's drawing:
Fig. 7 Superimposed image of Fig.6 and Leonardo's drawing.