# 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

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

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*

**…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).**

**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.

**If his cube has a SIDE= 6 x 6 x 6 = 216 is the volume**

**sphere Radius= 3.621**(but its off by less than a tenth) because,

**3.722 (to the 3rd power) x Pi (rounded to 3.142) x (4/3) = 216**

**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).

**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).

**solve the misinterpretation of the 3D objects depicted in the 2D rendering of the Vitruvian man**), enjoy…

[…] Note: This is Part 2 of our original post Vitruvian Man by Leonardo da Vinci […]