Proportions in Greek and Renaissance Art

Golden Rectangles

A Golden Rectangle is a rectangle in which the longer side is 1.618 times the shorter side, and the

shorter side is 0.618 times the longer side. Many shapes in nature fill a golden rectangle. A spruce

tree has golden proportions in height and width. The dragonfly s wingspan length to his body length

is a golden proportion.

Spruce tree

Golden Proportions

A proportion is the relation of one part to another. In a golden proportion, one length is 0.618 times

the other length. The exact formula is:

2

1 5

.

It can also be found from the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,

610,…). Dividing Fibonacci numbers (number gets closer and closer to the golden proportion):

1 1 = 1

1 2 = 0.5

2 3 = 0.667

3 5 = 0.6

5 8 = 0.625

8 13 = 0.615

13 21 = 0.619

21 34 = 0.618

Golden Proportions In Greek and Renaissance Art

Since the Ancient Greek times, artists have regarded the golden proportion as one of ideal beauty. It

can be found throughout in paintings (like the Mona Lisa

http://avline.abacusline.co.uk/pictures/jpeg/pics/mona.jpg, sculptures, and architecture (like the

Parthenon ). An easy way to quickly measure golden ratios is to use the Fibonacci numbers in

some measuring unit like centimeters or inches.

The following is quoted and paraphrased from

http://www.goldenmuseum.com/0305GreekArt_engl.html

As the main requirements of beauty Aristotle puts forward an order, proportionality and limitation in

the sizes.

Consequences in Greek Architecture building constructed on the basis of the golden section:

??? The antique Parthenon

??? “Canon” by Policlet, and Afrodita by Praksitle

??? The perfect Greek theatre in Epidavre and

the most ancient theatre of Dionis in

The theatre in Epidavre is constructed by Poliklet to the 40th Olympiad. It was counted on 15

thousand persons. Theatron (the place for the spectators) was divided into two tiers: the first one

had 34 rows of places, the second one 21 (Fibonacci numbers)! The angle between theatron and

scene divides a circumference of the basis of an amphitheater in ratio: 137?°,5 : 222?°,5 = 0.618

(the golden proportion). This ratio is realized practically in all ancient theatres.

Theatre of Dionis in Athens has three tiers. The first tier has 13 sectors, the second one 21

sectors (Fibonacci numbers)!. The ratio of angles dividing a circumference of the basis into two

parts is the same, the golden proportion.

From the Fibonacci series: 5, 8, 13 are values of differences between radiuses of circumferences

lying in the basis of the schedule of construction of the majority of the Greek theatres. The

Fibonacci series served as the scale, in which each number corresponds to integer units of

Greeks foot, but at the same time these values are connected among themselves by unified

mathematical regularity.

At construction of temples a man is considered as a “measure of all things: in temple he should enter

with a “proud raised head “. His growth was divided into 6 units (Greek foots), which were

sidetracked on the ruler, and on it the scale was put, the latter was connected hardly with sequence of

the first six Fibonacci numbers: 1, 2, 3, 5, 8, 13 (their sum is equal to 32=25). By adding or

subtracting of these standard line segments necessary proportions of building reached. A six-fold

increase of all sizes, laying aside of the ruler, saved a harmonic proportion. Pursuant to this scale

also temples, theatres or stadiums are built.

Golden Spirals

Make a small square. Then make a set of squares in which the length of the next size square is 1.618

times the length of the last square. The easiest way is to use the Fibonacci numbers — make one 1

cm on a side, then 2, 3, 5, 8, up to 55 cm on a side.

Draw a quarter circle in each square. Use a compass to make them exact.

Then arrange the squares to form a golden spiral.

There are many examples of golden spirals in nature — shells, horns of mountain sheep, ferns, pine

cones, pussy willows, elephant tusks, some spider webs, sea horse tails, hurricanes, and galaxies to

name a few.

http://www.goldenmuseum.com/0305GreekArt_engl.html

Examples on the application of the Golden Section in Greek Architecture: