Golden section

The golden section (Latin: sectio aurea) or also “divine division” (Latin: proportio divina) consists in dividing a line \overline{AB} into two segments \overline{AM} and \overline{MB}, so that the length of the longer segment \overline{AM} is related to the length of the shorter segment \overline{MB} in the same way as the length of the whole line \overline{AB} is related to the length of the longer segment \overline{AM} (cf. Figure 1):

Figure 1: Golden section as distance ratio

Expressed as an equation, this is called

    \[\lvert\overline{AM}\rvert:\lvert\overline{MB}\rvert=\lvert\overline{AB}\rvert:\lvert\overline{AM}\rvert\quad (\ast).\]

This division of a line in the ratio of the golden section has been known for millennia and is still of particular importance in architecture and art as an aesthetic principle. The first preserved exact description of the Golden Section is by Euclid (ca. 300 BC). The Franciscan friar Luca Pacioli di Borgo San Sepolcro (1445–1514), who taught mathematics in Perugia, Italy, called this division the “Divine Division”. The first known calculation of the Golden Section as an approximate 1.6180340 for the ratio \lvert\overline{AM}\rvert:\lvert\overline{MB}\rvert (see Figure 1) was communicated by the Tübingen professor Michael Maestlin to his former student Johannes Kepler (1571–1630) in 1597.

However, the term Golden Section, which is commonly used today, is first found in a mathematics textbook published in 1835 by Martin Ohm (1792–1872).

The Golden Section plays a significant role in architecture as well as in the fine arts. Thus, many buildings of antiquity (e.g. the front of the Parthenon temple of the Athenian Acropolis, built around 440 BC), but also famous buildings in later centuries (e.g. the gate hall of the monastery of Lorsch (around 770), the cathedral of Florence (1294 start of construction), Notre Dame in Paris (1163–1345) and the Old Town Hall in Leipzig (1556–1557)) follow in their proportions the Golden Section (in the latter case it is the distances of the entrance under the tower to the two sides of the building).

In addition, numerous sculptures by ancient Greek sculptors are considered to be evidence of the use of the Golden Section, as are a number of Renaissance paintings, including those by Raphael, Leonardo da Vinci, and Albrecht Dürer (e.g., his self-portrait of 1500 and the copperplate engraving “Melencolia” of 1514).

The golden ratio also finds admiration again and again because it can be observed in nature, as for example in the five-fold symmetry of bellflowers, in the ivy aralia leaf and the starfish. A number of proportions associated with the golden ratio can also be found on the human body. They were first systematically studied by Adolph Zeisig in the middle of the 19th century. In particular, the fact that on the human body the belly button divides the body size as the golden ratio has become famous (see exhibit in MATHEMATICS ADVENTURE LAND).


And now … the mathematics of it:

Figure: Calculation of the golden section

If we set a\coloneqq\lvert\overline{AM}\rvert and b\coloneqq\lvert\overline{MB}\rvert, then \lvert\overline{AB}\rvert=a+b. So from the above equation (\ast) it follows

    \[a:b=(a+b):a.\]

Now set \Phi\coloneqq a/b, then this gives

    \[\Phi=a/b=(a+b)/a=1+b/a=1+\Phi^{-1}.\]

Multiplying this by \Phi, we get the quadratic equation \Phi^2-\Phi-1=0, which has zeros \Phi_{1,2}=\frac{1\pm\sqrt{5}}{2}. But since \Phi is obviously positive, \Phi=\frac{1+\sqrt{5}}{2} must hold. This value is often called the golden number.


Literature

[1] Pacioli, L.: Divina Proportione, Venedig, 1509.

[2] Beutelspacher, A., und Petri, B.: Der Goldene Schnitt, Heidelberg, Berlin, Oxford, 1996.

[3] Hemenway, P.: Divine Proportion. Phi in Art, Nature and Science, New York, 2005.