Historians are more precise and consider the period from the death of Alexander the Great (323 B.C.) to the establishment of the Roman Empire (31 B.C.) as the Hellenistic Age. We may mark almost the same period by Euclid's
- the Let us see first the Greek concept - - In an earlier paper I demonstrated that the meadival and especially the Renaissance study of the Vitruvian text is the "missing link" between the Greek proportio [6]. Interestingly, Vitruvius' book has a Chinese parallel in the case of some questions of theory of proportions and architecture: the Zhou Li or Chou Li (Record of the Rites of the Zhou/Chou Dynasty, 11th-3rd cc. B.C.). It was compiled probably in the Former Han period (2nd-1st cc. B.C.) and gives the description of the posts and duties of government officials. Indeed, Vitruvius' book and the Chinese work were written in almost the same time and both of them survey practices of earlier ages, including the works of architects and technicians. The human figure as a standard of measure was suggested by both works. However, there is an interesting difference between them: while Vitruvius uses not only arithmetical, but also geometrical proportions, as we can see in the case of the description of "homo ad quadratum" and "home ad circulum" (which later inspired Leonardo's drawing of the "Vitruvian man"), the descriptions in the Chou Li are always arithmetical with ratios of numbers. Another interesting example is the ideal Chinese city that was built on the basis of geometric symmetry principles. The city was shaped in a square, each side measuring nine li, which is an old unit of measurement, about 3.9 km. This square was divided by nine vertical and nine horizontal lines (avenues) into smaller units, while the palace stood at the center. This scheme resembles the layout of the artificially built Roman cities, which were modelled according to the military camp. The new territories were first occupied by the army and later their camp became the basis of the first Roman cities there. Note, however, that those geometric principles that were essential in the Greek description of the universe played much less role in the case of Chinese models, as we discussed earlier. Remaining at buildings and architecture, the original version of the almost 2,400-kilometer (1,500-mile) long Great Wall, the largest artificial object ever made, was built at the Northern border of China Proper in a period that is more or less identical with the Hellenistic Age. The Wall separated the fertile farm-lands from the Northern grasslands (now Inner Mongolia, an autonomous region of modern China), where nomads lived. Although most work was done in the 3rd century B.C., it was continued until the 1st c. B.C. Later it was several times repaired and even extended by about 300 kilometers: what we see now is basically the restoration made during the Ming Dynasty (1368-1644). Now we went too far in space and time: let us return to our actual focus, the Hellenistic Age. We may conclude that the term symmetria played an important role in both Greek mathematics and art.However, a similar statement on the golden section, using this modern term, is very problematic and perhaps incorrect. In the surviving body of ancient Greek texts, there are very few references to this proportion and all of these are in mathematical works. The earliest known examples are in Euclid's book, but these could have had preliminaries (Theaetetus is the most likely candidate, while others point to Eudoxus and even to the Pythagoreans). It is especially striking that the used expression - Geometrical proportions: Artists could have used not only arithmetical ratios of small integers, but also geometrical methods for constructing incommensurable ratios. There are many data about the first possibility, but just a few ones for the latter and this is usually the ratio of a side and a diagonal of a square (1/Ö2 = 0.707...). For example, Vitruvius suggests this, together with 2/3 and 3/5, for the ratio of the breadth to the length of an atrium ( - Rational approximation of the golden number: The ratios 2/3 (= 0.666...) and 3/5 (= 0.6) that we just mentioned at Vitruvius can be interpreted as approximate values for the irrational golden number (0.618...). Note, however, that Vitruvius did not refer to the golden section or a related geometrical construction here, while 1/Ö2 was introduced geometrically. In the case of catapults Vitruvius also used, among others, the ratio 5/8 (= 0.625; Book 10, Chap. 10, Paras. 4-5; Schramm's interpretation of the symbols for fractions). Parallel, there are some related mathematical developments. Thus, we may suspect that the mathematician Hero (Heron) of Alexandria (1st c. A.D.) replaced deliberately the golden number by 5/8 (= 0.625) when he calculated the area of a regular pentagon approximately ( F_{n-1} + F_{n-2}; 13th c.) tend to the golden number. Theon of Smyrna (2nd c. A.D.) described a method for the successive approximation of Ö2 by 3/2, 7/5, 17/12, ... (diagonal numbers divided by side numbers) , which can be easily extended for the case of the golden number although he did not mention it [7]. Of course these works (1st-2nd cc. A.D.) are too late for shaping the artistic traditions of the Hellenistic age, but we may believe that similar methods were in use earlier. Still there is a gap between these mathematical results and the artistic motivations. If we are looking for external inspiration for the use of 2/3, 3/5, and 5/8 in art, we have more reasons to point to the Pythagorean musicological traditions that we discussed in the case of Polykleitos: 2/3 - fifth; 3/5 - major sixth; 5/8 - minor sixth [3].- Pentagram: One may suggest that artists could have been involved in the golden section via the pentagram that was used as a symbol by the Pythagoreans. It is true that this figure "manifests" the golden section, the sides intersect everywhere according to this proportion, but I feel that this argument is not practical. If we should sketch a pentagram, we would not use any mathematics. Even is a very precise drawings is required, we would measure the sides and the angles and hardly will draw it via the golden section. - Human proportions: Although this is a challenging idea, all of the available sources refer to simple ratios of small integers. Again, we need some "bending" of the records to speak about the golden section. In short, there is no convincing evidence for the artistic use of the golden section in the Hellenistic sources. It is true, however, that there was some preference for the ratios of 2/3 and 3/5 that approximates the golden number, without referring to this fact. We may believe that musicological or other unknown ideas were behind, not the golden section. What is the basis of the wide-spread "golden legend" that the Greek artists used the golden section frequently? The very term was introduced in German |