- the Greek symmetria, common measure (which originally refers to commensurability of line-segments, not to mirror symmetry!) and

- the sectio aurea (golden section), a proportion associated with the human body,
a/b = b/(a+b); its numerical value, the golden number, is (-1 + Ö5)/2 = 0.618....

Let us see first the Greek concept symmetria. With great probability this term was introduced by the Pythagoreans during the study of commensurable versus incommensurable lengths. Then, the expression symmetria was adopted by the sculptor Polykleitos and later used by Plato and Aristotle in both senses: commensurability in geometry and good proportion in the visual arts. Since Aristotle considered symmetria (proportion) as one of the three main species of beauty, together with order (taxis) and limitation (horismenon), this concept became prominent in aesthetics (Metaphysica, 1078 a 35 - b 1). There were some interesting debates that helped to refine the aesthetical aspects of symmetria by extending its measurable elements with subjective ones. Then, at the end of the period, Vitruvius used the Latinized symmetria and its derivatives 85 times in his book on architecture (here we consider the modern reconstruction of the text on the basis of the surviving medieval manuscripts). Vitruvius made a slight distinction between symmetria and proportio, the theoretical and the practical aspects of the same question. Parallel, Euclid and the mathematicians also dealt with symmetria. Their focus was the symmetria versus asymmetria dichotomy in the sense of commensurability versus incommensurability. They extended the topic of "linear symmetry" (commensurability in length) with "dynamic symmetry" (commensurability in squares). The study of regular and later semi-regular polyhedra had a climax in the works of Euclid and Archimedes although these figures were not linked to the Greek symmetria. On the other hand, both topics, commensurability in squares and regular polyhedra, contributed to the modern geometrical concept of symmetry, where a figure is analyzed in terms of its equivalent parts (from the 17th c.). At the end of the Hellenistic Age the term symmetria almost disappeared from the language of mathematics and aesthetics. The original meanings of the Greek symmetria were usually described not by the Latinized term, but by other expressions, for example by

- commensuratio (mathematics and philosophy)

- proportio (art and aesthetics).

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 symmetria and the modern symmetry. From the mid 15th century the translators of Vitruvius "reintroduced" the term symmetry into the Western culture: they had not too much choice since the Roman author forced them to distinguish symmetria and 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 akros kai mesos logos (the extreme and mean ratio) or, in a longer version, eis akron kai meson logon temnein (to make a section, or a cut, in the extreme and mean ratio) is not a technical term, but a periphrasis. Greek mathematicians coined a large number of mathematical terms, but, seemingly, they had no such interest here. This statement is valid not only for Euclid, but also for the later authors who commented his work (we shall see the case of Proclus, 5th c. A.D., in the followings). Let us turn to art and the humanities. The expression akros kai masos logos or an alternative term for the golden section is not available in the rich philosophical-aesthetical literature. Note that some modern authors confuse the golden section with the geometric mean: a/b = b/c. Obviously, the latter is more general and does not specify the golden section without a further requirement (c = a + b). The frequently mentioned statement that Plato had a special interest in the golden section is not supported by any of his surviving works. The roots of this claim is due to the mathematician Proclus who lived about 700 years later and remarked, in his commentary to Euclid's book, that Plato originated some propositions concerning the "section" (tomê). However, the meaning of this term is not clear: it could be a reference to the Euclidean "section" according to the extreme and mean ratio (golden section), but many historians of mathematics have different views (section of solids, section of lines, etc.). Note that in the same book Proclus uses the expression extreme and mean ratio: he did not propose a new terminology for this concept. Of course, the lack of records does not prove that this proportion was not used at all, but it illustrates that the golden section, with great probability, had no special importance in art. Moreover, if we consider the artists' preference for symmetria (good proportion, commensurability), we may say that the golden section is excluded. Indeed, the latter is an incommensurable proportion (the golden number is irrational) and cannot be linked to symmetria (commensurability). I immediately admit that this argument is too mathematical: although the artists' concept of symmetria is rooted in the mathematical one (commensurability), they could have relaxed its original meaning by considering any kind of visually "good proportions", including icommensurable ones. Let us see this possibility with various suggested backgrounds:

- 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 (De architectura, Book 6, Chap. 3, Para. 3).

- 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 (Metrica, Book 1, Paras. 17-18). Interestingly, we have here an "anticipation" of the 16th century observation that the ratios of the neighboring Fibonacci numbers (1, 1, 2, 3, 5, 8, ...   F_n = 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 (der goldene Schnitt) and it appeared in the mathematical-educational literature in the 1830s (F. Wolff, 1833; M. Ohm, 1835). Then, the German expression was translated into Latin (sectio aurea) and Greek (chrysê [or khrusê] tomê) to illustrate that the concept, not the term, is ancient. Almost in the same time German and French botanists described the role of neighbouring Fibonacci numbers in the case of leaf arrangement (phyllotaxis). In 1854 A. Zeising published a comprehensive book on the golden section in the context of human proportions and he discussed the importance of this topic in art and nature [8]. Although he emphasized that this is his own theory, the attractive illustrations by him led to "golden sectionism". The aesthetical significance of the golden section gained an additional support from G. T. Fechner, a leading physiologist and psychologist, who demonstrated a preference for this proportion in his experimental-psychological tests. Suddenly many people started to search for the golden section in art works using measured data, including the strongly damaged building of the Partheon and the sculptures of Pheidias and Polykleitos, which are not original, but late copies (sic!). Obviously measured data and their rounded ratios cannot "prove" the conscious usage of the golden section. (It is true that we cannot exclude the unconscious usage of the golden section by artists, but the modern experimental-psychological tests indicate just a limited preference for this proportion.) Even if the ratio of some measured lengths gives a good approximation to the golden number, we cannot decide whether the artists intended to manifest this number or some much simpler ratios that slightly differ from this. In fact, 2/3, 3/5, and even 5/8 could have had, beyond the visual preferences, musicological roots, as we have discussed. The modern mathematical notation for the golden number as phi from Pheidias name or tau from Proclus's tomê (section) is misleading from a historic point of view; they canonize a 20th century believe and an unclear statement from the 5th century A.D., respectively. Perhaps it was far from the Greek culture to limit the freedom of artists and to consider just one irrational number as the key to design in general. Some prejudice against irrational numbers survives in the very term (non-rational), while the Greeks preferred the power of ratio (logos). They may have used various ratios as the combination of artistic intuition and theoretical considerations. In addition to this, the Greeks dealt with various symmetric figures, from the pentagram to regular polyhedra (although they did not call it symmetria), and developed a special sensitivity for balance and harmony. It is interesting to compare the bilateral symmetry of the Greek temples with the rotational symbols (and style of thinking) in Taoism and Buddhism, including the circular motifs of the Chinese Yin-Yang and later the Indian mandalas. On the other hand, the basic idea of dialectics are very different in the Western culture and China: the yes-no (left-right, black-white) dichotomy strongly differs from the traditional Chinese way of thinking where there is no "yin" without "yang", and vice versa, and the two are interlocked in the circular symbol. We may add, however, that bilateral symmetry and two-fold rotational symmetry in the plane are equivalent in the 3-dimensional space. Both of them can be interpreted as a rotation around an axis, as a nice analogy between these two types of symmetries.