In the 5th century B.C. not only the Pythagorean school flourished, but this is also the age of the famous sculptors Pheidias, Myron, and Polykleitos (Polyclitus) and some great achievements in architecture, including the building of the Parthenon in Athens. Pythagoras himself was born around 560 B.C. in Samos, an island very close to the costal line of modern Turkey, but he moved to "Magna Graecia" (Southern Italy) and founded a religious-philosophical school at Croton (Krotôn), now Crotone in Calabria. Although at the beginning of the 5th century B.C., following some conflicts with the local leaders, he was forced to leave and retired at the nearby Metapontum (now Metaponto), the members of his school, the Pythagoreans, remained active in Croton. In the mid 5th century, after further troubles there, they moved partly to the close city Tarentum (now Taranto), partly to the Greek mainland. Obviously, these events contributed to the spread of Pythagorean ideas in wider circles, despite the secrecy of the movement. Originally the Pythagorean school had a very strong hierarchy with two groups of members, the akousmatikoi, the "listeners", and the mathêmatikoi, the "learned disciples" who were allowed to put questions and discuss their views. (The verb akouein, "to hear", is also the root of the modern expression "acoustic", while manthanein means "to learn".) Following the Pythagoreans' growing interest in numbers and geometrical figures, the word mathêma gained a new meaning beyond "something learned" and became associated with a new field of study: mathêmatikê or, in neutral plural form, mathêmatika. Interestingly, the Pythagorean theorem on the sides and the hypotenuse of a right-angled triangle had been known in Babylonia since the time of Hammurabi (ca. 1,800 B.C.), it was used in practical questions in Egypt, and also discovered by the Chinese in ancient times. Perhaps the Pythagorean contributions was the general formulation of the theorem and a mathematical proof, which was not definitely made by Pythagoras himself. They also had an interest in the pentagram, or five-pointed star, and discovered three of the five regular polyhedra: the tetrahedron, the cube, and the dodecahedron. We always should add "probably" since there are no surviving sources from this period and the later surveys are interwoven with legends. For example, the naive story from the 2nd century A.D. that the pentagram, which was known in various earlier cultures, became the secret recognition-symbol among the Pythagoreans is questionable. How did a relatively simple and widely known symbol serve as a "secret" recognition symbol? However, we may accept that part of the story, which is also supported by other sources, that the Pythagoreans preferred the pentagram as a symbol and associated it with health and other ideas. Indeed, the Pythagoreans' interest was not limited to mathematical questions, but these were amalgamated with musicological, astronomical, philosophical, aesthetical, and religious ideas. They discovered that musical harmonies, intervals where two tones sound well together ("consonance"), can be described by simple ratios of lengths of a vibrating string ("monochord"): 1/2 (octave), 2/3 (fifth), 3/4 (forth), 3/5 (major sixth), etc. [2]. Following this success, they extended the emphasis on numerical relationships for all things, including the movement of heavenly bodies. Did the Pythagorean mathematicians cooperated with visual artists? The secrecy of the school was against this, but the mathematicians' interest in art could have encouraged some informal discussions. Thus, Polykleitos' canon, which was summarized in a book and illustrated by a sculpture that are not extant, might have had some influence by the Pythagorean doctrine on the importance of numbers. Somehow Polykleitos started to think in terms of commensurable lengths (symmetria) with appropriate ratios of integers, as it was discussed in the Pythagorean theory of musical harmony, and this approach helped him to establish his canon. Parallel, the Pythagoreans were happy to see a further justification of their theory in the field of sculpture. The hypothesis of this mutual relationship is based on external evidence [3]. On the other hand, we should not be surprized that the "missing link", a convincing document, is not available. Neither sides had too much interest to overemphasize this connection. Polykleitos had a great credit for his canon, and, indeed, this was a novelty in the field of art. At the other side, the surviving documents by Pythagorean authors, which were written in later periods, are famous for overemphasizing their own success: perhaps there was not too much room in their writings for giving credit to an outsider.

Let us see the period of the 5th century B.C. at a broader cultural level, focusing not only on Greece. First of all, we should remark that both Greek mathematics and art were fertilized by the earlier achievements in Babylonia and Egypt. The Greeks, however, went beyond the practical mathematics of the earlier ages by theorems and proofs and renewed the existing forms of sculpture and architecture by a new aesthetics. The 5th century B.C. is also the age of many developments in the Jewish culture. Shortly before the beginning of this century, the Jews rebuilt the destroyed Jerusalem and the Temple, which event marks the beginning of the Second Temple Period. The revival of the spiritual life was started in the mid 5th century B.C. when many Jews, including Nehemiah and Ezra, the Scribe, returned from the Babylonian exile to Jerusalem. The Torah, which contains the Pentateuch, or Five Book of Moses, became the moral basis of the Jews. The revelation of the existence of one true God in the Second Commandment (which verse became the First Commandment in the Christian tradition) is followed by a statement on the prohibition of image-making (Exodus 20:4 and Deuteronomy 5:8). This led to a restriction of visual arts (iconoclasm), but the figural representation was not totally excluded [4]. On the other hand, this prohibition gave rise to the use of some geometrical motifs, including the pentagram (five-pointed star) and the hexagram (six-pointed star). There is no evidence that these figures had any special significance beyond ornamental purpose in the early periods. The association of the hexagram with the shield or the seal of King David and his son King Solomon (11th-10th cc. B.C.) has no clear basis, since the first known example of a hexagram on a seal is from the 7th century B.C. This symbol is not available in the oldest sources of Jewish magic. However, the pentagram and the hexagram became popular in the Second Temple period [5]. Of course their use was not limited for the Jews, but appeared in various cultures. Still there was an important difference: most other cultures had many figural symbols that were applicable as shields, seals, or coats of arms (which name immediately suggests a pictorial design), while the Jewish iconoclasm partly prevented the formation of such traditions. The pentagram was used already, among others, as a pictograph in Babylonia and, as we have seen, a symbol in the Pythagorean circles. (Although this is a much later example, the Japanese call it the "Seal of Abe Seimei", referring to the famous yin-yang geomancer in the mid-Heian period, around 1,000). Perhaps these circumstances contributed to the fact that the hexagram gradually became a symbol of the Jews. However, this did not happen in ancient times and its popular name, the Magen David (Shield of David), is not ancient. The first known usage of this term is of the 12th century, but in that time it did not refer to the hexagram, but to inscriptions on a shield: 72 holy names (Exodus, 14: 19-21), later Psalm 67 written in the shape of the menorah (candelabrum). Interestingly, the first case where the name of David, without mentioning the shield, and a hexagram are connected is a 6th-century tombstone in Tarento, Southern Italy [5]. (Was there any influence from the Pythagorean traditions, considering the fact that the same place was one of their centers in earlier ages?) The name "Seal of Solomon", which occasionally refers to both the pentagram and the hexagram, is probably from Christian sources. While the pentagram is mathematically a unicursal figure, we may sketch it without lifting the pencil from the paper, the hexagram is the union of two equilateral triangles. This fact gives ample opportunities for various symbolical and mystical interpretations. I can imagine that such "visual-mathematical" ideas also contributed to the fact that in the Medieval cabalistic literature the hexagram was frequently used. The cabala (or kabbalah, tradition), the Jewish mysticism and theosophy from the 12th century, continues ancient traditions by studying the esoteric and mystical values in the scriptures, including the magical meanings of numbers and letters. Obviously, the iconoclasm helped the formation of this interest, while the Christian traditions developed various forms of "visual interpretation" of the Bible in paintings and sculptures. However, the cabalistic books also required some illustrations and this tendency led to a new interest in geometrical symbols and their magical interpretation. The six lines of the hexagram divide the figure into seven parts: a larger internal part (hexagon) and six outer ones (triangles). The cabalist authors frequently wrote letters into these "cells". Sometimes even quotations from the scriptures were written in the form of a hexagram (anticipating Apollinaire's "calligrams" and the 20th century idea of "visual poetry"). In the mid-14th century the hexagram appeared on the flag of the Jewish community of Prague (King David's Flag). The only other visual symbol that may compete with the hexagram in the modern Jewish culture is the image of the menorah, the candelabrum. It is much older as a symbol and also has, although in a less abstract form, mystic and cosmic dimensions (for example, the seven branches correspond to the seven heavens and the seven planets).

Turning to other regions, in India there was an interesting geometrical-ritual tradition of constructing altars of various shapes. Earlier the Sulvasutras, texts associated with the Vedas, but perhaps not older than the 7th century B.C., described this activity in detail. These texts are considered as the earliest surviving works of mathematical nature in India. It is interesting to note that Zoroaster (Zarathustra) in Persia, Lao Tzu and Confucius in China, Buddha in India, and Pythagoras in "Magna Graecia" were probably contemporaries for a short period in the 6th century B.C. The three younger among them, Confucius, Buddha, and Pythagoras, passed away in almost the same time around 480 B.C. This coincidence, or, as the psychologist Jung would say, synchronicity, is remarkable, even if we admit that there are some question marks at these biographical data. Following the teachings of these men, Zoroastrianism, Taoism, Confucianism, and Buddhism became important religious systems where philosophical and ethical questions are also addressed, adding that Confucianism can be better described as a moral force than a formal religion. The basic part of I Ching or Yìjing (Book of Changes, here i or yì also refers to divination or oracle) was written earlier, but some traditions assign the appendices to Confucius. The latter view is questionable, but perhaps some of the appendices were written by members of the Confucian school. The I Ching provides a repository of universal concepts and a cosmic numerology (using here Joseph Needham's terminology). The listed concepts and the related numbers are visualized in a binary form. Specifically, the classification is made according to the 64 (= 26) hexagrams, six-line figures where each line is either "opened" (- -) or "whole" (---).

(We should not confuse this hexagram of parallel lines with that hexagram where the six lines form a six-pointed star, the earlier discussed Shield of David.) These methods of divination could have contributed to the existing numerological interest of the Chinese, while in the same time the Pythagorean and other Greek oracles had a focus on both numbers and shapes. The influence of I Ching went far beyond the practice of divination and became one of the "Five Classics" of Confucianism. Taoism also benefited from the I Ching. At a later stage, Taoism supplemented Confucianism and helped the growth of Buddhism.

The religious ideas of the Pythagoreans, however, did not lead to a new system (and related scriptures), but contributed the birth of modern geometry, number theory, musicology, as well as philosophy. During a relatively short period the philosophical schools of Socrates (ca. 470-399 B.C.), Democritus (ca. 460-371 B.C.), Plato (427-347 B.C.), and Aristotle (384-322 B.C.) were born, the beginning of dialectics and mathematical logic was initiated by the paradoxes of Zeno of Elea (ca. 490-430 B.C.). (Note that the main principle of Zoroastrianism, the fight between good and evil, can be also associated with dialectical thinking.) The Greek philosophical debates and the dialectics, the refutation of opponents by drawing unacceptable conclusions from their views (reductio ad absurdum), also had an impact on mathematical thinking and led to the method of indirect proof. For example, we can demonstrate that Ö2 is not a rational number by supposing that we still may write it in the form of a/b (opposite case). Then, manipulating this fraction, we produce a contradiction and conclude that a/b does not exist. In the framework of this type of proofs we manipulate non-existing objects in an "imaginary" world. From this time we may see a bifurcation of mathematical ideas: those that are used in the practice by craftsmen and artists (Vitruvius, 1st c. B.C., gave an insight into this) versus those ones that led to abstract mathematics with axioms, definitions, theorems, and proofs (Euclid, ca. 300 B.C.). The Greek philosophy and mathematics significantly contributed to the Western way of knowing Nature by scientific hypotheses and mathematical models. The Chinese approached Nature by direct observation and aesthetical intuition. The latter also has a great value, but remained less successful in explaining the phenomena of nature and predict them. The Chinese produced some similar results to the Greek achievements, for example the School of Forms and Names (Hsing Ming Chia), or the Logicians, which flourished in the 4th-3rd centuries B.C., can be compared with Zeno and his group in Elea (5th century B.C.), but these results were not developed into a coherent methodology. The Greek focus on geometry was more useful for making visual-mathematical models in astronomy and mechanics than the Chinese emphasis on numbers. Last, but not least, the social backgrounds in Greece also contributed to the great achievements in the 5th century B.C.: the end of the Persian wars, the era of Pericles with peace and prosperity, and the formation of the Greek democracy. Considering the period of more than 1,200 years from the poet Homerus to mathematician Proclus, it is quite remarkable that so many developments in mathematics and art happened in the 5th century B.C.