2. Mathematics and the visual arts: Interesting
First, I propose a thesis about the general characteristics of the developments in mathematics and in the visual arts from the point of view of two basic types of "structures", dominating forces, in history:
Of course this is an oversimplified picture, but still useful to comprehend some tendencies. Obviously, the listed preferences are not exclusive, just dominant (for example the Greek city-states had more interest in geometry than algebra, but of course they dealt with both at some level). Let us see the "logical" backgrounds of these preferences although, we must admit, history is not really a field that is associated with logic. A large empire usually needs a centralized administration with many local offices that perform the financial and economical management. The duties of measuring and recording the economical data, the collection of tax, the distribution of goods and services from central places, and so on, require a special focus on numerical data and their treatment. Mostly the empires are also interested in reforming the existing calendars and making their own version, which is based on astronomical observation (and some geometry) and numerical calculations. Parallel, an empire needs a large number of buildings of various categories: palaces and offices for the local leaders and administrators, centers of cultural and religious activities, the objects of a large-scale infrastructure, and so on. This means that artists are commissioned to build and decorate very many similar buildings. Often the speed is more important than the artistic quality. On the other hand, a small country or a relatively isolated region has a strong focus on the careful division of the limited territories and this is based on "geo + metry". Here the artists are not forced to make a large number of "standardized" objects of art; they have better possibilities to represent individual views.
Babylonia, Egypt, China, India, the Mayas, the Islamic Abbasid Empire, and the British Empire produced remarkable algebra and mathematical analysis, respectively. Here we also think about the introduction of number systems with advanced symbols for numerals and place-value (positional) notation, including the Babylonian (Sumer) sexagesimal system (base-sixty, ca. 2,400 B.C.), the Chinese decimal system (before 250 B.C.), the Maya base-twenty system with the idea of zero (ca. 250 B.C.), and the ancient Hindu decimal system and numerals, originally used by the priests (Brahmans), which later became the basis of the modern Hindu-Arabic notation. The term zero or cipher (also see decipher, decoding) came from the Arabic sifr, zero (or, as adjective, empty), which is the translation of the Sanskrit sunya, empty. Interestingly the Greeks had an alphabetical notation (alpha = 1, beta = 2, gamma = 3, etc.) with no place-value, which was so ineffective for calculation that their astronomers frequently turned to the Babylonian system. Although the Roman Empire did not make important algebraic advances, but shifted from the Greek notation to the better Roman numerals. The Incas developed an intricate system of quipu, knotted cords, to keep numerical records and perform some calculations. All of these empires or large countries, from Babylonia to the British Empire, produced a wide-ranging network of buildings with many similar objects of arts and crafts. Of course the wealth of emperors or other leaders produced remarkable pieces of art at central places, but these were not available at other parts of the empire. On the other hand, the ancient Greek city-states, the Italian dukedoms during the Renaissance, the small German states and some other European countries in the modern age produced remarkable geometry and visual art at various places. In some sense India is between the two categories: a large country, but not an empire and, indeed, they contributed not only to algebra, but also to geometry and they have both standardized and individual artistic tendencies. The case of the Abbasid Empire is very interesting from the point of view of the visual arts. Although there were some signs of standardization, but religious factors contributed to a geometric art with a unique richness, as we shall see later. Parallel, the algebra-oriented Arabic mathematicians occasionally used their skills to deal with geometrical problems, including some questions related to visual arts. In Europe the successor states of the Roman Empire did not make new geometrical discoveries in a scholarly sense, but they produced the Gothic architecture, geometry in stone. Our thesis also "works" if we compare China and Japan, an empire and a nearby large island country. The Chinese mathematics is number-oriented, while geometry plays much less role. The Japanese mathematics wasan had a Chinese influence at the beginning, but also produced a very exciting "discrete geometry", using this modern Western term, with a special interest in dense packing of circles and balls in various figures. Of course algebra was also an important field, as it should be in a large country. Turning to the visual arts, the Japanese art had a strong influence from China, but the Japanese garden art, ikebana, the art of tea ceremony, origami, and some other fields represent many individual achievements. We should also note that these are not "standardized" fields. For example there about 3,000 schools in ikebana. Finally, turning to more recent developments in the Western culture, the non-Euclidean geometry was born not in the "centrum", but at a remote place of Hungary (Bolyai) and a distant city of the Russian Empire (Lobachevskii). The latter is not a counterexample to our theory since the place is not St. Petersburg or Moscow, but Kazan, which was more provincial than imperial. These results of non-Euclidean geometry paved the way to a new space-time concept that was born in Zürich (Einstein) and Göttingen (Minkowski) and had a great cultural impact in wide circles.
Of course the social context and the interest of patrons do not fully determine the focus of scholars and artists, but strongly influence it. For example, Newton was appointed as Warden, later Master, of the Mint and he executed the introduction and standardization of the new coins. The scholars who built the first modern electronic computer in the late 1940s and early 1950s, including von Neumann, worked under a contract by the Ballistic Research Laboratory of the U.S. Army. We should even see the heroic effort of János Bolyai, an officer in the Army of the Austrian Empire, who chose an early retirement and the financial difficulties in order to work on geometry and other scholarly questions at his home in Hungary. This is in a sharp contrast to the possibilities in the Italian Renaissance or in the 19th century German states. Artists and scholars attracted patrons and if they had troubles they moved to other places and sponsors.
This brief survey gives immediately some hints about periods when both geometrical and artistic achievements flourished. We may suspect that during these periods there were some "visual-mathematical" links between the two sides. However, let us go according to a chronological order and survey the "climaxes" of mathematics and the visual arts from the prehistory. Since the earliest times numbers, which are based on abstraction and difficult to keep in memory, were represented visually. Counting rods were exactly such tools: they helped to record the numbers of people, animals, and various things. We may guess that footprint-literacy , which is an essential skill for the hunting and gathering society, also contributed to idea of numerical counting and linear decoration. Man, the toolmaker was also the man, the footprint-reader, and he or she had immediate access to various linear symmetry groups (frieze groups). Indeed, the same four-footed animals "print" different symmetry groups if they are just walking slowly or running quickly. Footprints and tracks offer a tremendous amount of information not only on the location of particular animals, but also on their actual behavior. Man, the toolmaker had a special interest is shaping artificial objects, from small cutting edges to choppers, from tiny talismans to large shelters. Obviously, this activity can be linked to both mathematical and artistic ideas. (We should not go into terminological tricks, but feel free to replace "mathematics" with "proto-mathematics", if you wish.) In prehistoric times, the mathematical and artistic ideas were not separated, but formed a syncretic unity. Let us see the process of separation and then some interesting steps in the development of mathematics and the visual arts, including some "coincidences" between them. We will see that these coincidences are not by chance.