Laterally related topics: The Stone Builders, The Middle Ages, The Renaissance, The 1600s, The 1800s, The 1700s, The 1400s, The 1500s, The Paleolithic Era, The 1900s, and The Late Bronze Age.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Jablan, Slavik. Geometry in the pre-scientific period. Geometry in the pre-scientific period; ornament today, 1--32, Hist. Math. Mech. Sci., 3, Math. Inst., Belgrade, 1989. SC: 01A10, MR: 91i:01004.
Discusses geometric ornamentation in Paleolithic and neolithic mathematics, focusing on the symmetries in the ornamentation. The author gives many examples. The only possible symmetry groups of the rosettes are Cn and Dn. There are infinitely many of these, of course, but the basic types occur in both the Paleolithic and the Neolithic. There is a somewhat wider variety in the Neolithic. In addition, neolithic artists have also explored some of the corresponding antisymmetry (or bichromatic) groups. It turns out that all 7 of the frieze already occur in the art of the Paleolithic; thus not surprisingly they occur in the art of the Neolithic as well. The examples show that there are interesting differences in the ways that the frieze patterns are applied. 14 of the 17 bichromatic strip patterns (antisymmetry groups) occur in neolithic ornamental art. 14 of the 17 plane patterns occur in the Neolithic. The author discusses reasons why the artists may have explored the patterns that they did. The author also finds 23 of the bichromatic plane patterns, and gives an example of each. (He classifies these using the Coxeter group/subgroup notation.) Closely related topics: The Paleolithic Era, Frieze Patterns, Plane Patterns, Bichromatic Strip Patterns, Bichromatic Plane Patterns, and Rotational Symmetry Groups (Rosettes).
Mathews, Jerold. A Neolithic oral tradition for the van der Waerden/Seidenberg origin of mathematics. Arch. Hist. Exact Sci. 34 (1985), no. 3, 193--220. (Reviewer: M. Folkerts.) SC: 01A10 (01A25), MR: 88b:01005.
Abraham Seidenberg advanced a theory that mathematics arose from a common origin, and that some the mathematics was preserved by an oral tradition, and very likely a religious tradition, perhaps one like the one seen in the Indian Sulvasutras. Van der Waerden's book Geometry and Algebra in Ancient Civilizations takes a similar views, and in fact van der Waerden credits Seidenberg for making him look at the history of mathematics a new way. As Mathews notes, the Chinese Chiu Chang Suan Shu is very important in van der Waerden's work. Mathews relies heavily on this work as well to "give a small, coherent, and basic core of geometry concerning rectangles and their parts, ..., which may serve as what van der Waerden has called an 'oral tradition current in the Neolithic age.'" He states the he hoped "to give this hypothesized ancient core some credence through its relation to the Chiu Chang and its explanatory power. After giving a thorough discussion of this geometry, he then carefully analyzes the ninth chapter of the Chiu Chang in terms of this core. He is able to find a strong match, though his conclusions on one of the problems (Problem 20) are not consistent with those of some other researchers, who find in problem 20 instead suggestions of something like Horner's method. A very interesting article. Hopefully future papers will discuss how well the author's geometry agrees with the ancient geometry of other cultures. As he notes, "Until I can thoroughly test his conjecture on, say, the Babylonian corpus, I can argue for the merits of my conjecture only on such grounds as the simplicity of explanation it allows, or its congruence with received results or figures." Closely related topics: Religion, Geometry, The Sulvasutras, The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art), and Abraham Seidenberg.