To refine search, see subtopics The Zuņi, The Pueblo Indians, The Anasazi, The Blackfoot Indians, The Sioux, Hopewellian Indians, California Indians, The Navajo, and The Iroquois. To expand search, see The United States and Indigenous Mathematics of North America. Laterally related topics: African American Mathematics, The United States in the 1700s, The Inuit, and The Bellacoola.
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.
Ascher, Marcia and Ascher, Robert. Ethnomathematics. Hist. of Sci. 24 (1986), no. 64, part 2, 125--144. (Reviewer: Jens Høyrup.) SC: 01A10 (92A20), MR: 88a:01005.
Discusses the danger of identifying non-literate mathematics with "primitive" mathematics. Warns against assuming that because a group has two sets of number words (as in the Blackfoot Indians, who are said to use different sets of numbers for the living and the dead), the group therefore doesn't understand the underlying identity between the different words. Regarding logic, when asked the question "All Kpelle men are rice farmers. Mr Smith is not a rice farmer. Is he a Kpelle man?", one Kpelle respondent answered "If you know a person, if a question comes up about him you are able to answer. But if you do not know the person, if a question comes up about him, its hard for you to answer." The authors emphasize that a response like this doesn't show a lack of ability in logical reasoning, but just differences in views in talking about people you don't know and about 'playing along' with a questioner. The authors discuss how the Sioux viewed the circle as a more natural shape than the (western) line. Kinship systems of the Aranda of Australia, and in Ambrym in the New Hebrides. How elders in Ambrym used diagrams to elucidate the kinship systems, and explicitly explained the patricycles of degree 2 and the matricycles of degree 3. An interesting question for a student might be to investigate if the Aranda system (with six groups) is optimal in ruling out certain types of marriages that are too close. Closely related topics: Ethnomathematics General, Number Words, Logic, Kinship Systems, The Aranda, Ambrym, New Hebrides, The Blackfoot Indians, The Sioux, and The Kpelle of Guinea.
Aveni, A. and Hartung, H. The observation of the Sun at the time of passage through the zenith in Mesoamerica. Archaeoastronomy No. 3 suppl. J. Hist Astronom. 12 (1981), S51--S70. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82k:01003.
A careful analysis suggests that two near-vertical tubes in central America were used to mark the zenith sun, and possibly the June solstice. At about the time that the second tube was used, transits of the Pleiades could be observed as well. Arguing for the significance of this is the recent knowledge of the importance of the Pleiades in the Aztec calendar. In the structure containing the second tube is also a doorway making a place on the horizon. At the period in question, Capella underwent helical rise at that place on the same day as the first zenith sun. Further evidence of the importance of zenith sun watching is found in a clay model of a temple offering and in two clay figurines. The author gives an example of how observations of the sun through an aperture were also important to the Zuņi (in this case through the roof of a Zuņi chief or priest). Closely related topics: The Maya, Astronomy, The Aztec, and The Zuņi.
Campbell, P. J. The geometry of decoration on prehistoric Pueblo pottery from Starkweather Ruin. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 731--749. (Reviewer: M. P. Closs.) SC: 01A12 (92A90), MR: 90h:01003.
Starts by introducing the mathematical principles behind classifications of symmetry groups for strip or frieze patterns and the plane patterns, and briefly discusses some other symmetry groups. Next, reviews the literature of the papers that have used symmetry patterns to classify and analyze designs. All an excellent introduction. The remainder of the article applies these methods to the later Pueblo pottery at Starkweather Ruin (Tularosa black-on-white and Reserve black-on-white). Ends with a discussion of to what extent the work of these and similar potters was mathematical. Closes with a quotation by Schattschneider on the work of "amateurs": "The mind and spirit are the forte of all such amateurs---the intense spirit of inquiry and the keen perception of all they encounter. No formal education provides these gifts. Mere lack of a mathematical degree separates these 'amateurs' from the 'professional'. Yet their dauntless curiosity and ingenious methods make them true mathematicians." Closely related topics: Archaeology, Frieze Patterns, Bichromatic Strip Patterns, Plane Patterns, Pottery, and The Pueblo Indians.
Crowe, D. W. and Washburn, D. K. Groups and geometry in the ceramic art of San Ildefonso. Proceedings of the conference on groups and geometry, Part A (Madison, Wis., 1985). Algebras Groups Geom. 2 (1985), no. 3, 263--277. (Reviewer: H. S. M. Coxeter.) SC: 05B45 (00A05 01A12 20F32 52A45), MR: 87k:05055.
Discusses the types of frieze patterns and bichromatic strip patterns occurring in the pottery of the pueblo of San Ildefonso in New Mexico. The people of San Ildefonso are Tewa speaking and are thought to be of Anasazi descent. However, it should be noted that the pottery has apparently been influenced by the Spanish and by attempts to make it more readily salable. All 7 of the strip patterns and 14 of the 17 possible bichromatic strip patterns are exhibited. (The authors supply the missing 3 bichromatic strip patterns in a similar style. The authors supplement their discussion with an explanation of the appealing Coxeter notation for classifying the bichromatic patterns (the standard classification system is cumbersome) and give a table of the correspondences between various systems. A historical aside briefly discusses the study of plane patterns in the context of the Alhambra, where there is still some disagreement on which patterns are represented. Closely related topics: The Pueblo of San Ildefonso, Frieze Patterns, Bichromatic Strip Patterns, Plane Patterns, Pottery, Archaeology, The Islamic World, and Spain in the Middle Ages.
Hively, Ray and Horn, Robert. Geometry and astronomy in prehistoric Ohio. Archaeoastronomy No. 4 (1982), S1--S20. (Reviewer: C. R. Fletcher.) SC: 01A10, MR: 84f:01002.
The geometrically designed earth-works near Newark, Ohio have been the subject of curiosity for centuries. They are Hopewellian, and are now dated at approximately 0-250 AD. From a purely geometric point of view the site is interesting because of its use of a circle and an almost equilateral octagon. The authors have carefully analyzed the available survey data. They first determined that the site was constructed using a standardized unit of length, and then considered possible astronomical alignments in the site. They found no convincing evidence of solar or planetary alignments, but they did find quite a bit of evidence for lunar alignments. Important lunar points include the minimum and maximum north and south extremes for the Moon's rise and set points, and there is in fact the possibility that all 8 of these points are recorded, though the evidence for some is stronger than the evidence for others. It appears that some deviation from symmetry in the octagon may have resulted from efforts to incorporate the given alignments. This study suggests that the builders may have been interested in the 18.61 year lunar cycle. The authors do not consider stellar alignments, since uncertainties in the date of the site make effects of precession unacceptably large. A related Hopewellian earth-works construction is discussed in Hively, Ray and Horn, Robert, Hopewellian geometry and astronomy at High Bank. Closely related topics: Hopewellian Indians, Astronomy, and Polygons.
Hively, Ray and Horn, Robert. Hopewellian geometry and astronomy at High Bank. Archaeoastronomy No. 7 Suppl. J. Hist. Astronom. 15 (1984), S85--S100. (Reviewer: M. P. Closs.) SC: 01A12, MR: 86f:01005.
This paper continues the investigations that the authors started with Hively, Ray and Horn, Robert, Geometry and astronomy in prehistoric Ohio. In the present article, the authors discuss the Hopewellian earthworks construction at High Bank in Ohio. Like the Newark construction, this includes a circle and an equilateral octagon. This site is oriented roughly 90o differently, however, and the octagon is on a different scale than at Newark. Nevertheless, both sites were apparently constructed using the same standard of length. [The octagon may have been constructed using a different procedure than the octagon at Newark.] There are possible alignments to the same lunar events as at Newark, and there are also possible alignments to sunrise and sunset on both the summer and the winter solstice. All may differ, of course, in their likelihood of being intentional. Like its predecessor, a very interesting article. Some suggestions for future research are given. Closely related topics: Hopewellian Indians, Astronomy, and Polygons.
Hughes, Barnabas B. The earliest known record of California Indian numbers. Historia Math. 1 (1974), no. 1, 79--82. SC: 01A15, MR: 57 #15836.
The author discusses a document from 1775 which is now thought to be the earliest written record of the number system of a California Indian tribe. The document includes numbers for one through 14. One interesting feature is that some of the words suggest a base 4 number base. Also interesting is the fact that some of the number words that were recorded are different from the ones recorded by Dixon and Kroeber for related Costanoan Indians in 1907. These differences between this these number words (from Angel Island) and the others (from Mission Santa Clara) may indicate the influence of other tribes. The author notes that since Angel Island is nearly a centerpoint of various waterways, this influence is not surprising. The document was written by Fr. Vincente Maria, the chaplain of a Spanish expedition, and the author of the article seems to suggest that the confessional practices of the time may have encouraged Indians to use a decimal system for numbers larger than ten. This was because sins had to be identified at confession both by kind and by number, and because the Indians were not likely to be otherwise understood. Closely related topics: Number Words, Number Systems, California Indians, and Religion.
Katz, Victor J. Essay reviews of Ethnomathematics [Brooks/Cole, Pacific Grove, CA, 1991; MR: 92c:01006] by M. Ascher and The crest of the peacock [Tauris, London, 1991; MR: 92g:01004] by G. G. Joseph. Historia Math. 19 (1992), no. 3, 310--315. SC: 01A07 (00A30), MR: 1 177 496.
Katz reviews and contrasts Marcia Ascher's book Ethnomathematics: A Multicultural View of Mathematical Ideas and George Gheverghese Joseph's book The Crest of the Peacock: Non-European Roots of Mathematics. He finds that both correct serious omissions in the literature (and in particular, in Morris Kline's Mathematical Thought from Ancient to Modern Times). Joseph focuses on the history of mathematics in the large civilizations of ancient Egypt, Babylonia, China, India, and the Islamic World. He wanted to highlight "(1) the global nature of mathematical pursuits of one kind or another; (2) the possibility of independent mathematical development within each cultural tradition; and (3) the crucial importance of diverse transmissions of mathematics across cultures, culminating in the creation of the unified discipline of modern mathematics." Katz seems disappointed only in the third thesis, "because the documentary evidence for transmission of mathematical ideas is lacking." (For example, he notes that "whether Diophantus was directly influenced by the Babylonian tradition is a subject of scholarly debate." Joseph's treatment of Indian mathematics seems to be particularly good "especially since it is difficult to find this material in other sources." The focus of Ascher's book is completely different. She looks at traditional non-literate peoples. As Katz notes, "She has no intention of claiming that the mathematics developed in the cultures she discusses had any influence on developments elsewhere. Her main goal is simply to show that mathematical ideas, even if not developed by those called mathematicians, can be found in many societies if one only knows where to look." Katz reports examples as coming from the Inuit, Navajo, Iroquois, and Incas of the Americas, the Malekula, Warlpiri, Maori and Caroline Islanders of Oceania, and the Tshokwe, Bushoong, and Kpelle of Africa. This very useful review concludes by highly recommending both books. Closely related topics: Ancient Egypt, Sumerians and Babylonians, China, India, The Islamic World, The Inuit, The Navajo, The Iroquois, The Inca, The Malekula of Vanuatu, The Warlpiri, The Maori, The Caroline Islands, TheTshokwe, The Bushoong, and The Kpelle of Guinea.