Journal of American Indian EducationVolume 27 Number 2
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MATHEMATICS-LIKE PRINCIPLES INFERRED FROM THE PETROGLYPHS Charles G. Moore The petroglyphs created and carved on the surfaces of caves, cliffs and stones by the ancestors of the American Indians have long been a source of wonder and fascination to modem man. What was the motivation for their creation? Is there a message for us to be found in them? Can we read anything in them that will enable us to begin to understand the thought processes of those ancient peoples? These are intriguing questions that have never been answered. In recent years the writer has been involved in the development of learning modules in mathematics as part of the NSF sponsored Outdoor World Science and Mathematics Project. References to the petroglyphs encountered while conducting background research for the modules rekindled an enthusiasm for the ancient art form. That research, in conjunction with recent developments in mathematics, make it possible to identify certain principles of mathematics-like thought participated in by the carvers of the petroglyphs. Those principles are: 1) iteration, 2) recursion, 3) similitude, 4) tiling, and 5) symmetry. The intent of this paper is to provide a rationale for the hypothesis: Elements of mathematics-like thought can be inferred through a study of the petroglyphs. Mathematics may be divided into two major categories: abstract and applied. In like manner, the petroglyphs may be categorized under the two major headings of abstract and applied. The abstractions are geometrical designs while the applied are the representational glyphs depicting easily recognizable forms such as deer or mountain sheep. The claim has been made that the application of the figures of animals was to contact their spirits for the purpose of enhancing the tribe's probability of a successful hunt. A subcategorization of the abstract petroglyphs will now be proposed. 1) Iteration - Iteration refers to a repeated process. A basic form was repeated in the petroglyphs presumably for the purpose of presenting an aesthetically appealing pattern. An example is shown in Figure 1.
The iterative pattern shown in Figure I was photographed near Flagstaff, Arizona and is approximately six inches in height. 2) Recursion - Recursion is described as a repetitive process but with a systematic amplification of the basic pattern at each repetition. The amplification may be negative resulting in a reduction of the pattern. Examples of petroglyphs utilizing the principles of recursion are shown in Figure 2.
Figures 2a and 2c occur frequently in sites where petroglyphs; abound and Figure 2b was photographed by Weaver (1986) in Wupatki National Monument. A standard dictionary will list iteration and recursion as being synonymous. The preceding distinction was made by Roberts (1986). It is improbable that this classification of petroglyphs as iterative or recursive could have been made in previous literature. 3) Similitude - The importance of the principle of similitude was described in detail by West (1987). Similitude refers to the process of repeating a figure with a systematic expansion or contraction at each repetition with the added constraint that each new figure be similar to the preceding one. The expanding chambers of a Nautilus shell are a prime example and the successively branching and reducing of the tubes of the bronchial system are another. See Figure 3.
If the progression of similitude is from larger to smaller, one encounters the problem of an infinite number of similar figures becoming infinitely small. The writer has an example on a potsherd. When confronted with the problem the decorator solved it by painting out the troublesome part. See Figure 4.
4) Tiling - Important in advanced geometry is the principle of tiling. The problem is to fill the plane with a repeated pattern. A checkerboard is a simple example. The problem becomes increasingly complex with the complexity of the primary pattern. In Figure 5 is shown an admirable tiling of the plane achieved in a petroglyph again photographed by Weaver (1986) at Wupatki National Monument.
The writer was never successful in attempts at sketching the tiling shown in Figure 5 with all of the geometric properties exhibited in the petroglyph. He then identified the essential properties mathematically and found solutions to the resulting equation. There appear to be four variables involved: the width (x) and length (y) of the large repeated rectangle, the side of the filler square (z), and the width (w) of the uniform spaces between the squares and rectangles. The function and requirements of the variables are shown in Figure 6.
Assigning a value of I to the smallest dimension w there is only one set of integer solutions satisfying equation (4) and they are w = 1, z = 2, x = 3 and y = 9. With the aid of engineering graph paper the tiling shown in Figure 7 was completed. It coincides fairly accurately with the tiling of the petroglyph.
The maker of the petroglyph, possibly being more motivated by artistic license than engineering accuracy, imposed upon himself or herself the added problem of carving the entire figure with all of its right angles on a bias of forty-five degrees. 5) Symmetry - Two types of symmetry are easily identified in the petroglyphs. These are axial symmetry (symmetry with respect to a line or axis) and central symmetry (symmetry with respect to a point or center).
In analytic geometry the graph of an equation relating x and y is said to possess axial symmetry (with respect to the y-axis) if the replacement of x by -x leaves the equation unaltered. Similarly, if replacement of both x by -x and y by -y leaves the equation unaltered, then the graph of the equation is said to possess central symmetry with respect to a center (the origin).
In Figure 8a the vertical or y-axis is the axis of symmetry of the parabola and in Figure 8b the origin is the center of symmetry of the ellipse. Commonly found types of glyphs through which the carvers strove for and achieved axial symmetry are shown in Figure 9. Of special interest when considering the property of symmetry are petroglyphs with axial symmetry such as those shown in Figure 10. In each of the glyphs the axis of symmetry is explicitly exhibited. There are numerous examples of rock art which exemplify the principle of central symmetry. Examples are shown in Figure 11. It is not necessary to attempt to analyze the thought processes of the petroglyph carvers in order to recognize and identify in their work the mathematical principles discussed in this article. It can be argued that the creators of the petroglyphs did appreciate those properties now called mathematical to a degree of sufficient intensity for them to be motivated to express those propertiers through the images they carved in stone. They had discovered and utilized those properties and modem mathematicians labeled them and wrote the equations that describe them. The author invites correspondence from readers who know of examples of rock art, whether it be petroglyphs, pictorgraphs or pre-historic pottery decorations utilizing the principles discussed in the article.
Suggested Classroom Activities Activity 1) Hand the students dittoed copies of the petroglyph diagrammed in Figure 5. Ask them to look at it and make a figure like it. Discuss the question of whether their sketches captured all of the essential geometrical properties of the petroglyph. The activity will help the students gain an appreciation for the talents of the creator of the original tiling. Activity 2) Provide the students with a sheet of graph paper. Ask them to use the dimensions used in Figure 7 to tile a half page by repeating the figure. The resulting "basketweave" effect is remarkably attractive. Activity 3) Use inked blocks of dimensions 2 units by 2 units and 3 units by 9 units with a spacer of width I unit to print a cloth with the pattern suggested by that long-ago geometric artist. Activity 4) After classroom discussions indicate that the students understand the five principles described, take the class for a field trip to a known site of petroglyphs. Interest in the glyphs will be heightened if the students can identify examples of the principles they have studied. Charles G. Moore received the Ph.D. at The University of Michigan and is presently professor of mathematics at Northern Arizona University. In recent years he has been involved in the writing of culturally related materials in mathematics for use in high schools having predominantly American Indian student bodies. REFERENCES <Barnes, E.A. (1982). Canyon Country Prehistoric Rock Art. Wasatch Publishers, SaIt Lake City. Bradley, Claudette (1984). "Issues in Mathematics Education for Native Americans and Directions for Research." Journal for Research in Mathematics Education, 15 (2). Grant, Campbell (1967). Rock Art of the American Indian. Thomas Y. Crowell, New York. Roberts, Eric S. (1986). Thinking Recursively. John Wiley and Sons, Inc., New York. Schaafsma, Polly (1980). Indian Rock Art of the Southwest. School of American Research, Southwest Indian Art Series, Santa Fe. Turner, Christy (1963). "Petroglyphs of the Glen Canyon Region." Museum of Northern Arizona Bulletin, No. 38, Flagstaff. Weaver, Donald E. Jr. (1986). Images on Stone. The Museum of Northern Arizona. West, Bruce, J., and Goldberger, Ary L. (1987). "Physiology in Fractal Dimensions." American Scientist, 75, July-August.
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