To expand search, see Symmetry. Laterally related topics: Frieze Patterns, Plane Patterns, Five Fold Symmetry, Penrose Tilings, Pattern, The Regular Solids, Double Frieze Patterns, Two Sided Frieze Patterns, Rotational Symmetry Groups (Rosettes), Bichromatic Plane Patterns, and Dynamic Symmetry.
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.
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, Plane Patterns, Pottery, and The Pueblo Indians.
Cromwell, Peter R. Celtic knotwork: mathematical art. Math. Intelligencer 15 (1993), no. 1, 36--47. SC: 01A07 (00A69), MR: 1 199 275.
Cromwell discusses a theory for the construction of Celtic knot friezes. These knot patterns may have been inspired by basketry (or maybe by textiles). He then analyzes the patterns in the knot friezes using a notion of a two-sided frieze pattern. There turn out to be 31 such patterns; 7 of these are the standard monochromatic strip patterns; 17 are exactly analogous to the bichromatic strip patterns; and 7 are like the monochromatic strip patterns but require the two sides to be identical. These last 7 "grey" patterns can't occur in knotwork, since the two sides of a crossing are not identical. Of the 24 monochromatic and bichromatic patterns, 12 cannot occur in Celtic knotwork because they would require strings that don't tie up, and 2 require a string straight through the centerline (and also don't occur). The other 10 can theoretically appear. Of these 10, two do not seem to occur at all, and one occurs but with an apparently different constriction technique (an example of this type is thought to be Scandinavian). The author is able to explain the rareness of these symmetry types in terms of the theory for their construction and from the fact that Celtic know friezes were generally finite and had their ends knotted together; these constraints require construction with an even grid, and the three problematic patterns require construction with an odd grid. This explains the type which does occur appears to use a different construction technique. In fact, the author found only one Celtic pattern that uses an odd grid. (And of course it can't be used in a bounded way, though it can be used in a kind of border.) All 7 of the monochromatic frieze patterns were apparently used in generating the existing know patterns, assuming the theory of construction is true (the author makes no claims that it is). The author includes examples of his own for the 3 problematic odd-grid know patterns. Excellent article. The author includes a good bibliography of related topics. It goes as far as Norwegian peasant art, for example. Not inordinately technical, in spite of the way it might sound. Closely related topics: The Celts, Knots and Knotwork, Two Sided Frieze Patterns, Frieze Patterns, Weaving, and Basket Making.
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, Plane Patterns, Pottery, Archaeology, The Islamic World, and Spain in the Middle Ages.
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, The Neolithic Era, Frieze Patterns, Plane Patterns, Bichromatic Plane Patterns, and Rotational Symmetry Groups (Rosettes).
Jablan, Slavik. Ornament today. Geometry in the pre-scientific period; ornament today, 33--65, Hist. Math. Mech. Sci., 3, Math. Inst., Belgrade, 1989. SC: 01A10, MR: 92g:01008.
The author discusses how a wide variety of mathematical notions can be used to help describe and understand the patterns occurring in art. One of the most important is, of course, the notion of symmetry, including those in the rotational symmetry patterns, frieze patterns, plane patterns, and their bichromatic (or antisymmetry) variants. More complex types of patterns also occur in art, and as Grünbaum, Grünbaum, and Shephard observed in their article Symmetry in Moorish and other ornaments, many of the problems originating from these are still unsolved. Examples are given from the Paleolithic to the 20th century. The author touches on (to give a few examples) interlace patterns (often considered to be connected with weaving), similarity symmetry, symmetries in higher dimensional spaces, and on some of the ideas of the theory of tilings, including Penrose tilings and hyperbolic tilings. The author also gives examples from the work of artists including M. C. Escher, B. Riley, and R. Neal. A fine article. A fine article. It could easily take a class an entire semester to examine in detail all the ideas presented. Closely related topics: Art, Pattern, Symmetry, Frieze Patterns, Plane Patterns, Bichromatic Plane Patterns, Rotational Symmetry Groups (Rosettes), Penrose Tilings, Weaving, Similarity, and M. C. Escher.