To refine search, see subtopics Symmetry, Analytic Geometry, Trigonometry, Pattern, Geometric Theorems, The Pyramid, Similarity, The Triangle, The Method of Exhaustion, Projective Geometry, Algebraic Geometry, Non-Euclidean Geometry, The Parallel Postulate, The Regular Solids, The Pentagram, The Sphere, The Conic Sections, Polygons, Topology, Spirals, Line-Point Duality, Geometric Fixed Point Principles, The Cycloid, Tilings, and The Square. For more material on this topic, see subtopic Irrationals. Laterally related topics: Religion, Time and Space, Mathematics in Recreation, Art, Language and Literature, Music, Measurement, Arithmetic, Mathematics and Mysticism, Discrete Mathematics, Optimization, Philosophy, Calculus, Statistics, Social Science, Logic, Computation, Probability, Applied Mathematics (General), Education, Algebra, Number Theory, Optics, Archaeology, Medicine, Creativity, Business, Fractals, and Science.
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.
Altshiller-Court, Nathan. The Dawn of Demonstrative Geometry. Mathematics Teacher 57 (1964), 163--66.
The author argues that it seems unlikely that the Greeks could have invented their notion of proof so rapidly and in isolation. Instead, he suggests that the notion of geometric proof was a secret that was jealously guarded from all but the "inner sanctum" of the Egyptian priesthood. (Of course, since his argument implies by its very nature that Egyptian proofs were unlikely to have been written down, this will be a hard argument to either prove or disprove.) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Proof, Ancient Egypt, and Greece.
Anagnostakis, Christopher and Goldstein, Bernard R. On an error in the Babylonian table of Pythagorean triples. Centaurus 18 (1973/74), 64--66. (Reviewer: E. M. Bruins.) SC: 01A15, MR: 58 #20994.
The authors explain a well-known mistake in the Babylonian tablet Plimpton 322 (column I, entry 10) as a consequence of a certain method of computation and of the neglect of a medial zero. It is a very appealing theory, and could give us some insight the way Babylonians did their mathematics. Other solutions have also been proposed. A good example of how we can learn from mistakes! Closely related topics: Sumerians and Babylonians, Pythagorean Triangles and Triples, and Algorithms.
Angell, Ian O. Megalithic mathematics, ancient almanacs or neolithic nonsense. Bull. Inst. Math. Appl. 14 (1978), no. 10, 253--258. (Reviewer: C. R. Fletcher.) SC: 01A10, MR: 80f:01002.
Discusses different explanations for the shapes of megalithic stone rings. The author briefly discusses some of the theories of Alexander Thom, which involve an astronomical calendar and an effort to make the circumference equal to 3 times the "diameter" rather than the irrational pi. He then discusses two new theories of his own. One explains the shapes of the stone rings as extensions of the ellipse, generated with three or four pegs and a string rather than with just the usual two. The other explains the shapes as an effort to store shadow lengths. Neither theory may be given entirely in earnest. A theme of the paper is how theories may start as intellectual games, go out of control, and be changed into pseudo-science. Closely related topics: The Stone Builders, Astronomy, The Calendar, The Ellipse, Pseudoscience, and Alexander Thom.
Archibald, Raymond Clare. Babylonian Mathematics. With Special Reference to Recent Discoveries. Mathematics Teacher 29 (1936), 209--19. (Originally delivered at a joint meeting of the National Council of Teachers of Mathematics, the American Mathematical Society, and The Mathematical Assocation of America, at St. Louis, Mo., on January 1, 1936.)
Surveys some of Neugebauer's remarkable discoveries on Babylonian mathematics, at a time when many of these discoveries were just made. Discusses notation, tables of squares, cubes, and n3+n2. Also exponentials, approximations to compound interest problems where we would use logarithms, a sum of a finite geometric series and a finite sum of squares. Geometric results, including the Pythagorean theorem, proportionality of sides in similar right triangles, a perpendicular bisecting the base in an isosceles triangle, the angle in a semicircle being a right angle, formulas for the circumference and area of a circle (using pi = 3), formulas for the frustum of a square pyramid (at least one incorrect). The relation between chords and sagitas in a circle. Approximations to the square root of a2+b2; both the well known a+b2/2a and the still hypothetical a+(2ab2)/(2a2+b2). An approximation to a square root by comparing with other solutions to an equation x2+D=y2. (The value isn't especially accurate, but the method is interesting.) Equations in five or more unknowns. Problems requiring solutions to apparently general cubic and biquadratic equations. Were the solutions just guessed, or, as Neugebauer suggests, did the Babylonians have some general methods? If so, the most likely theory is that the cubics were solved by effectively reducing them to the form x3+x2, and then using the n3+n2 table. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, Cubics, Quartics, Solutions of Linear Equations, Logarithms, Exponentials, Square Roots, Interpolation, Geometric Theorems, The Circle, and The Pyramid.
Artmann, Benno. The cloisters of Hauterive. Math. Intelligencer 13 (1991), no. 2, 44--49. SC: 00A69 (01A99), MR: 1 098 219.
The author discusses geometric principles behind Gothic tracery. The Gothic style developed in France about 1150, but spread widely in the next few centuries. Examples are taken from Reims, Haina, Strasbourg, and Esslingen. The geometric principles are by no means trivial; some make rather challenging exercises. The author discusses the windows of the cloisters of Hauterive in some detail. Hauterive is a Cistercian monastery near Fribourg in Switzerland, and the cloister dates from 1320-1328. The windows there are unusually geometric, and the author advances the theory that the windows amount to a kind of commentary on Book IV of Euclid's Elements. One window, however, can not be constructed with straightedge and compass: it involves the construction of a regular 9-gon. The author notes that a regular 15-gon may have originally been envisioned, but that "esthetic considerations overwhelmed mathematics." Interesting article. A number of illustrations, a few of which appear in Artmann, Benno; Swetz, Frank J., The Geometry of Gothic Church Windows. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, Euclid, and Religion.
Artmann, Benno; Swetz, Frank J. The Geometry of Gothic Church Windows. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. 228.
Illustrations adapted from Artmann, Benno, The cloisters of Hauterive. The tracery in European Gothic churches uses arcs of a circle, fitted together in ingenious ways. Some of the ingenious ways have mathematical principles underlying them. Although this brief excerpt does not mention it, it is not uncommon for the construction to be repeated in the same tracery in a different scale---a kind of reaching to infinity that is reminiscent of fractals. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, and Religion.
Bérczi, Sz. Symmetry and technology in ornamental art of old Hungarians and Avar-Onogurians from the archaeological finds of the Carpathian Basin, seventh to tenth century A.D. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 715--730. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 92K99), MR: 91a:01058b.
Analysis of symmetries can be very helpful in better understanding archaeological art and artifacts. The types of symmetries not only show what the author describes as "intuitive mathematical development in ornamental art" but can also help trace relationships between different communities. Such studies are now relatively new, but with time should become "an accepted, standard part of the description of archaeological finds". In this article, the author discusses how all 7 types of strip/frieze patterns occur in Old Hungarian ornamental art, and develops a notion of a double frieze pattern, which is intermediary between frieze patterns and plane patterns. A number of these patterns occur (sometimes individualized) in Avar-Onogurian artifacts. The author's classification of double frieze patterns focuses on how the patterns are generated horizontally and vertically, and may be more useful for archaeological purposes than classification by the related plane patterns. The author gives examples of some plane patterns that came up somewhat naturally, including patterns from weaving, chained ring structures, and the optimal fitting of furs (a pmg plane pattern). The author compares the frequencies of certain symmetry patterns in collections from several cultures. Closely related topics: Hungary in the Middle Ages, Frieze Patterns, Plane Patterns, Double Frieze Patterns, Archaeology, and Metal Work.
Brendan, Brother T. How Ptolemy Constructed Trigonometry Tables. Mathematics Teacher 58 (1965), 141--49.
Discusses how Ptolemy may have constructed his trigonometry tables, which in effect give a table of sines for every quarter degree between 0o and 90o correct to four decimal places. Ptolemy's first theorem shows how he could have constructed the chords of 36o and 72o. Ptolemy's second theorem can be used to find sum and difference angle formulas, and a half angle formula. Since the chord of 60o is simple, he can thus find chords of 12o, 6o, 3o, 3/2o, and 3/4o. The sticky part is then to find the chord of 1o [one sees this also in the Islamic world, where in one instance an approximate solution was found to a cubic]. Ptolemy uses a clever argument and the values for 3/2o and 3/4o to find an accurate answer for the chord of 1o. The table also includes a method to interpolate values of chords at every minute of arc (in effect, sines of every half minute). The author does not discuss the method of interpolation in detail. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ptolemy (Claudius Ptolemaeus), Trigonometry, and Interpolation.
Bruins, Evert M. The division of the circle and ancient arts and sciences. Janus 63 (1976), no. 1--3, 61--84. (Reviewer: J. L. Berggren.) SC: 01A15 (01A20), MR: 57 #12015.
One Etruscan cup, made in Caere about 500 BC, and now in the Museum of Fine Arts in Budapest, has both an 11-gon and a 14-gon inscribed on it. As the author notes, one possible reason why both were given together could be that the sum of the sides of an 11-gon and of a 14-gon imperceptibly deviates from the radius of a circle inscribing them. Moreover, methods known in the old Babylonian period could be used to provide excellent approximations to the lengths of the sides. All this raises questions about the level of Etruscan mathematical development, about which little is still known (their language still being poorly understood). The author also discusses Heron's rather accurate method for approximating the area of a circle. The article is very interesting, but the reader should be forewarned that it is a bit technical. Closely related topics: The Etruscans, Sumerians and Babylonians, The Circle, Polygons, and Heron.
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.
Chorbachi, W. K. In the tower of Babel: beyond symmetry in Islamic design. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 751--789. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A30 92K99), MR: 91a:01058c.
An interesting and personal account of how the author discovered geometric manuscripts written for Islamic artisans. With this discover, the author gives a new historical and scientific basis to the study of certain kinds of Islamic art. Much work preceding the author's had focused on religious, mystical, or perceptual interpretations of the work. Many ideas were primarily hypothetical, such as the (incorrect) idea that all Islamic art derives from the circle. The author suggests that many religious and mystical interpretations of Islamic geometric art should not be regarded as being historically based. Instead, the author shows how some Islamic art is highly mathematical, showing concerns with such topics as Pythagorean triangles and the notion of similarity (he gives an example where a shape appears in three different scales, each similar shape being derived from the last by a clever process). Much of the article discusses these in the context of a cyclic quadrilateral appearing in Islamic art with sides 1, 2, 2, 71/2. The author even noted an Islamic anticipation of a shape used to produce Penrose tilings. The author suggests that symmetry groups, while useful, can not alone give a full understanding of Islamic art. Closely related topics: The Islamic World, Art, Plane Patterns, Pythagorean Triangles and Triples, Penrose Tilings, Religion, and Mathematics and Mysticism.
Court, Nathan Altshiller. Mathematics in the History of Civilization. The Mathematics Teacher 41 (1948), 104--11.
How different concerns of society influenced mathematics. How the development of the concept of number is reflected in language. How the concept of how many led to arithmetic. How the concept of how much led to geometry. (Taxation and agriculture also contributed to both.) Efforts to keep time led to trigonometry. Navigation and associated astronomical problems led to logarithms [and more trigonometry]. Problems in artillery led to graphs. Both required an understanding of motion. Analytic geometry and calculus were invented in part to better understand motion. Statistics developed to understand problems in the social sciences. Also discusses the nature of mathematics: mathematics for its own sake and the axiomatic method. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Why Study History Of Math, Mathematics in Language, Number Systems, Arithmetic, Taxation, Agriculture, Astronomy, The Reckoning of Time, Trigonometry, Artillery, Graphing, Navigation, Dynamics, Force, and Motion, Analytic Geometry, Calculus, Statistics, Social Science, and Proof.
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, Bichromatic Strip 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, Bichromatic Strip Patterns, Plane Patterns, Pottery, Archaeology, The Islamic World, and Spain in the Middle Ages.
Crowe, Donald W. Erratum to: "The geometry of African art. I, II" (J. Geometry 1 (1971), 169--182; Historia Math. 2 (1975), 253--271). Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics (Boston, Mass., 1974). Historia Math. 2 (1975), no. 4, 617. (Reviewer: M. P. Closs.) SC: 01A15 (20H15), MR: 58 #9986c.
The articles Crowe, Donald W., The geometry of African art and Crowe, Donald W., The geometry of African art interchange the names of the symmetries p3m1 and p31m in several places. Closely related topic: Plane Patterns.
Crowe, Donald W. The geometry of African art. III. The smoking pipes of Begho. The geometric vein, pp. 177--189, Springer, New York-Berlin, 1981. (Reviewer: M. P. Closs.) SC: 01A10 (51M20), MR: 84b:01004.
Introduces the strip and plane patterns. Gives a useful flowchart for recognizing them (and some examples). Then classifies the patterns appearing in smoking pipes from the Krama quarter of Begho, in Ghana. The most common strip pattern is the one usually referred to as pmm2 (number 7 in the author's own system). The most common plane patterns are pmm and p4m. As the author notes, both of these can be easily created as rows of pmm2 strips. Representatives of all 7 strip patterns were found, but only 7 of the 17 possible plane patterns occurred. The author also considered questions on the relative preponderance of the various strip types by four different levels in the dig; no noticeable differences were found. Closely related topics: Ghana, Frieze Patterns, Plane Patterns, and Archaeology.
Crowe, Donald W. The geometry of African art. II. A catalog of Benin patterns. Historia Math. 2 (1975), 253--271. (Reviewer: M. P. Closs.) SC: 01A15 (20H15), MR: 58 #9986b.
Discusses the strip patterns and plane patterns occurring in Benin art. All 7 strip patterns and 12 of the 17 frieze patterns occur, though about five of the frieze patterns which do occur are rare: two may only occur once, and one of these may be based on a European model. The author compares the Benin patterns with the Bakuba patterns. Glide reflections are more rare in Benin art than in Bakuba art, possibly because glide reflection symmetries may arise most naturally from weaving patterns. Benin art also tents to be more representational, Bakuba art more abstract. The author also considers Benin patterns to be less varied than Bakuba patterns. However, it appears that the bronzework itself is nearly unsurpassed. A catalog is given with most of the strip patterns the author has found in Benin art, along with one example of each of the 12 plan patterns that occur. The author does not discuss this, but some patterns combine elements of different symmetries: the authors example of a p1 symmetry would have been classified differently if either of its two motifs were removed. Also see the erratum, Crowe, Donald W., Erratum to: "The geometry of African art. Closely related topics: BeninCity, Nigeria, Frieze Patterns, Plane Patterns, The Bakuba of Zaire, Weaving, and Bronzework.
Crowe, Donald W. The geometry of African art. I. Bakuba art. J. Geometry 1 (1971), 169--182. (Reviewer: M. P. Closs.) SC: 01A15 (20H15), MR: 58 #9986a.
Discusses strip and plane patterns occurring in Bakuba art, particularly in textiles and woodcarving. The inspiration for many of these patterns seems to be from weaving, but at least one pattern may originate in the technique of sewing together triangles to make bark cloth. All seven strip patterns occur, and 12 of the 17 possible plane patterns. Discusses the relative proportions of some of these patterns, and gives an example of each. In all but one of the strip patterns, the author gives both cloth and carved examples (the other is given in cloth only, being rare in wood). The author includes an appealing claim about one of the patterns, made by an earlier researcher (too enthusiastic in the view of the authors): "it is probably the most remarkable example of this kind... its discovery is certainly a mathematical accomplishment of the first magnitude." Also see the erratum, Crowe, Donald W., Erratum to: "The geometry of African art. Closely related topics: The Bakuba of Zaire, Frieze Patterns, Plane Patterns, Weaving, and Wood Carving.
Dibble, William E. A possible Pythagorean triangle at Stonehenge. J. Hist. Astronom. 7 (1976), no. 2, 141--142. (Reviewer: C. R. Fletcher.) SC: 01A10, MR: 58 #20990a.
Dibble notes that one triangle at Stonehenge is rather close to a 5,12,13 Pythagorean right triangle. The conclusion is bound to be controversial, and Dibble is cautious about making definite claims. Closely related topics: The Stone Builders and Pythagorean Triangles and Triples.
Doczi, György. Seen and unseen symmetries: a picture essay. Symmetry: unifying human understanding, I. Comput. Math. Appl. Part B 12 (1986), no. 1-2, 39--62. SC: 92A27 (01A99 52-01), MR: 838 136.
Certainly an unorthodox essay. It may be hard to understand the author's terms dinegy and dinergic symmetry (involving the union of complementary opposites), at least in a concrete mathematical sense, but the discussion and pictures do emphasize how mathematical proportions can pervade both art and the natural world. Closely related topics: Proportion and the Golden Ratio and Symmetry.
Emmer, Michele. Art and mathematics: the Platonic solids. The Visual Mind, 215--220, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
The author begins by mentioning some ancient representations of Platonic solids. These include a pair of Egyptian die from the Ptolemaic dynasty, an Etruscan dodecahedron (at least 2500 years old), two Celtic dodecahedra, and a West German dodecahedron from the 2nd century BC. The author continues with a discussion of the regular solids in Plato's Timaeus. The author notes that Dürer's Melancholia, which includes a truncated rhombohedron, is sometimes thought to show the influence of Luca Pacioli. The magic square in the painting gives some evidence for this; Dürer's engraving may be one of the earliest depictions of a magic squares in the West, but an earlier manuscript by Pacioli showed an interest in them. On the other hand, Luca Pacioli's De Divina Proportione relied heavily on, and perhaps even appropriated the work of Piero della Francesca. The book is also notable for its pictures of the regular solids, attributed to Leonardo da Vinci. Also discusses work on the regular solids due to Johannes Kepler, including Kepler's recognition of a duality and his idea of a combination of two tetrahedra called a stella octangula. The author notes that the notion of the stella octangula also appears in Pacioli's De Divina Proportione. In addition, Kepler's stellated dodecahedron occurs in mosaics in the San Macro Cathedral in Venice; this work is thought to have been done by Paolo Uccello. Regarding Uccello, the author quotes Donatello as saying to his close friend "Ah Paolo, this perspective of yours makes you neglect what we know for what we don't know. These things are no use except for marquetry." (The source is Vasari's Vita di Paolo Uccello.) The author, Michele Emmer, collaborated on the film Art and Mathematics. Closely related topics: The Regular Solids, Plato, Art, The Etruscans, Germany in Ancient Times, The Celts, Albrecht Dürer, Luca Pacioli, Magic Squares, Piero della Francesca, Leonardo da Vinci (1452-1519), Paolo Uccello (1397-1475), Johannes Kepler (1571-1630), and Perspective.
Engels, Hermann. Quadrature of the circle in ancient Egypt. Historia Math. 4 (1977), 137--140. (Reviewer: L. Guggenbuhl.) SC: 01A15, MR: 56 #5124.
Explains the Egyptian formula for the area of a circle in terms of the practices of Egyptian stone masons. In order to form a relief, the stone masons covered their designs with a grid. The hypothesized construction involves an error which would confirm the now commonly held view that the ancient Egyptians did not properly understand the Pythagorean theorem. Closely related topics: Ancient Egypt, The Circle, Coordinates, and Pythagorean Triangles and Triples.
Eves, Howard. Omar Khayyam's Solution of Cubic Equations. Mathematics Teacher 51 (1958), 285--86.
Shows how Omar Khayyam solved the equation x3+b2x+a3=cx2 using the intersection of a circle and a rectangular hyperbola. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Omar Khayyam (abu-l-Fath Omar ibn Ibrahim Khayyam), Cubics, and The Conic Sections.
Eves, Howard. On the Practicality of the Rule of False Position. Mathematics Teacher 51 (1958), 606--8.
Eves shows how the method of false position can be simpler than our own methods by giving one example from the Ahmes Papyrus, three from the Greek Anthology of c. 500 AD, and two of his own. One of his examples is from surveying, and Eves says that it is the method a surveyor would probably use. In the other example of his own, he likens the rule of false position to the method of similitude in geometric constructions. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Method of False Position, Ancient Egypt, Medieval Europe, and Surveying.
Fields, Margaret. Practical Mathematics of Roman Times. Mathematics Teacher 26 (1933), 77--84.
Surveys Roman mathematics. Some of the most interesting examples come from the De Architectura of Vitruvius, which discusses principles of symmetry and proportion and how to use them in architecture. Vitruvius goes as far as how to correct for an optical illusion on the capitals of columns. He also discusses geometric procedures to be used in laying out a town (to shut out winds), and various Roman instruments, including leveling instruments and an instrument for measuring distance called a hodometer. The hodometer is used for "telling the number of miles while sitting on a carriage or sailing by sea", and is particularly ingenious. Second to Vitruvius, the most important source on Roman engineering may be the Urbis Romae of Frotinus, which includes mathematical rules (not entirely successful) to determine the flow of an aqueduct. Surviving Roman bridges show a high level of skill; there were surely mathematical principles behind their design, but no detailed study has survived. Roman tunnels are equally impressive. Heron discusses how to use an instrument called the "dioptra" to survey for tunnels, measure the width of a river, and so on. Roman sundials were relatively unsophisticated. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Vitruvius, Architecture, Symmetry, Proportion and the Golden Ratio, Optics, Leveling, The Measurement of Distance, Frotinus, Heron, Surveying, and The Sundial.
Fletcher, E. N. R. The area of the curved surface of a hemisphere in ancient Egypt. Math. Gaz. 54 (1970), no. 389, 227--229. SC: 01A15, MR: 58 #9987.
Problem 10 of the Moscow papyrus discusses the surface area of a basket and is thought by some to compute the surface area of a hemisphere. The author analyzes which units may have been used in the problem, and advances the theory that the basket in question was, in fact, hemispherical, and was designed to hold 100 Hekat of corn. He notes that the units used in ancient Egypt appear to have some interesting geometrical properties. For example, a circle with a radius of 1 pes (or "foot", equal to 16 digits) was approximately equal in area to a square with sides measuring 1 royal cubit. These are all fascinating possibilities. Closely related topics: Ancient Egypt, Surface Area, The Sphere, and The Measurement of Area and Volume.
Gerdes, P. Reconstruction and extension of lost symmetries: examples from the Tamil of South India. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 791--813. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 92K99), MR: 91a:01058d.
Gerdes discusses the designs drawn (or formerly drawn) by Tamil women in South India during the harvest month Margali. The author shows that some of the diagrams may be degradations of earlier patterns that display more symmetry and/or are constructed according to the cultural ideal of having only one line. Gerdes also discusses drawing algorithms; many algorithms work by applying a series of simple transformation rules to a simpler motif. The function of these diagrams appears to be religious. As the author explains, "Margali is the month in which all kinds of epidemics were supposed to occur. Their designs serve the purpose of appeasing the god Siva who presides over Margali." Closely related topics: The Tamil of South India, Continuous Tracing Problems, Symmetry, and Religion.
Gerdes, Paulus. Fivefold symmetry and (basket) weaving in various cultures. Fivefold symmetry, 245--261, World Sci. Publishing, River Edge, NJ, 1992. SC: 52B99 (01A07), MR: 1 178 750.
Gerdes suggests that five-fold symmetries arose from efforts to solve problems in basketweaving rather than in observations of five-fold symmetry in natural phenomena (such as starfish). One way five-fold symmetries can arise is by modifying the more obvious six-fold symmetries (such as those used by peasants in Mozambique) to fit a curved surface. The author reports that "these pentagonal-hexagonal baskets are, for instance, also woven by the Ticuna and Omagua Indians (northeastern Brazil), by the Huarani Indians, by the Kha-ko in Laos, and by the Menda in India. One sees them also in China, Japan, and Indonesia." The Malaysian sepak tackraw ball is similar to the soccer ball and is woven in the same way. The author reports that the peasants of the island Roti (Indonesia) may have discovered a way to fold a regular pentagon as a kind of a thimble. The author shows how a similar pentagonal weaving pattern is used in weaving brooms in Mozambique. (A near pentagram then appears inside the knot.) The author notes that a similar method is used in Angola to hold together the bars of a cage. The author in addition discusses how hat weaving techniques can lead naturally to three- and five-fold symmetries. The author's main example is with the hats of the Belu of central Timor, but he notes that related techniques are used in northern Mozambique, southern Tanzania, and by the Kuva of Congo. The author also shows a Chinese hat with five-fold symmetry. Two other particularly interesting examples are "a burden basket ... from the Papago Indians (Arizona) which combines beautifully a global sevenfold symmetry with local fivefold symmetry", and the "center of a Japanese basket, which combines global ninefold symmetry with local fivefold symmetry." Closely related topics: Five Fold Symmetry, Basket Making, Mozambique, Malaysia, and The Belu of Central Timor.
Gerdes, Paulus. On mathematics in the history of sub-Saharan Africa. Historia Math. 21 (1994), no. 3, 345--376. SC: 01A13, MR: 95f:01003.
This paper broadly surveys the recent research in sub-Saharan mathematics (and some related areas as well). Areas discussed include prehistoric mathematics (e.g., the Ishango and Border Cave bones), number systems and symbolism (including algorithms and education), games and puzzles (for example, a leopard-goat-cassava leaf river crossing problem and a "topological" puzzle), symmetry in African art, graphs or networks (e.g. Tschokwe sand drawings), architecture (one case involving magic squares; also a brief reference to fractals). Gerdes mentions string figures as a possibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies on language and mathematical concepts. A goal of the studies mentioned is apparently to better understand mathematics learning in Africa. Some studies focus on logic. Questions on interaction with ancient Egypt are still largely open. A better understanding of Islamic mathematics in sub-Saharan Africa is desirable as well. The author also touches on factors connected with the slave trade; e.g., the remarkable but not perhaps entirely atypical abilities of Thomas Fuller. Includes an extensive bibliography. Closely related topics: Sub-Saharan Africa, TallySystems, Games, Puzzles, Topology, Symmetry, Continuous Tracing Problems, Architecture, Magic Squares, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Education, Mathematics in Language, Logic, The Islamic World, and Thomas Fuller (1710-1790).
Gerdes, Paulus P. J. On ethnomathematical research and symmetry. Symmetry in a kaleidoscope, 2. Symmetry Cult. Sci. 1 (1990), no. 2, 154--170. SC: 01A07, MR: 1 188 949.
Gerdes begins with a discussion of why symmetry is such a common phenomenon in human culture. He notes that some symmetries which are rare in nature (e.g., rotational symmetries of order 2) are common amongst us. Gerdes gives the example of rotational symmetry being used in the tattoos of the Makonde of northern Mozambique. Gerdes explains how symmetries such as the rotational symmetry of order 2 can arise naturally in solving problems in such areas as weaving. Gerdes then turns to the geometry of the line drawings made by the Tamil women in South India (during harvest month) and those made by the Tshokwe. These drawings have some strong similarities, and in both cases show an interest in tracing out a figure with a single continuous line. They also show a strong interest in symmetry, and Gerdes gives examples of how designs which fail to follow the one-line cultural norm may also fail to display the expected symmetries, suggesting that such drawings are degradations of more symmetric ones drawn with one line. The author advances a construction principle that can be used to construct both the Tamil and Tshokwe patterns. (Although the author doesn't note this, it is interesting that this principle is very similar to another principle that has been advanced for Celtic knot friezes!) Gerdes then discusses some mathematical properties of curves made using his construction principle. He also discusses some other interesting topics in his ethnomathematical research. For example, the author mentions that he has a found a new hypothesis on the origin of the Egyptian formula for the volume of a truncated pyramid, and has also found an infinite series proof for the Pythagorean theorem. Closely related topics: Symmetry, The Tamil of South India, TheTshokwe, Continuous Tracing Problems, The Celts, Ancient Egypt, and Pythagorean Triangles and Triples. Also possibly relevant: Mozambique, Tattoos, and Weaving.
Gerdes, Paulus and Bulafo, Gildo. Sipatsi. Technology, art and geometry in Inhambane. Translated from the Portuguese by Arthur B. Powell and Gerdes. Instituto Superior Pedagógico, Ethnomathematics Research Project, Maputo, 1994. 102 pp. (Reviewer: J. S. Joel.) SC: 01A07 (00A08 00A69 01A13 51M20), MR: 95f:01002.
The authors discuss the construction and mathematical properties of the Mozambican sipatsi, which are essentially woven handbags. They are generally decorated with strip or frieze patterns, and in fact all 7 possible types of strip patterns occur in the sipatsi from Inhambane province in Mozambique. This book includes a description of the processes used to create the sipatsi, a catalog of the strip patterns found, and a chapter designed for people using the sipatsi to teach mathematics. The authors also give just a few examples of strip patterns on wooden spoons (also from Inhambane province) and on vases and pots (from Maputo). Closely related topics: Mozambique, Basket Making, Frieze Patterns, and Education.
Gillings, R. J. The Volume of a Truncated Pyramid in Ancient Egytian Papryi. Mathematics Teacher 57 (1964), 552--55.
Gillings gives a clever way to derive the formula V=1/3(a2+ab+b2) for the volume of a truncated pyramid, using only the formula for the volume of a complete pyramid and other methods that the Egyptians had at their disposal. As he shows, fairly simple arguments suffice when b=a/2,a/3,..., and also when b=2/3a. Since to the Egyptians, every number could be represented as a finite sum of unit fractions, the demonstration is now complete. Of course we (or the Greeks) would require something like the method of exhaustion. (Even without it, the jump to a general number is a difficult step, and not trivial geometrically.) (Since in the Moscow papyrus, b=a/2, one might wonder if perhaps the Egyptians did not know the general case after all.) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Ancient Egypt, The Pyramid, The Measurement of Area and Volume, and The Method of Exhaustion.
Grünbaum, Branko. The emperor's new clothes: full regalia, G-string, or nothing? With comments by Peter Hilton and Jean Pedersen. Math. Intelligencer 6 (1984), no. 4, 47--56. (Reviewer: H. S. M. Coxeter.) SC: 01A15 (01A60 05B45 20F32 52A45), MR: 86d:01004.
Grünbaum's article: The author discusses the common misconceptions that the Egyptians and the artists of the Alhambra had used all 17 types of plane patterns. In fact, the Egyptians appear to have missed the five symmetry groups which have three-fold rotations. The sources for these misconceptions are discussed as well. The author has done fairly extensive research on the subject, and has concluded that two of the four plane patterns missing from the Alhambra seem not to appear at all in Islamic art (these are pg and pgg; the two missing at the Alhambra but present elsewhere are p2 and p3m1). A final theme of the author's is that the language of symmetry groups may at times be inadequate to discuss patterns, and can also be misleading in connection with the intentions of the artists themselves.The response by Peter Hilton and Jean Pedersen: The author's acknowledge Grünbaum's correction about the Egyptians. The authors note that the Egyptians and Moore's between them only missed one symmetry group, p3m1. They comment briefly on Chinese and Japanese designs, and quote Schattschneider, who notes that Chinese and Japanese artwork features rotations and glide reflections much more strongly than Islamic art does. Schattschneider also cites an illustration from a Japanese book that seems to suggest that underlying lattices of squares, equilateral triangles, rhombuses, and parallelograms were consciously used in developing symmetry patterns. The authors acknowledge the limitations of group theory in discussing symmetry, but also emphasize its usefulness. Closely related topics: Plane Patterns, Ancient Egypt, The Islamic World, Penrose Tilings, Japan, and China.
Groemer, H. The symmetries of frieze ornaments in Maya architecture. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 203 (1994), 101--116 (1995). (Reviewer: J. S. Joel.) SC: 01A12 (00A69 01A07 51M20 52C20), MR: 96b:01006.
The author discusses the frieze patterns that occur in Mayan architecture, with occasional references to the frieze patterns found on Mayan pottery. All seven basic types of frieze patterns occur, though one (the one with only glide reflections) is rather rare. The author notes that many of the symmetries appear to be derived from the symmetries of the same base motif, which is merely translated; it is acknowledged that this distinction is not a mathematical one. The author also distinguishes between discrete and continuous patterns. One interesting pattern is classified as having only translations and vertical reflections, but as the author notes, the "negative space" has an upside-down version of the same ornament This particular pattern could be classified as a bichromatic strip pattern, but apparently the "negative space" symmetry is lost in some other examples of the motif. The author finds, to his surprise, that there seems to be little Toltec or Zapotek-Mixtec influence in the Mayan frieze patterns. Closely related topics: The Maya, Architecture, Frieze Patterns, and Pottery.
Grünbaum, Branko, Grünbaum, Zdenka; Shephard, G. C. Symmetry in Moorish and other ornaments. Computers \& Mathematics with Applications. Part B 12 (1986), no. 3--4, 641--653.
It is observed that 13 of the 17 plane patterns are represented at the Alhambra. Two of the four missing groups have been found in Toledo, Spain, and dating from about the same period (one, p3, was found in a church, and the other, p3m1, was found in a synagogue). The authors note that the remaining two patterns (pg and pgg) seem not to appear in Islamic art at all. The authors note that features of Islamic art are not always fully described by the symmetry groups alone; such features can include color changes and interlace patterns. The color-symmetry groups are only a partial solution to the former, since colors are often in ratios "2:1:1, 4:2:1:1, 6:2:1, 6:3:1:1:1 or some similar ratio... The mathematical theory of such colorings still awaits development." The authors also attack the commonly held view that the artists of the Alhambra exhausted the possibilities of symmetry in art, and illustrate their points with pictures. Moreover, the authors suggest that ideas of local structure are as important as ideas of global structure. "The various kinds of symmetry groups are useful in the description of many of the artifacts, but more general approaches (based on 'adjacency relations' or other 'local' criteria) are necessary for a better understanding of the ornaments and artwork, and of the ways their creator thought about them." Closely related topics: The Islamic World and Plane Patterns.
Grünbaum, Branko; Shepard G.C. The geometry of fabrics. Geometrical Combinatorics, 77--97, Pitman, Boston, 1984.
Symmetry groups have already been used in discussing mathematical properties of textiles, but may not be appropriate for all kinds of fabrics. In this article, the author discusses primarily isonemal fabrics. In these, the symmetry group acts transitively on the strands of the fabric. Many fabrics that actually occur in nonmathematical discussions are actually isonemal. This article (and for example its classification of isonemal fabrics into 5 genera) could form part of the foundation for a new ehtnomathematical research area. In addition, a number of interesting combinatorial questions arise. The article focuses primarily on traditional fabrics, with two perpendicular layers. The terminology of the article may be less appropriate for the three layer fabrics that sometimes occur in basketweaving. "It is interesting to note that in the case of 3-way 3-fold fabrics some 'partial fabrics', that is parts of fabrics that do not hang together, are used in basketry... The classification of such fabrics seems to be a totally unexplored area of the subject." With regard to 3-way fabrics, he notes that "it seems that such fabrics are more stable under diagonal strain than 2-way 2-fold fabrics, and so have been used in such practical applications as parachutes." Similar concerns may of course account for their occurrence in basket making as well! Closely related topics: Weaving, Combinatorics, Symmetry, and Basket Making.
Grünbaum, Branko; Shephard G. C. Interlace patterns in Islamic and Moorish art. The Visual Mind, 147--155, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
Many Islamic and Moorish patterns exhibit what the authors call interlace patterns, where the patterns seem to be made of strands that alternately go over and other strands. This is a phenomenon that makes these Islamic artworks appear something like a 2-D extension of the Celtic knot friezes; the over/under rule is of course also common in weaving. The authors focus on the seemingly curious phenomenon that many of the Moorish and Islamic interlace patterns can be viewed as being made of a small number of basic shapes, often one or two. The authors analyze this phenomenon for the symmetry groups p4m and p6m, and find that it arises in a mathematically natural way, especially if artists used stencils, as is sometimes now thought. The article gives give propositions without proof; proofs of these should be within reach of a good undergraduate with the requisite knowledge of group theory. Closely related topics: The Islamic World, Plane Patterns, Knots and Knotwork, and Weaving.
Hargittal, István and Lengyel Györgi. The seven one-dimensional space-group symmetries illustrated by Hungarian folk needlework. Journal of Chemical Education 61 (1984), 1033.
All 7 frieze patterns can be found in Hungarian needlework. The authors give an example of each pattern. A related article by the same authors is Hargittal, István and Lengyel Györgi, The seventeen two-dimensional space-group symmetries in Hungarian folk needlework. Closely related topics: Frieze Patterns, Hungary, and Needlework.
Hargittal, István and Lengyel Györgi. The seventeen two-dimensional space-group symmetries in Hungarian folk needlework. Journal of Chemical Education 62 (1985), 35--36.
The Hungarians of the late 1800s may be among the earliest people known to have "discovered" all 17plane patterns. The authors give an example of each pattern from Hungarian needlework. For the related article on frieze patterns, see Hargittal, István and Lengyel Györgi, The seven one-dimensional space-group symmetries illustrated by Hungarian folk needlework. Closely related topics: Frieze Patterns, Hungary in the 1800s, and Needlework.
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.
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 Strip 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 Strip Patterns, Bichromatic Plane Patterns, Rotational Symmetry Groups (Rosettes), Penrose Tilings, Weaving, Similarity, and M. C. Escher.
Jones, Phillip S. Irrationals or Incommensurables. I. Their discovery, and a "Logical Scandal". Mathematics Teacher 49 (1956), 123--27.
The discovery of irrationals. Discusses an appealing theory, due to Kurt von Fritz, that the discovery of irrationals grew out of a study of the pentagram. Von Fritz is in support of the traditional theory that discovery or irrationals was due to Hippasus of Metapontum. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Irrationals, The Pentagram, and Hippasus of Metapontum.
Jones, Phillip S. Irrationals or Incommensurables. III. The Greek solution. Mathematics Teacher 49 (1956), 282--85.
Shows how Eudoxus' Method of Exhaustion is used to prove that circles are to one another as the squares on their diameters. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Method of Exhaustion, Eudoxus, The Measurement of Area and Volume, and The Circle.
Jones, Phillip S. Recent Discoveries in Babylonian Mathematics. II. The Earliest Known Problem Text. Mathematics Teacher 50 (1957), 442--44.
Continues Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. I.. Discusses a very old Babylonian problem text (c. 2000 BC), that seems to show an understanding of the proportionality of sides in similar right triangles. Continued in Jones, Phillip S., Recent Discoveries in Babylonian Mathematics. III., which has a different character from both of its predecessors. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, Similarity, and The Triangle.
Kemp, Martin. Spirals of life: D'Arcy Thompson and Theodore Cook, with Leonardo and Dürer in retrospect. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 37--54. SC: 01A99 (92-03), MR: 96j:01047.
Discusses theories of how art appears in biology. The author starts with St. Augustine, who concluded "If, then, we argue from the facts, first, that as everyone admits, not a single visible organ of the body serving a definite function is lacking in beauty, and, second, that there are some parts which have beauty and no apparent function, it follows, I think, that in the creation of the human body God put form before function." The author then discusses and compares the investigations of D'Arcy Thompson and Theodore Cook into the mathematical/biological manifestations of the spiral. Thompson and Cook agreed on many issues, though Thompson didn't approve of the "mystical conceptions" that he found in Cook's work. Specific topics discussed include the appearance of the golden ratio in biological systems (often in the guise of the Fibonacci series), turbulence, and transformations that take one biological object into a related one (one of Thompson's examples compares the skulls of Hyrachyus agrarius and Aceratherium tridactylum). In the process, the author touches on the work of Albrecht Dürer and Leonardo da Vinci (as the title suggests). Obviously, this article can not to be comprehensive, and the author himself tells us that the article is itself intended as a preface; it serves this function well. Both Thompson and Cook were well aware of the mathematical difficulties involved in thoroughly understanding the phenomena they wrote of. Cook wrote "It would only be possible to imagine life or beauty as being 'strictly' mathematical" if we ourselves were such infinitely capable mathematicians as to be able to formulate their characteristics in mathematics so extremely complex that we have never yet invented them." And Thompson wrote "And just as in the very simplest of actual cases we meet with a departure from such symmetry as could only exist under conditions of ideal simplicity, so do we pass quickly to cases where the interference of numerous, though still perhaps very simple, causes leads to a resultant which lies beyond our powers of analysis." The author writes that Thompson ended his book with "a plea for biological mathematicians and mathematical biologists to cultivate 'a field which few have entered and no man has explored'". He continues "Thompson's plea did not fall upon deaf ears, but it is only recently that new techniques of computer modeling have begun to realize something of the potential of some of his techniques." Closely related topics: Art, Biology, Spirals, Topology, Proportion and the Golden Ratio, Albrecht Dürer, and Leonardo da Vinci (1452-1519).
Knight, Gordon. The geometry of Maori art---Rafter Patterns. New Zealand Math. Mag. 21 (1984), no. 2, 36--40.
The Maori have been fond of carving patterns on their rafters. The author wondered if all seven possible strip or frieze patterns occur in the work of the early Maori, and he found in fact that they do. There is also a brief discussion of the seven types of strip patterns and a flowchart for recognizing them. The author's source was the book Maori Art by A. Hamilton (N.Z. Institute, Wellington, 1901), which is now reprinted by Holland Press, London, 1972. Hamilton's book illustrates 29 rafter patterns, and these turned out to have had only six of the seven patterns; fortunately the one with only vertical reflections turned out in a photograph elsewhere in the book, in "part of the porch of a large house of the Ngati-Porou at Wai-o-Matatini". The author lists two questions that he does not have answers to: What was the relative frequency of each group, and did this vary from one tribal region to another? Also, is there a geometrical difference in character between the early Maori patterns and those produced after the influence of the Pakeha? Closely related topics: The Maori, Wood Carving, and Frieze Patterns.
Knight, Gordon. The geometry of Maori art---spirals. New Zealand Math. Mag. 22 (1985), no. 1, 4--7. (Reviewer: H. S. M. Coxeter.) SC: 51N20 (01A10), MR: 87m:51060.
The Maoris frequently use spirals in their tattoos and wood carvings. These appear very much like the spirals of Archimedes, but often interlace two or more such spirals. Although the easiest way to construct a spiral similar to the spiral of Archimedes may be to use sets of concentric semicircles (or other segments of circles) offset with respect to one another, the author believes that the Maoris didn't use this technique. "In Spirals of Archimedes, and, it seems, in Maori spirals, there is a gradual, rather than an abrupt, change in curvature." The author gives several examples from Maori artwork; there are examples with 2, 3, and 4 interlaced spirals. The author notes that the 3 spiral form is more common in tattooing patterns than in carving. Apparently there was once a 6 spiral pattern on one of the figures guarding the gateway of Papawai Pa. The center of the spiral can be varied somewhat; for example, two spirals can come together in an S-curve. In one case, "the plain ridges, which form an S-curve, are made to cross over the notched spirals, giving a woven effect. According to Phillips this was chiefly an Arawa modification." The author concludes with a note that the spiral of Archimedes should perhaps have a Maori name instead. He suggests that an investigation of these spirals might be useful in mathematics education (when polar coordinates are studied). Closely related topics: Spirals, The Maori, Tattoos, Wood Carving, Archimedes, and Education.
Knight, Gordon. The geometry of Maori art---weaving patterns. New Zealand Math. Mag. 21 (1984), no. 3, 80--86. (Reviewer: H. S. M. Coxeter.) SC: 51N20 (01A10), MR: 87m:51059.
If one restricts only to 90 degree weaving, only 12 of the 17 plane patterns are possible as symmetry groups. 10 of these 12 plane patterns are represented in Maori art. The article gives an example of each. There is also a simple flowchart for recognizing the 17 symmetry groups of the plane patterns. As an additional aid in recognition, the author also includes a couple of examples of plane patterns which he labels with possible translation vectors, points of rotation, and lines that can be used in reflections and glide reflections. The author does not discuss whether weaving of the 120 degree type occurs in Maori art. Closely related topics: The Maori, Weaving, and Plane Patterns.
Knorr, W. R. The geometer and the archaeoastronomers: on the prehistoric origins of mathematics. Review of: Geometry and algebra in ancient civilizations [Springer, Berlin, 1983; MR: 85b:01001] by B. L. van der Waerden. British J. Hist. Sci. 18 (1985), no. 59, part 2, 197--212. SC: 01A10, MR: 87k:01003.
The reviewer discusses van der Waerden's book Geometry and Algebra in Ancient Civilizations. Although the reviewer clearly admires van der Waerden for his work in algebra and in the history of mathematics in general, he is highly critical of the conclusions reached in van der Waerden's book. A basic theme of the book is that there is a pre-Babylonian ancestor to mathematics in Babylonia, ancient Egypt, Greece, China and India; thus the book can therefore be thought of in part as a further development of Abraham Seidenberg's theories on the ritual origins of ancient mathematics. The reviewer takes issue with several facts cited in the book, and in addition with three assumptions that he sees van der Waerden using explicitly or implicitly in the book: "(1) independent discovery is so rare that it may effectively be discounted as a working hypothesis for relating technical traditions; (2) derivative traditions are inferior to their source traditions; (3) borrowing from one tradition to another is not selective, but entails the adoption of whole bodies of technique." (The phrase "inferior to" in (2) could just as well be replaced by "degraded in".) The reviewer suggests in addition that van der Waerden has not been sufficiently critical in accepting claims by Alexander Thom and others about advanced mathematics in megalithic monuments, and sees these claims as forming "the veritable linchpin of van der Waerden's thesis". The author briefly discusses some of Thom's work in megalithic mathematics, and concludes that he finds no real evidence of the Pythagorean theorem, the ellipse, or a standard unit of distance in neolithic times. The review concludes with the statement "I fear even more the regrettable impact on credulous nonspecialists who may not know to distinguish between the general enterprise of scientific research and the reckless notions of some scientists." Closely related topics: Sumerians and Babylonians, Ancient Egypt, Greece, China, India, The Stone Builders, Alexander Thom, and Pythagorean Triangles and Triples.
Llyod, Daniel B. Further Evidences of Primeval Mathematics. Mathematics Teacher 57 (1966), 668--70.
A tablet from a dig at Tel Dhibayi near Baghdad shows how to find the dimensions of a triangle from its diagonal and area. The solution requires a knowledge of the Pythagorean theorem, and artfully sidesteps the difficulty of solving a quadratic equation by solving a pair of simple linear equations. Many other articles discuss similar tablets and solutions, but few so concisely as this. However, note that in the context of other Babylonian sources, the method of solution may be less obscure than the author seems to suggest. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Sumerians and Babylonians, The Quadratic Formula, and Pythagorean Triangles and Triples.
Loeb, A. L. The magic of the pentangle: dynamic symmetry from Merlin to Penrose. Symmetry 2: unifying human understanding, Part 1. Comput. Math. Appl. 17 (1989), no. 1-3, 33--48. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 52-03), MR: 91a:01058a.
In this interesting and entertaining article, Merlin the magician assists Arthur and Key in exploring the secrets of dynamic symmetry (in a problem with four beetles in a square always walking towards each other), in the logarithmic spiral (the curve generated by the beetles), the golden rectangle (and its own associated spiral), and the Fibonacci numbers. The article closes with a discussion of the pentangle, which the author says "is central to the late fourteenth-century 'Sir Gawain and the Green Knight', to medieval sign theory as well as to recent research in quasi-periodic alloy crystals. The Socratic discussions here could be turned used as active learning exercises for talented students. Highly recommended. Closely related topics: England in the Middle Ages, Dynamic Symmetry, Spirals, Proportion and the Golden Ratio, Leonardo of Pisa (Fibonacci), The Pentagram, and Education.
Lumpkin, Beatrice. Note: the Egyptians and Pythagorean triples. Historia Math. 7 (1980), no. 2, 186--187. SC: 01A15, MR: 81c:01004.
The author notes that some ancient Egyptian problems suggest a knowledge of certain Pythagorean triangles. For example, in the Berlin Papyrus there are problems where a given square is to be written as the sum of two squares in a given ratio. The solutions involve the fact that 62+82=102 and 122+162=202; these facts are familiar to us from our knowledge of the (3,4,5) right triangle. She also notes that the Egyptian units of measurement suggest a knowledge of the Pythagorean theorem in the special case of an isosceles right triangle. "The double remen is the diagonal of a square whose side was one cubit. By changing the units of measurement from cubits to double remens, the area of a figure would be doubled." Closely related topics: Ancient Egypt and Pythagorean Triangles and Triples.
MacDonnell, Joseph. Jesuit geometers. A study of fifty-six prominent Jesuit geometers during the first two centuries of Jesuit history. With a foreword by John W. Padberg. Studies in Jesuit Topics. Series IV, 11. Institute of Jesuit Sources, St. Louis, MO; Vatican Observatory Publications, Vatican City, 1989. iv+106 pp. ISBN: 0-912422-94-7. (Reviewer: J. V. Field.) SC: 01A99 (01A40 01A45 01A50 51-03), MR: 92e:01082.
One can find in this book that Jesuit mathematicians and scientists made interesting and important contributions not only in geometry, but in a number of other areas, some closely related and some not. One particularly interesting figure in the book is Anthanatius Kircher (1602-1680). He, for example, did such diverse creative things as design a water organ, observe tiny animals under a microscope that he connected with the bubonic plague, and showed the Egyptian Coptic language "was a vestige of early Egyptian. ... In fact it was because of Kircher's work that scientists knew what to look for when interpreting the Rosetta stone." Not every person discussed made discoveries that are quite so sensational, of course, but those who didn't still help give a more full view of mathematics in the periods the the author is interested in. Naturally, in a book of this length it is impossible to go into all Jesuit contributions in detail (and the reviewer was perhaps a little too harsh in his criticisms). Closely related topic: Religion.
Mainzer, Klaus. Symmetry and beauty in arts and mathematical sciences. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 91--103. SC: 01A99 (00A69), MR: 96h:01043.
As this article explains, symmetry appears in a variety of disciplines over a variety of ages. The author begins by briefly discussing the natural and philosophical reasons for studying symmetry (starting in ancient Greek times). He then discusses the appearance of the 7 frieze groups and 17 ornamental groups of the plane and related groups in mathematics and crystallography. Next, he discusses appearances of symmetry and symmetry breaking in modern physics, in the theory of relativity, and in quantum mechanics and superstring theory. He finds that symmetry considerations are important in chemistry and biology as well: "In biochemistry macromolecules (for example L-amino acids or D-sugars) possess a characteristic homochirality ('dissymetry') which is assumed to be caused by parity violations of weak atomic forces." He also explains that "The emergence of pattern structure can be described by symmetry breaking not only in chemistry, but in biology. Since the pioneering work of the famous English logician and mathematician A. Turing on the chemical basis of morphogenesis in biology (1952), there has been an increasing interest in this topic." He then proceeds to discuss "Symmetry and Symmetry Breaking in the Computer World", focusing on dynamical systems. For example, he write, "Nevertheless the Feigenbaum diagram is self-similar. Every part of the tree contains the Feigenbaum diagram infinitely often like Russian dolls. It follows that mathematical chaos can be highly symmetric." He closes with a discussion of modern architecture, where he finds that symmetry concerns are important as well: "But the variety of historical reminiscences and asymmetrical elements in architecture does not mean a movement back to historicism or eclecticism. It is the expression of a sceptic and ironic view of the world which no longer believes in an omnipotent technical rationality and its claim to solve all human problems. It underlines individuality and the importance of accidental details, and has doubts about universal harmony and rationality. So it prefers symmetry breaking as a chance of variety, pluralism, and individual freedom." And this is a theme that nicely rounds of his article: "But variety and pluralism need not be in conflict with unity. It was Leibniz who suggested that the unity of the world can only be experienced by man under special aspects. So his motto was 'unity in variety.' It dates back to the old philosophical idea of Heraclitus that even symmetry breaking is related to a sometimes hidden symmetry." Interesting and thought-provoking article. Closely related topics: Symmetry, Philosophy, Greece, Physics, Chemistry, Biology, Alan Turing, Computation, Fractals, and Architecture.
Mamedov, Kh. S. Crystallographic patterns. Symmetry: unifying human understanding, II. Comput. Math. Appl. Part B 12 (1986), no. 3-4, 511--529. SC: 00A69 (01A99 20H15 51F15), MR: 87e:00008.
This article discusses how crystallographic patterns "and their distribution and connection with natural phenomena and subjects of pure and applied art." It is written as an essay from a personal point of view. As the author tells us "I have made no effort to restrict the style of my meditations. I have presented a flow of free and sincere statements, and have not attempted to impose on them a style which might conceal their individuality. A great advantage of such statements is that one's 'falsehoods' are merely considered to be delusions, thus somehow mollifying the anger of those strict critics who feel obliged to adhere to absolute truths." The author himself is a chemist, so it is not surprising that there is some discussion of how crystallographic patterns in art are similar to those in chemistry. However, his observations on art from his own background in a nomadic family from Azerbaijan may be at least as valuable. The author notes that M. C. Escher is often identified with the applied art of crystallographic patterns, but these ideas are common in many cultures. Crystallographic patterns involving elements such as colored symmetry "are very characteristic of ancient and medieval decorations of Siberia, Kazakhstan, Central Asia, Azerbaijan, and Asia Minor." Quite a few examples of the art in this article use Islamic khufic script, and as he notes it is common to attribute the rise of patterned art rather than representational art to religious demands. The author does not seem entirely sympathetic with this idea, writing "The the problem was 'explained with God's help.' It is evident that in such cases it is much easier for the representatives of some other tradition to invent a new explaining theory than to examine the artwork using the language of its own traditions." The author gives some examples of crystallographic patterns in his own art and that of his associates. Interesting and enjoyable article. Closely related topics: Symmetry, Plane Patterns, Religion, Language and Literature, and M. C. Escher.
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: The Neolithic Era, Religion, The Sulvasutras, The Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art), and Abraham Seidenberg.
Meserve, Bruce E. The Evolution of Geometry. Mathematics Teacher 49 (1956), 372--82.
Discusses the history of geometry starting with the Egyptians and Babylonians and continuing into modern times. The rise and decline of Greek geometry, the logical structure of Greek proofs. Contributions by the Islamic world on the parallel postulate. Contributions of Renaissance artists studying perspective. Analytic geometry. More on the parallel postulate. Non-Euclidean geometry. The development of projective geometry and algebraic geometry. The article concludes with a discussion of how computational technology might change the nature of mathematics. Reprinted in edited form in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Analytic Geometry, Projective Geometry, Algebraic Geometry, Greece, The Islamic World, The Parallel Postulate, and Perspective.
Nagy, Dénes. Symmet-origami (symmetry and origami) in art, science, and technology. Symmetry Cult. Sci. 5 (1994), no. 1, 3--12. SC: 00A69 (01A99), MR: 1 309 239.