To expand search, see Geometry. Laterally related topics: 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, Irrationals, The Pentagram, The Sphere, The Conic Sections, Topology, Spirals, Line-Point Duality, Geometric Fixed Point Principles, The Cycloid, Tilings, and The Square.
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.
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), 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), The Circle, and Religion.
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, and Heron.
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 and Astronomy.
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 and Astronomy.