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by Brian J. McCartin


Welcome to Mysteries of the Equilateral Triangle (MOTET), my collection of equilateral triangular arcana. While at first sight this might seem an idiosyncratic choice of subject matter for such a detailed and elaborate study, a moment’s reflection reveals the worthiness of its selection.

Human beings, “being as they be”, tend to take for granted some of their greatest discoveries (witness the wheel, fire, language, music,...). In Mathematics, the once flourishing topic of Triangle Geometry has turned fallow and fallen out of vogue (although Phil Davis offers us hope that it may be resuscitated by The Computer [70]). A regrettable casualty of this general decline in prominence has been the Equilateral Triangle.

Yet, the facts remain that Mathematics resides at the very core of human civilization, Geometry lies at the structural heart of Mathematics and the Equilateral Triangle provides one of the marble pillars of Geometry. As such, it is the express purpose of the present missive, MOTET, to try and salvage the serious study of the equilateral triangle from the dustbin of Mathematical History [31].

Like its musical namesake, MOTET is polyphonic by nature and requires no accompaniment [10]. Instead of being based upon a sacred Latin text, it rests upon a deep and abiding mathematical tradition of fascination with the equilateral triangle. The principal component voices are those of mathematical history, mathematical properties, Applied Mathematics, mathematical recreations and mathematical competitions, all above a basso ostinato of mathematical biography.

Chapter 1 surveys the rich history of the equilateral triangle. This will entail a certain amount of globetrotting as we visit Eastern Europe, Egypt, Mesopotamia, India, China, Japan, Sub-Saharan Africa, Ancient Greece, Israel, Western Europe and the United States of America. This sojourn will bring us into contact with the religious traditions of Hinduism, Buddhism, Judaism, Christianity and Scientology. We will find the equilateral triangle present within architecture, sculpture, painting, body armour, basket weaving, religious icons, alchemy, magic, national flags, games, insects, fruits and vegetables, music, television programs and, of course, Mathematics itself.

Chapter 2 explores some of the mathematical properties of the equilateral triangle. These range from elementary topics such as construction procedures to quite advanced topics such as packing and covering problems. Old chestnuts like Morley’s Theorem and Napoleon’s Theorem are to be found here, but so are more recent rarities such as Blundon’s Inequality and Partridge Tiling. Many of these plums may be absorbed through light skimming while others require considerable effort to digest. Caveat emptor: No attempt has been made either to distinguish between the two types or to segregate them.

In Chapter 3, we take up the place of the equilateral triangle in Applied Mathematics. Some of the selected applications, such as antenna design and electrocardiography, are quite conventional while others, such as drilling a square hole and wrapping chocolates, are decidedly unconventional. I have based the selection of topics upon my desire to communicate the sheer breadth of such applications. Thus, the utilization of the equilateral triangle in detecting gravitational waves, the construction of superconducting gaskets, cartography, genetics, game theory, voting theory as well as many other areas have all been included.

The subject of Chapter 4 is the role of the equilateral triangle in Recreational Mathematics. Traditional fare such as dissection puzzles appear on the menu, but so do more exotic delicacies such as rep-tiles and spidrons. Devotees of the work of Martin Gardner in this area will instantly recognize my considerable indebtedness to his writings. Given his extensive contributions to Recreational Mathematics, this pleasant state of affairs is simply unavoidable.

Chapter 5 contains a collection of olympiad-caliber problems on the equilateral triangle selected primarily from previous Mathematical Competitions. No solutions are included but readily available collections containing complete solutions are cited chapter and verse. Unless otherwise attributed, the source material for the biographical vignettes of Chapter 6 was drawn from Biographical Dictionary of Mathematicians [144], MacTutor History of Mathematics [230] and Wikipedia, The Free Encyclopedia [330]. Finally, we bid adieu to the equilateral triangle by taking a panoramic view of its many manifestations in the world about us. Thus, MOTET concludes with a Gallery of Equilateral Triangles that has been appended and which documents the multifarious and ubiquitous appearances of the equilateral triangle throughout the natural and man-made worlds.

I owe a steep debt of gratitude to a succession of highly professional Interlibrary Loan Coordinators at Kettering University: Joyce Keyes, Meg Wickman and Bruce Deitz. Quite frankly, without their tireless efforts in tracking down many times sketchy citations, whatever scholarly value may be attached to the present work would be substantially diminished. Also, I would like to warmly thank my Teachers: Harlon Phillips, Oved Shisha, Ghasi Verma and Antony Jameson. Each of them has played a significant role in my mathematical development and for that I am truly grateful. Once again, my loving wife Barbara A. (Rowe) McCartin has lent her Mathematical Artistry to the cover illustration thereby enhancing the appearance of this work.

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