MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Preface v
PREFACE
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 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. N.B.:
Circa 1000 A.D., Gerbert of Aurillac (later Pope Sylvester II) referred to
the equilateral triangle as “mother of all figures” and provided the formula
A ≈ s 2 ·3/7 which estimates its area in terms of the length of its side to within
about 1% ( N. M. Brown, The Abacus and the Cross, Basic, 2010, p. 109).
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.