Upcoming Book...
DIVINE
PROPORTIONS: Rational Trigonometry to Universal Geometry by N J
Wildberger
"A unique and revolutionary text
which establishes new foundations for trigonometry and Euclidean
geometry'
AVAILABLE SEPTEMBER 2005
Download (pdf)
Revolutionary content
This text introduces a new and
simplified approach to trigonometry and a major restructuring of Euclidean
geometry. It replaces cos, sin, tan and all those other transcendental trig
functions with rational functions and elementary arithmetic. It develops a
complete theory of planar Euclidean geometry over a general field without any
reliance on `axioms'. And it shows how to apply this new theory to a wide range
of practical problems from engineering, physics, surveying and calculus.
New formulas and theorems
Divine Proportions: Rational
Trigonometry to Universal Geometry contains dozens of new and important
formulas, and almost a hundred theorems. The formulas are mostly polynomial or
rational, and the basic ones, such as the Triple Quad formula,
the Spread Law, the Cross law and the
Triple Spread formula are quadratic in any variable. The theorems are
often based on existing results, but they hold in much greater generality, and
their formulations are sometimes unusual. For example the new Spread
polynomials replace the closely related Chebyshev polynomials of the
first kind, but extend to arbitrary fields.
Increased accuracy
Classical trigonometry relies on
transcendental functions, so that the numbers output by calculators are only
approximations to the true values. This means round off errors are ever present
in classical trigonometry, and for complicated computations these can become
significant. Rational trigonometry is more accurate, since ordinary arithmetic
carries along all the information as it goes. Decades ago this would have been
unwieldy for large scale applications, but with the development of computer
algebra systems, such as MAPLE, MATHEMATICA, MUPAD and so on, it can be made
fast and accurate.
New areas of research
Divine Proportions: Rational Trigonometry to Universal
Geometry opens up new areas of research not only in Euclidean geometry, but
also in algebraic geometry, number theory,
combinatorics, sprecial functions, Lie theory
and non-Euclidean geometries. It proposes a return to
the classical algebraic geometry of Fermat, Newton, Euler and their
contemporaries, where metrical aspects are encoded in the algebraic concepts of
quadrance and spread.
Opportunities for mathematics education
Trigonometry has for centuries been
difficult for students to learn. Part of the reason is that the foundations of
the subject are problematic, relying on vague pictorial definitions of the key
concepts of angles and trigonometric functions. That means students rarely come
to grips with the logical structure of the theory. Rational trigonometry is much
simpler, relying on the fundamental quadratic notions of
quadrance and spread, and with basic
laws that can be derived completely rigorously using only elementary
algebra.
Quality book
Divine Proportions: Rational
Trigonometry to Universal Geometry will hopefully be published this year by
Wild Egg, an innovative new publishing company specializing in ground breaking
mathematical titles. The first edition will be hardcover, 320 pages on 100%
acid-free paper, with section sewn binding. It will be printed by BPA, one of
Australia's foremost quality printers. The cover design is by Alex Snellgrove.
It will be available online at http://wildegg.com at the end of September
2005.
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