N J Wildberger

I am an Associate Professor in mathematics at UNSW (University of New South Wales) in Sydney Australia.


School of Maths UNSW Sydney 2052 Australia


Tel:61 (02) 9385 7098

Fax:61 (02) 9385 7123

A new mathematics 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 NOW from Wild Egg Books : http://wildegg.com

Order here

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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 is published by Wild Egg, an innovative new publishing company specializing in ground breaking mathematical titles. The first edition is hardcover, 320 pages on 100% acid-free paper, with section sewn binding. It was printed by BPA, one of Australia's foremost quality printers. The cover design is by Alex Snellgrove. It is available online at http://wildegg.com.