A revolutionary new text


DIVINE
PROPORTIONS: Rational Trigonometry to Universal
Geometry
by N J Wildberger (Assoc Prof UNSW Sydney)
provides a complete
and powerful alternative to classical trigonometry
and Euclidean geometry. Now available here at
wildegg.com.
Order
now
Authors
corner: FAQ, Errata and other stuff



Reviews
Review by
Prof Jan Aarts (Delft Univ) for NAW (Nieuw Archief voor Wiskunde) 5/11 Sept. 2010
Review by Prof
Michael Henle for The
American Mathematical Monthly (Dec 07)
Review
by Prof Robert
Dawson for CMS
Notes
Review
for Newsletter of the European Mathematical Society
(June 07)
Review
by Dr Laura Wiswell for The
Proceedings of the Edinburgh Mathematical Society
Review
by
Dr Rod Gover for The
Gazette of the Australian Mathematical Society
Review
by Prof
James Franklin for The
Mathematical Intelligencer
Review
by Prof. S. L. Arlinghaus for Solstice:
An Electronic Journal of Geography and Mathematics
Review
by Prof. Michael Hardy for Amazon.com
Review by Prof. Joseph C. Varilly for Amazon.com
Downloads
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(pdf) : Preface /
Contents / Introduction
/ Chapter1
Book
Description
Rational
trigonometry
This new theory, described
here for the first time, dispenses with all the usual
complicated trigonometric functions and relations and
establishes a trigonometry based entirely on elementary
arithmetic and algebra. The essential quadratic aspects
of geometry are brought to the fore, with the key concepts
of quadrance and spread replacing
distance and angle. The two subjects of geometry and
periodic motion are cleanly separated; it is neither
necessary nor desirable to have a theory which merges
the two.
In Divine Proportions:
Rational Trigonometry to Universal Geometry, Associate
Professor N
J Wildberger of the School
of Mathematics at the University
of New South Wales shows how to solve even complicated
geometric problems without tables or calculators, with
more accurate answers. Many engineering, surveying and
physics problems are solved in this book, illustrating
in detail the advantages of this new approach. For large
scale problems, rational trigonometry harnesses modern
computational power more efficiently.
Because of the rational
form of the basic laws, the theory extends to general
fields, opening up numerous new areas of investigation
for number theory, algebraic geometry and combinatorics.
A lovely new family of orthogonal polynomials called
spread polynomials extends the theory of special
functions to general fields.
Universal
geometry
Using rational trigonometry
as a basis, a new form of metrical Euclidean geometry
is constructed, incorporating most of high school geometry,
but extending in exciting new directions. This new Universal
Geometry contains similiar triangles, Heron's formula,
centroids, circumcenters, orthocenters, cyclic quadrilaterals
and Brahmagupta's theorem, circles, parabolas, the Euler
line, tangent lines and tangent conics, nine point circles,
and much moreall over a general field!
Examples are taken not
only from the usual decimal numbers, but also finite
fields and the complex numbers.
Education
Divine Proportions
argues that there is a good reason why the standard
high school mathematics curriculum turns so many people
off mathematics, often permanently. The reason is that
the mathematics being currently taught is logically
weak and overly complicated for what it accomplishes.
It confuses the study of triangles with the subject
of uniform circular motion. These are separate areas
of mathematics, requiring different concepts and techniques.
To understand the geometry
of a triangle, one doesn't require transcendental trigonometric
functions such as sin, cos, tan and their inverse functions.
Until now, there has been no sensible alternative to
focus the minds of educators, even though most students
intuitively understand that the theory is too complicated.
This book finally addresses the failure of modern geometry
to win the minds of young people, and provides the mathematical
foundation for a dynamic and elegant new approach to
teaching trigonometry.
Applications
 Two dimensional geometry
problems
 Surveying problems
(Elementary)
 Surveying problems
(Advanced) Resection and Hansen's problems
 Boxes, pyramids, wedges
and pagodas
 PhysicsProjectile
motion, Snell's law, Algebraic dynamics, Lorentzian
addition of velocities (as in Einstein's special theory
of relativity)
 Platonic solids
 Folium of Descartes
and Bernoulli's lemniscate
 Five fold symmetry
and regular polygons
 Calculus problems involving
rotational and spherical symmetry (volumes, surface
areas, centroids, moments of inertia of spheres, caps,
hyperboloids, toroidal rings etc.)
 New formulas for classical
curves using rational polar coordinates.
Features
of the book
 Introduction and overview
chapter motivate and explain the key features of the
new theory, along with some of the weaknesses of traditional
trigonometry
 Preliminary chapter
provides review of necessary background material
 More than one hundred
theorems (most of them new) over four parts
 More than one hundred
and seventy illustrative diagrams
 Index with separate
table of theorems
 Two appendices.
Intended
audience
The book's content is
largely elementary, but is presented concisely. It requires
mathematical maturity and skill at algebraic manipulation,
along with an interest in geometry and its applications.
It will be especially valuable to
 Professional mathematicians,
especially those with an interest in geometry (including
algebraic geometry and differential geometry), number
theory, combinatorics and special functions
 Scientists with an
interest in mathematics, i.e. physicists, chemists
 Engineers and some
computer scientists
 Mathematically talented
high school students
 Undergraduate
mathematics or physics majors
 High school and college
mathematics teachers and lecturers
 Amateur mathematicians
with strong algebraic skills and an interest in geometry.
General members of the
public who do not fit into one of these categories may
well find the book too technical to be easily accessible.
Dr Wildberger intends on writing a companion book at
a more elementary level which explains the subject to
the general public.
Author
Dr
N J Wildberger (BSc University of Toronto, PhD Yale
University) has taught at the University of Toronto
(as an undergraduate), Yale University (as a graduate
student), Stanford University and the University of
Toronto.
He is currently Associate Professor in the School of Mathematics
and the University
of New South Wales (UNSW) in Sydney Australia. His
main research interests are Lie theory, hypergroups,
representation theory and
geometry. He has also done work in number theory, combinatorics
and mathematical physics. This is his first book.
Wildberger is now using Rational Trigonometry as a springboard to redesigning metrical geometry in a largescale way. He has recently completed a trilogy of papers on Universal Hyperbolic Geometry. Rational Trigonometry is also supported by the YouTube series WildTrig available at Wildberger's channel Insights into Mathematics.
First
edition
This
is a quality hardcover
book of 320 pages. It uses 100% acid free paper, section
sewn and printed by BPA group, one of Australia's foremost
printers, with a striking cover design by designer Alex
Snellgrove. This book is available here at this website,
for the special low price of $49.95 (AUS) plus shipping.
To
Order
Divine Proportions:
Rational Trigonometry to Universal Geometry
is now available online here at wildegg.com for the
special low price of $49.95 (AUS)
plus shipping. Order now
The book
is also available at amazon.com.
