Rational
Trigonometry Site
These pages will attempt to provide an
overview of Rational Trigonometry and how it allows
us to reformulate spherical and elliptic geometries,
hyperbolic geometry, and inversive geometry, and leads
to the new theory of chromogeometry, along with many
practical applications.
Reference: The
main reference is the book
`Divine Proportions: Rational Trigonometry
to Universal Geometry'
by N J Wildberger, ISBN 097574920X
(hardcover), Wild Egg Books, Sydney 2005 available at
http://wildegg.com
and also at amazon.com.
Here are free downloads of sections of
this book (all in pdf format)
 Preface  a one
page outline of the book
 Contents  table
of contents
 Introduction 
a detailed description of the book
 Chapter1  an
overview of rational trigonometry and justification
 Chapter27 
polar and spherical coordinates done rationally
YouTube Videos: I have posted
a series of YouTube videos on rational trigonometry
and geometry (about 50 so far) in the WildTrig
series under user name njwildberger.
These videos are meant for a general audience with an
interest in geometry, and are packed with information
about rational trigonometry and its applications, and
about geometry more generally.
Here is a list of the WildTrig
videos with brief descriptions.
My YouTube channel is called `Insights into Mathematics',
and also has another series called MathFoundations,
which attempts to lay proper foundations for mathematics,
and criticizes the current sloppy approach to logical
difficulties. This series is meant for a general audience.
There is a third series which will be devoted to mathematics
seminars I have given, these are at a more advanced
level (be warned!)
Hyperbolic Geometry:
Here you can find an
explanation of how rational trigonometry gives a new
approach to hyperbolic geometry, as well as a series
of The Geometer's Sketchpad worksheets that illustrate
various constructions and theorems of universal hyperbolic
geometry.
The main reference paper is the paper
`Universal Hyperbolic Geometry I: Trigonometry'.
Additional Elementary Papers
(by N J Wildberger)
The following papers are meant for a high school audience.
They provide a resource for teachers and students. You
may download as pdf.
Additional Advanced Papers (by
N J Wildberger)
The following papers are meant for an academic audience,
meaning mathematicians, university math majors, and
enthusiastic high school teachers. You may download
as pdf.
 N J Wildberger, Universal Hyperbolic Geometry I:
Trigonometry (http://arxiv.org/abs/0909.1377)
ABSTRACT: Hyperbolic geometry is developed in
a purely algebraic fashion from first principles,
without a prior development of differential geometry.
The natural connection with the geometry of Lorentz,
Einstein and Minkowski comes from a projective point
of view, with trigonometric laws that extend to `points
at infinity', here called `null points', and beyond
to `ideal points' associated to a hyperboloid of one
sheet. The theory works over a general field not of
characteristic two, and the main laws can be viewed
as deformations of those from planar rational trigonometry.
There are many new features.
 N J Wildberger, Affine
and projective universal geometry (pdf) (http://arxiv.org/abs/math/0612499)
This paper establishes the basics of universal geometry,
a completely algebraic formulation of metrical geometry
valid over a general field and an arbitrary quadratic
form. The fundamental laws of rational trigonometry
are here shown to extend to the more general affine
case. Also a corresponding projective version, which
has laws which are deformations of the affine case
is established. This unifies both elliptic and hyperbolic
geometries, in that the main trigonometry laws are
identical in both!
 N J Wildberger,
One dimensional metrical geometry (pdf) (http://arxiv.org/abs/math/0701338)
The basics of universal geometry are already visible
in the one dimensional situation, the great blind
spot of modern geometry. There is both an affine and
a projective version. The affine version is interesting
especially with regard to the quadruple quad formula,
the relation between quadrances from four points,
which anticipates the formula of Brahmagupta for cyclic
quadrilaterals. The projective version depends on
the one dimensional analog of a quadratic form. Chromogeometry
already makes an appearance. The spread polynomials,
which are rational equivalents of the Chebyshev polynomials
of the first kind, contain already the seeds of two
dimensional symmetry.
 N J Wildberger, Chromogeometry
(pdf) (http://arxiv.org/abs/0806.3617)
My favourite discovery. A three fold symmetry in planar
metrical geometry, that ends up transforming almost
every aspect of the subject. Euclidean geometry meets
two hyperbolic or relativistic geometries, and all
three interact in a lovely way. This introductory
paper illustrates applications to triangle geometry,
in particular it `explains' in a new way the Euler
line and introduces a new triangle of coloured orthocenters
which controls all three Euler lines, along with circumcenters
and nine point centers. There is much more to be said
about this subject!
 N J Wildberger Chromogeometry
and Relativistic Conics (pdf) (http://arxiv.org/abs/0806.2789)
This paper discusses how chromogeometry sheds new
light on conics. It gives a novel formulation of an
ellipse involving two canonical lines called the diagonals
of the ellipse, together with an associated corner
rectangle, and shows how this concept applies both
in blue (Euclidean), red and green geometries. Red
and green foci come in two pairs, determined geometrically
in a simpler fashion than the usual foci. Hyperbolas
are also discussed. The parabola plays a special role,
as metrically it is the same in all three geometries,
and the interaction between the three colours is particularly
striking.
 N J Wildberger, Neuberg
cubics over finite fields (pdf)
(http://arxiv.org/abs/0806.2495)
The Neuberg cubic is the most famous triangle cubic,
organizing more than a hundred known triangle points
and many lines. This paper uses the framework of Universal
Geometry to extend triangle geometry to the finite
field setting, and studies an example of a Neuberg
cubic over the field of 23 elements. Many, but not
all, of the usual real number properties hold. The
paper also discusses tangent conics to elliptic curves
in Weierstrasse form, showing them to be disjoint
or identical if 3 is not a square.
Additional Papers (by others)
To post your comments/papers/developments involving
Rational trigonometry here, send them to me at n.wildberger
(at) unsw (dot) edu (dot) au
Links, news and reviews
Here are links to various places on the internet where
you can read about Rational Trigonometry.
