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 0-9757492-0-X (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'.

The following papers are meant for a high school audience. They provide a resource for teachers and students. You may download as pdf.

• The Wrong Trigonometry -- Facing up to the true difficulties of classical trigonometry, usually hidden behind the power of calculators.
  
  
• A Rational Approach to Trigonometry-- Statements and brief proofs of the 5 main laws of rational trigonometry, together with illustrative examples.
  
  
• Survivor: the Trigonometry Challenge -- Classical or rational trigonometry---which is superior? This paper poses some geometrical challenges to both.
  
  
• Pythagoras, Euclid, Archimedes and a new Trigonometry --How the basic laws of rational trigonometry follow naturally from the work of the ancient Greeks.
  

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

• The Impossibility of Trisecting an Angle with Straightedge and Compass: An Approach Using Rational Trigonometry by David Poole, Trent University

• A Comparison of Rational and Classical Trigonometry by Michael Gilsdorf
  
  
• Rational Trigonometry: Computational Viewpoint by Olga Koshaleva

Links, news and reviews

Here are links to various places on the internet where you can read about Rational Trigonometry.