AUTHOR'S CORNER:`Divine Proportions'
This page will be updated and maintained by the author
of Divine Proportions, Dr
N J Wildberger. Here Norman will post additional resources for Rational Trigonometry, both in the Euclidean setting and in the hyperbolic and spherical settings. You can also find an Errata for the book and answers to FAQ.
Norman's
UNSW website is at http://web.maths.unsw.edu.au/~norman/.
Norman's
YouTube channel Insights into Mathematics is at http://www.youtube.com/user/njwildberger. One of the playlists is WildTrig, which supports this book. Another, called UnivHypGeom builds from the content of this book, to give a dramatic reformulation of hyperbolic geometry, built up purely rationally without transcendental functions.
His recently
established blog njwildberger: tangential thoughts is at http://njwildberger.wordpress.com.
Errata
Errata
[Jan 27/2006] (pdf)  For pointing out errors and
typos, many thanks to Pierre Abbat, Peter Pleasants
and in particular Rupert McCallum!
Extra
topics and additional papers
An
introduction to finite fields [Oct 28/2005] (pdf)
Gives a brief background to fields in general and in
particular to the finite prime fields
The
Wrong Trigonometry  Explains why the classical
trigonometry taught in schools is overly complicated
and logically weak. Meant for high schools students
and teachers.
A
Rational Approach to Trigonometry  An elementary
summary of the basic laws of rational trigonometry with
proofs accessible to high school students and teachers.
This article also gives some examples of how rational
trigonometry provides pleasant answers to real life
problems.
Survivor:
The trigonometry challenge  A light hearted look
at both classical and rational trigonometry when applied
to fundamental geometrical questions involving a specific
triangle.
Projective
and spherical trigonometry  This paper shows
how to extend the ideas of rational geometry to spherical
trigonometry. The many traditional complicated formulas
are replaced by much simpler and more elegant ones.
The geometry of the projective plane is given its proper
metrical structure, thereby implicitly revolutionizing
algebraic geometry. In this paper the main applications
are to the Platonic solids, which are seen in a new
light.
The
ancient Greeks present: Rational Trigonometry
This paper reevaluates three fundamental results of
ancient Greek geometry. The theorems are Pythagoras'
theorem, the area of a triangle as one half the length
of the base times the height, and Heron's formula on
the area of a triangle in terms of the side lengths.
By recasting these with quadrance instead of distance,
more natural and general formulations are obtained,
and rational trigonometry falls out almost immediately.
Universal Hyperbolic Geometry I: Trigonometry (Geometriae Dedicata 2012) This paper gives a thorough reformulation of classical hyperbolic geometry by applying a projective version of Rational Trigonometry. The result is a major reevaluation of the subject. Using the Klein Beltrami model, but outside the null circle as well as inside, and hyperbolic versions of quadrance and spread, the main structure of hyperbolic trigonometry is greatly simplilfied and expanded.
Universal Hyperbolic Geometry II: A pictorial overview
(KoG 14 2010) This paper complements the previous one by giving a largescale overview of Universal Hyperbolic Geometry, with many pictures illustrating many new theorems. [By the way, this particular issue of KoG must be one of the most beautiful mathematics journal issue of modern times!]
Universal Hyperbolic Geometry III: First steps in projective triangle geometry (KoG 15 2011) This paper introduces major new advances and directions in triangle geometry in the hyperbolic setting and beyond. There are many pictures and new results. [By the way, this particular issue of KoG is close to being as beautiful as the previous one!]
FAQ
Why
hasn't rational trigonometry been discovered before?
Does
rational trigonometry define an angle in a new way?
Isn't
angle a natural concept?
What
is the spread between two lines?
How
do you replace adding adjacent angles?
Why
is quadrance more natural than distance?
What
about rotations and circular motion?
Won't
students have to learn the old trig to pass exams?
Is
Divine Proportions a high school text?
How
did you stumble upon rational trigonometry?
Why
hasn't rational trigonometry been discovered before?
This is best understood
by a brief look at the history of trigonometry, which
has its origins in astronomy. The Alexandrian Greeks,
Hindus, Persians, Arabs, Chinese and indeed most major
civilizations were fascinated by the night sky, and
in particular the motion of the planets. Trigonometry
was developed to study the celestial spherean imaginary
sphere containing the stars with us at the center, on
which the planets move. The basic notion here is the
spherical (angle) between two stars, defined as the
distance between the stars on the surface of this distant
sphere divided by the radius of the sphere.
For 1500 of the roughly
2000 years of trig, spherical trigonometry was the main
interest, and the planar case was hardly of significance.
Now of course the situation is totally reversed, but
tradition still keeps in place that relic from astronomythe
angle. So by the time people began investigating triangles
seriously, they simply took the spherical notions and
modified them to the planar case. Thus circles are nowadays
important in discussing triangles, and the subject is
more complicated than it needs to be.
Does
rational trigonometry define an angle in a new way?
No. Rational trigonometry
replaces the notion of an angle with a related
but simpler notionthat of the spread. A big difference
that you must get used to is that the spread is truly
defined between lines, not rays. This turns out to be
more natural algebraically, and allows the theory to
extend to more general fields.
Isn't
angle a natural concept?
Actually it is not. The
ancient Greeks, who were keen and careful geometers,
struggled mightily with the concept. Euclid does not
use angles in his classic treatment (right angles and
a few special related cases excepted).
Here is a simple test
to see for yourself how complicated the notion of angle
REALLY IS. Take any three random points in the plane
(say A=[1,2], B=[5,3] and C=[4,0]). Now without a calculator
or tables, see if you can determine the angle <ABC
to say 4 decimal places in degrees. My bet: even if
you are a professional mathematician, this will take
you at least several hours, and possibly several days!
The next time someone
tries to convince you that the idea of an angle is completely
elementary, give them this little challenge, and watch
them wriggle and squirm!
What
is the spread between two lines?
The spread between two
lines turns out to be the square of the sine of the
usual angle between them, but that's not how you should
think about it. The definition doesn't involve the notion
of angle, nor the understanding of the trig function
sine.
The spread between two
lines is defined informally in terms of the ratio of
two quadrances: the numerator is the quadrance from
a point B on one of the lines to the closest point C
on the other line, and the denominator is the quadrance
between B and the intersection A of the two lines. All
you need to define the spread is the notion of quadrance
and the idea of dropping a perpendicular from B to the
other line to get C.
More formally it is defined
in terms of an algebraic expression involving the equations
of the two lines. A theorem asserts the equivalence
of these two approaches.
How
do you replace adding adjacent angles?
An angle of 35 degrees
beside an angle of 2 degrees gives an angle of 37 degrees.
Or perhaps 33 degreesit depends on the relative positions
of the rays. So mathematically there are still generally
two solutions, even if you only see one in any given
picture. That suggests that there is really a quadratic
equation around. With spreads, this quadratic equation
is one of the five main laws (the Triple spread formula)
and allows you to calculate the spread s(l,n) if you
know s(l,m) and s(m,n) for lines l,m and n.
However if you have a
problem involving lots of 'angle chasing' then you are
better off with angles. The vast majority of trig problems
that arise in practise are not of this kind.
Why
is quadrance more natural than distance?
Distance is important
from a practical point of view. A carpenter will still
want to know how long his next beam should be cut, and
a bush walker (hiker) will still want to know the distance
from one place on the map to another.
The point is rather that
the mathematical theory becomes simpler when you work
with quadrance rather than distance. Pythagoras' theorem
uses squares of distance, as do many, many theorems
of Euclidean geometry. Given two points in the plane,
finding the quadrance between them is easy, while the
distance involves a square root.
The recommended approach
is thus:
 convert the problem
to quadrances and spreads
 solve the problem with
rational trigonometry
 convert back to distances,
and if necessary angles.
This should in fact be
familiar to students of traditional trigonometry: it's
exactly what you do when working with angles, which
are easier to understand with degrees, but the mathematics
simplifies if you convert to radians. So typically students
start with degrees, convert to radians to solve the
problem, and then convert back.
What
about rotations and circular motion?
It turns out that rotations
can be handled perfectly with rational trigonometry
(the book Divine Proportions shows how). But
uniform circular motion cannot be handled with
rational trigonometrythe trig functions are necessary
for it. The point is that trigonometry (the study of
the measurement of triangles) and circular/periodic
motion are two separate subjects. It
is not appropriate to have a mathematical technology
that covers both topics at once, as the current approach
does. The reason is that the theory becomes unnecessarily
complicated to incorporate two overlapping but disjoint
subjects.
Here is an analogy: you
might have a lawn mower and a barbecue in your back
yard. How about a machine that both mowed your lawn
and cooked your steaks? Wouldn't that be a great idea?
Well, it probably would make both jobs more onerous,
and cost a lot more than two separate machines specially
designed for the separate tasks.
The knowledge of the trig
functions required to understand circular motion is
in fact rather minimal. Most of the identities students
learn are unnecessary. In fact (horrible secret) once
you do a bit more advanced mathematics you realize that
cos and sin are just shadows of the complex exponential
function. Use of this more natural and elegant
function at one stroke simplifies almost all trig identities
and integrals.
Won't
students have to learn the old trig to pass exams?
For a while, sure. But
slowly people will start to examine the effectiveness
of the new theory, engineers may start using it in graphics
packages and robotics, surveyors may change their instrumentation
software to accomodate quadrance and spreads, NASA might
start thinking about its uses for real time calculations,
and lo and behold! Gradually it will trickle down from
academics to teachers to students.
It won't happen overnight,
and until then students will still have to grin and
bear the current curriculum. At first perhaps rational
trigonometry will appear as an enrichment subject for
gifted students, then possibly it will be tried on very
weak students who can't hope to understand the usual
trig. Then slowly more and more students and parents
will start demanding change, and after a while it will
be inevitable.
The simplicity and power
of rational trigonometry will sink in. You just can't
argue with a theory that is at least three times as
easy to learn! And more accurate and powerful to boot!
My guess is that in twenty five years classical trigonometry
as a way of studying triangles will have gone the way
of the horse and buggy.
Is
Divine Proportions a high school text?
No. It is a thorough,
careful laying out of this new theory for academics
to be able to judge its merits. The treatment is concise,
requires mathematical maturity, and builds up Euclidean
geometry over a general field. While a very talented
high school student will be able to read much of it
and gain a lot from it, for most high school students
it is rather hard.
The book attempts to present
a lot of evidence in one spot to convince mathematicians
and teachers to rethink their current understandings.
How
did you stumble upon rational trigonometry?
It was an accidental byproduct
of some more arcane investigations I was making 4 years
ago in relativistic geometry. Although some of the ideas
had been bubbling away in the back of my mind for many
years before that. It all came together very quickly
when I realized that I didn't really HAVE to consider
angle and distance as the fundamental concepts.
The real surprise to me
was that this somewhat simple minded theory (ie rational
trigonometry) ended up not only solving all the standard
trigonometric problems I investigated, but often did
so more elegantly, accurately and quickly. That is a
very important point.
Order
now
