Customer Reviews

Divine Proportions: Rational Trigonometry to Universal Geometry

5 star:		(4)
4 star:		(0)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

If you have ever taken (or taught) a college-level course in
Euclidean geometry, you'll know that the hardest part is just to get
started. What exactly is a point, a line, a triangle, and how are they
related? About 100 years ago, several people such as Pasch, Hilbert
and Veblen thought seriously about this and came up with a bunch
of 14 "axioms" that extend and improve the prologue to Euclid's treatise.
The result is a nightmarish discussion about properties of incidence,
"betweenness", congruence and whatnot, before one can settle down
to match triangles according to size and shape.

Nowadays the usual approach is to cheat, and instead play games
with coordinates: a point in a plane is just a pair of numbers,
a line is a first-degree equation, and so on. Even so, the concept
of angle between lines and angle measurement is rather tough for
beginners, and needs a hard slog through the lore of sines and
cosines before one can get worthwhile results.

Now comes Norman Wildberger, a respected researcher at Sydney,
who proposes a radical departure that he calls rational trigonometry.
First, he replaces distances by their squares, which he calls
"quadrances", banishing the need to calculate with square roots:
this is a good idea but not very new. However, he also proposes
to replace the angle between two lines by their "spread", which
happens to be the sine-squared of the angle. In this way
Pythagoras' theorem becomes a first-degree relation: the two spreads
at the narrow corners of a right triangle add up to 1. The kicker is
that the classic properties of triangles, namely the sine law,
the cosine law, and the constant angle-sum, become simple formulas
expressible by polynomials of degree two or three.

Here is the big advantage: since only polynomials are needed, one can
use coordinates in any number system, such as a finite field. Many of
the examples in this book concern triangles and conics over finite fields.
There is a pitfall: in some finite fields the sum of two squares can
be zero, so one must watch out for things like "null lines". But this is
precisely what happens in relativity, where there are "light-like world
lines", and finite geometry provides excellent training for that. On the
other hand, much of classical geometry is still available; the book has
a fine chapter on the nine-point circle.

Who is the book for? It is not a textbook for beginners, one needs to
know some geometry already and to be ready to fill in the details
from time to time. But a bright high-school student can read it,
a college math major should read it, and any high-school geometry
teacher will surely love to have it on the bookshelf.

The author begins in a messianic style, and continues and ends that way
too, which is rather off-putting. But he is not a crank, the math is all good.
His terminology is sometimes bizarre (where on earth did he get the
words quadrola and grammola?) and hard to translate (what is a good
Spanish word for spread?). Some simple things become complicated: for
instance, collinearity of three points is a quadratic relation. Many will
resist the downgrading of trigonometry, and electrical engineers may
dismiss it as silly. All in all, however, this is a charming and
intriguing book that forces us all to think again about simple things
that we thought we knew. Have a look.

One of Wildberger's claims is that his approach to
trigonometry is both simpler and easier that the
traditional one ("simpler" is not synonymous with
"easier", but sometimes "simpler" entails "easier").

WARNING: This book will NOT make trigonometry easier
for non-mathematicians of the sort who are learning
trigonometry for the first time, since they are not
its intended audience. This is not a textbook. Wildberger
has said he will write that later. This book explains
his approach to mathematicians, who will find it easy to
read.

It would be very unfair to Wildberger and to this book to
judge it harshly for not making trigonometry easier to
those learning it for the first time. Wait until his
textbook is published and then judge THAT by that criterion.
Judge this book by its appeal to mathematicians and others
who are its intended audience.

Etymologically, trigonometry is "measurement of triangles".
Students learning trigonometry also spend a lot of time on
measurement of circles. An etymologically parallel term for
measurement of circles is "cyclometry", a seldom seen word
occasionally taken to refer to circle-squaring. Wildberger
"measures triangles" without "measuring circles", without what
are usually called trigonometric functions or any other
"transcendental functions", without talking about angles or
rotations. The angle at which two lines meet is determined
by what Wildberger calls the "spread", which is a rational
function of their slopes (this is a glimpse of the reason for
the word "rational" in the title), and in the language of
conventional trigonometry is the square of the sine.

All of that portion of conventional trigonometry that is
concerned with measuring triangles, whether applied to physics,
geography, land surveying, navigation, etc., is simpler if
done by the methods introduced by Wildberger.

Wildberger also covers a considerable amount of interesting
geometry not usually treated when trigonometry is taught.

The sine and cosine functions used in Fourier series and
harmonic analysis do not appear. For Fourier series, it
is of course simpler to use "cis(x)" = cos(x) + i sin(x)
= e^{ix} than to use the sine and cosine functions. Since
the trigonometric identity cis(x + y) = cis(x)*cis(y) is
simpler than the formulas for sin(x + y) and cos(x + y),
should we expect that if Wildeberger's proposal to separate
trigonometry from what I called "cyclometry" above becomes
the norm, then all those complicated identities normally
worked with in trigonometry will be forgotten? I will
indulge in wild speculation for a moment. Another place
where those functions can appear is as generating functions
in combinatorics. The coefficients in the not-too-well-known
power series for tan(x) enumerate "alternating permutations"
of the set {1, ..., n} when n is odd; those of sec(x) do the
same when n is even. Conventional trigonometry tells us that
the sum of a tangent and a secant is a tangent, i.e.
tan(x) + sec(x) = tan(x/2 + pi/4); thus, that function
enumerates all "alternating permutations". Will things like
that someday be a reason for all those complicated identities
to be remembered? And will they somehow become simpler if
someone succeeds in separating their study both from the topics
relied on in Fourier series and from the measurement of
triangles?

This is an exciting book, but it is not a textbook for learning this system of trigonometry (trig). It is a proof of concept book for those involved with math to study the concept and learn about it. The claim that it provides a simpler system of learning and using trig is based on the fact that simple algebraic equations, even with square roots, are more easily learned and used than are transcendental (sines, cosines, tangents, etc) functions, even when done by hand. (And such higher math was done by hand long before the advent of calculators or even slide rules.) In that respect, the book very much does live up to its claim.

This book struck a very deep note with me. It drives right to the core of one of the biggest frustrations in trying to understand and learn math - trigonometry. What Wildberger has done is to go back to the very roots of basic math concepts and rewrite trigonometry from the ground up. In doing this, he has dispensed with angles and transcendental functions, and used quadratic equations and straightforward algebra instead. The concepts of angle and distance are replaced by "spread" and "quadrance," which are logically derived in very clear terms from basic definitions of point and line. Wildberger avoids layers of unclear axioms in explaining and deriving "rational trigonometry." He discusses why the concepts and axioms necessary for dealing with angles and transcendental functions are so confusing and sometimes inaccurate.

The book begins with an overview and short discussion of the situation and Wildberger's solution. He goes through background material and relevant definitions in the first section of the book. The second section details rational trigonometry and derives the basic theorems in clear, understandable terms. The third section connects rational trig with a Universal Geometry that takes Wildberger's concepts into a truly expandable geometry that can easily move into higher dimensions with consistent rules and procedures. The forth section deals with sample applications, such as harmonic relation, Pythagoras' Theorem in 3 dimensions, projectile motions, surveying, Platonic solids, Beta function, spherical coordinates, and much more. Wildberger also shows how this new approach to trig might eliminate errors and make for more accurate calculations, as well as greater clarity of principles in the minds of those using trig. He delves into the errors generated by the approximations inherent in the calculation of angles and their transcendental offspring, and shows how rational trig can avoid those errors. While not a tour de force of all related mathematics, the book constitutes a very solid foundation for understanding and exploring this new and exciting approach to trigonometry.

Even though it has been some time since I did math any more involved than speaker impedance calculations, I found his examples and derivations clear and easy to follow (at least up to my limits - college level calculus). I found that I could understand the concepts of spread and quadrance easily from his instruction. This material could be accessible to some high school students. Those involved in college level and higher math should have no trouble with it, and will possibly even enjoy it. It is more clearly and succinctly written than most of the college math books I remember.

While the terms "groundbreaking" and "breakthrough" are thrown about fairly often these days, this is one of the few works that can actually retain that claim under close scrutiny. This is a major leap ahead in trig, and for math in general. Aside from the startlingly fresh ideas developed and presented here, just the simple fact that a radically new view of such an old (dare I say ancient?) field can exist should make everyone interested in math jump and shout with glee. (Well, okay, at least smile and nod approvingly.)

I immediately found myself wanting to explore the implications of this work, even though my math geek days are long past. I wonder how this different view of trig will impact fields using higher math? I wonder what young minds will be inspired by this discovery and launch into whole new realms of exploration? I wonder how many little discrepancies will come to light with a new tool to see them? I wonder how looking at a subject with "new eyes" will foster other new ideas and insights? When one begins to speculate what could come of this discovery, it seems clear that the possibilities are non-trivial.

I also started thinking about some of the more exotic areas that may feel the impact of this work. I wonder how Michael Leyton's work in higher dimensional geometry would blend with the concepts in this book. What are the implications for cutting edge physics? It seems to me that there might also be a distinct bridge here between hard science and more esoteric forms of math, such as so-called Sacred Geometry. (Bear in mind that these are my ideas. Wildberger does not stray from very staid mathematical realms and makes no claims to such excitement for his work.)

What else have we missed? What else might we see? What else might we discover? Advancements in knowledge are as much as function of new perceptions as new technology, perhaps even more so.

This is the main reason I am excited about this book - it gives us radically new eyes for looking at something to which we have grown quite accustomed. That new discoveries will emerge from this work is virtually guaranteed. The question is, how soon?

I believe this book could become a foundational piece in mathematics in the next century. If you enjoy seeing things from a new perspective and enjoy math, you will like this book. I encourage anyone interested in math and new frontiers to give it serious consideration. I also anxiously await Wildberger's next book on the subject, which I believe will be a textbook.

I discovered Dr Wildberger on youtube, where his excellent lecture series recorded at the University of New South Wales have been uploaded with permission.

My background is in mathematics, but I have never encountered this approach to trigonometry, nor many of the ideas that follow. They are no mere novelties, Dr Wildberger has completely changed the way I approach geometric concepts. I only wish I could have read this in undergrad.

I sincerely hope Dr Wildberger publishes again soon.

To get the most out of any review, and to evaluate the degree to which its author's opinions apply to, or will interest you, it is generally helpful to know where that author is coming from. If you don't care, you should skip the next paragraph.

About Me:

I have been a professional software engineer for 16 years and have spent the majority of my career developing video games. When I make something like a 3D hierarchical animation system, the majority of the work involved is designing the `mechanics', structure, and interfaces of the system so that it can be used in all the ways we might envision and can be extended in useful ways. The actual math is, to me, an implementation `detail', which I can get help with or which might already be handled by a good math library. However, there is too little focus on good engineering and too much focus on specialties, techniques, and trivia in this industry. As a result it is becoming increasingly inconvenient for me to not be able do discuss certain types of math problems off the top of my head. Trigonometry certainly won't cover the knowledge I need, but it is the right place to start.

The bottom line here is that I am both motivated and able to learn this subject from any good teacher (and have, in fact taken trigonometry years ago).


The Claim:

I have read what some mathematicians have to say about this book and its relevance mathematically, and their speculation on whether or not it is plausible to try to change the course of trigonometry at this point. But my main interest is in the author's (implied) CLAIM THAT HIS APPROACH MAKES THE SUBJECT EASIER TO LEARN. He says, in his introduction about how his `rational trigonometry' revolutionizes the subject, "Learning trigonometry and geometry should be easier than it currently is...", and "New laws now replace the Cosine law, the Sine law, and the dozens of other trigonometric formulas that often cause students difficulty." The implication being, that his system DOES make the subject easier to learn.


The Verdict:

It is my opinion that this book does not deliver on that promise. The simplest way to explain this is in terms of the previous quote, "New laws now replace the Cosine law, the Sine law, and the dozens of other trigonometric formulas..." So solving trigonometric problems is done in essentially the same way as before. Gather the known values, construct the relevant triangles, apply the (memorized) laws to create relevant equations, and solve the equations algebraically for the desired values (using a calculator to evaluate common, complex functions). Only now you use the new laws instead of old ones. As far as I can tell, the trigonometric TECHNIQUES have not changed. The author claims the advantage that a calculator is generally no longer needed because he doesn't use sine, cosine, tangent, etc., but he hasn't escaped the square root, so I don't consider that claim valid.

How might he have made the subject easier to learn then? Perhaps these nebulous transcendental functions have been replaced with well defined, meaningful concepts, so that we know when and how to use them intuitively, by virtue of their meaning, rather than by rote practice? Not as far as I can tell.

In fact I have a problem with the author's idea of good names and definitions. He replaces the concept of the angle, for describing the degree to which two intersecting lines diverge from each other, with that of `spread'. Now I could think of worse words to use, but I could instantly think of the word `divergence' which, though perhaps not the best possible word, seems to more completely capture the essence of the concept at hand. To me that's a bad sign. Things seem to go even further awry with `cross', and `twist', which are so poorly defined (and I can only assume, poorly named), that I'm still not sure what concept, if any, they're meant to represent. The best I can guess is that `spread', `cross', and `twist' were picked to correspond, in their first letters, with the sine, cosine, and tangent they replace.

And this, I think, is the essence of the problem I have with the book's claim. The author's definitions might have appeal to mathematicians, if they happen to agree that they are more concise, unambiguous, and fundamental than other alternatives. But that isn't why trigonometric formulas "cause students difficulty". What makes them difficult is that SINE, COSINE, AND TANGENT DON'T MEAN ANYTHING to the student. And on this front, I don't think he has made any headway.

In fact, in some ways, he seems to be going backwards. As a simple example, the author makes a point of the precision of his definitions without using any `hand waving' or pictures, such as, "A point ... is an ordered pair of numbers." Perhaps that is mathematically brilliant in its exactness, but to the student all he has done is AVOIDED DEFINING A POINT AT ALL, and instead described how he chooses to represent it for his mathematical purposes. This `definition' gives me no understanding of what it is or intuition of how I might want to use it. Maybe that doesn't seem like a big deal since I happen to have a good idea what a point is, but I have no idea what a `Determinant' is or what a `Grammola' is, and suffice it to say that his discussion of them has not enlightened me.