Sine of the times

September 25, 2005 9:34 PM

Norman Wildberger's New TrigonometryDr Norman Wildberger has rewritten the arcane rules of trigonometry and eliminated sines, cosines and tangents from the trigonometric toolkit. The First chapter of his new book, Divine Proportions, is online (.pdf).
posted by Kwantsar (21 comments total)

Interesting, but as a pedagogic method, it doesn't seem to scale well past the Cartesian coordinate system?
posted by Rothko at 9:51 PM on September 25, 2005

For those not bothered to check the PDF, instead of distance and angle, we have quadrance (square of the distance) and spread (square of the sine of an angle) as the fundamental operands.
posted by Gyan at 9:53 PM on September 25, 2005

Finally, math without all those damn exacting numbers!
posted by Balisong at 9:54 PM on September 25, 2005

this was discussed exhaustively on slashdot recently.
posted by snoktruix at 10:12 PM on September 25, 2005

This is way cool!

Solving any 2D trig problem without recourse to Pi, sine, cosine or tangent? Yes please! Unless there's some major problem not covered in Chapter 1 (where the free sample ends, sadly), this is surely the way of the future for school trig courses, and much else besides.

I find myself wondering if ray-tracing software already uses this type of math in practice... if it doesn't there are some glorious optimizations to be made.

I can just see the author's cunning little hands rubbing together when he wrote this:

Since rational trigonometry is so much simpler than the existing theory, why was it not
discovered a long time ago?
...
Perhaps these different clues have not been put together formerly because of the
strength of established tradition, and in particular the reverence for Greek geometry.
posted by cleardawn at 10:20 PM on September 25, 2005

Slashdot discusses potential optimization opportunities at length.
posted by cleardawn at 10:26 PM on September 25, 2005

Interesting. I'll have to read this tomorrow when I'm not so tired.

I skipped trig: just not enough there for a semester. I don't understand the difficulty with trig. The basic tenets can be learned in a day. From there on out it's mainly memorization.
posted by teece at 10:38 PM on September 25, 2005

The .pdf file isn't loading for me. Does anyone have a link to a summary and a rational, non promotional analysis of this so-called "new trigonometry"?
posted by muddgirl at 11:08 PM on September 25, 2005

Sorry muddgirl, it looks like the Prof's site is down. MeFi tourists wahay!

In the meantime, it's good for the soul to take an occasional trip through Slashdot, where the crudest stereotypes of geekdom are humbled by reality, and exclamation points are the new black.

For example, computing the derivative of the cosine function is not easy to understand if you restrict the definition of cosine to side adjacent over hypotenuse!
posted by cleardawn at 11:36 PM on September 25, 2005

I'd like to reiterate the basic point made on slashdot: while this approach is fine if you never plan on advancing past trig in your math studies, when you get to calculus the next year in high school (or first semester of college, or whatever), you are going to have to learn all the sin, arcsin, sin^(-1), sin', integral of sin, etc stuff anyway so you may as well start getting comfortable with them as early as possible.

By the time you actually graduate and go into graphics program and these tricks become useful again, you should be well enough versed in both calculus and trig to be able to figure out how to do angular measurments using sin and sqrt as little as possible, anyway.

I also found this pretty funny in the slasdot thread:

e^(ix)=cos(x)+i*sin(x)
=> cos(x)=(e^(ix)+e^(-ix))/2
=> sin(x)=(e^(ix)-e^(-ix))/(2i)

Once you know this, then integration and derivation of all sin/cos and derived functions boils down to algebra and derivation and integration of e, which is trivial.

I cannot tell you how angry I was with them for not teaching me this until well after integral calculus.

I, like the poster, was never taught this handy little gem, but figured it out proving e^(pi*i) = -1 for a class project in high school. Ever since could never understand why people had such a hard time with the integrals of trig functions.
posted by ChasFile at 1:24 AM on September 26, 2005

I didn't take the time to wade through the bulcomments on SlashDot, but MeFi is one of my favorite daily on-line staples, and this prompted me to sign up for an account!

This "Rational Trigonometry" is old stuff with a new, catchy label. Anyone who studied linear algebra should know part of it, and anyone who studied computational geometry should know most of it.

I did submit the PDF file for inspection to my former teacher of C.G., and he had the following comments to make (sorry for the jargon):

• Prof. Wildberger tries to get rid of roots by carrying the squares all the way through; this is not a great discovery from a computational standpoint and it is deprecable from a pedagogical one (roots are difficult, so let's get rid of them… and mankind is happier);
• At least in the first chapter, there is no mention of non-algebraic trascendental numbers (like pi), who have nothing to do with roots and as such cannot be cured that way (computational geometry did face this problem some fifteen years ago);
• it is most disappointing that he wants to get rid of trigonometry by making use of orthogonal projections, but without mentioning that both the concept of angle and the concept of orthogonality are children of the same mother: the scalar product, and thus pre-Hilbert spaces.

Again, leaving the jargon aside, nothing really new; these "non-trigonometrical" methods have been textbook material for a long time!

Our 4 cents :)
posted by pino.it at 1:36 AM on September 26, 2005

The bits about the history of math read like the Timecube guy wrote them. I'm having trouble convincing myself that the author is actually a professor of mathematics, despite the UNSW math department website.

Also, he acts like no one has ever thought of doing geometry over arbitrary fields before. This would come as a surprise to Hilbert, who Wildberger himself cites! Hilbert did fundamental work in Algebraic geometry, which deals with geometry over arbitrary fields. There are lots of other really weird things going on in this paper.
posted by samw at 1:48 AM on September 26, 2005

Also, Wildberger completely ignores the biggest problem with using quadrance and spread. They aren't additive. If I put two line segments end-to-end, the quadrance of the result is not the sum of the quadrances of the initial segments. Quadrance and spread aren't measures.
posted by samw at 1:58 AM on September 26, 2005

Aside from those who go on to be scientists or engineers, I just don't see a lot of usefulness in teaching as much trig as they do in high schools. Given how widely used statistics are in the media and in political debates, the extent to which most people can't identify their wanton misuse suggests to me that teaching some basic stats to the high school crowd might be a better use of class time. Personally, I am now working on my M.Phil at Oxford in the social sciences. While I've never once used the trig I learned in AP Calculus, I use first year statistics almost every day.
posted by sindark at 2:07 AM on September 26, 2005

This might be a good idea, if for no reason that an increasing number of math teachers do a really, really poor job of explaining those concepts which are being struck out of the curriculum in Wildberger's proposal.
posted by clevershark at 4:38 AM on September 26, 2005

er, "if for no reason *other* than..."
posted by clevershark at 4:39 AM on September 26, 2005

I read through the sample chapter a while ago, as far as I can tell the author only invented new terms for old concepts. During a boring summer job as a security guard while I was an undergraduate I decided to delve into graphics programming, especially 3D transofrmations and projections. In high school I had done it once before but I had directly calculated sin and cosine in order to do the math. Since I only had a Commodore 64 this was painful even with the relatively low resolution. In university I realized that actually calculating sin and cosine every time was a waste, even on my incredibly fast 25 MHz 486. The first optimization I did was to store sin and cosine values. The next was realizing I always used the square of them and storing those values.

That said it's more of an optimization technique in my opinion than a better way of teaching trigonometry. The circle and it's relationship to pi is terribly important for really understanding trigonometry.

sindark, in the US stats aren't taught in high school? I learned about basic statistics in either grade 12 or 13 in Canada.
posted by substrate at 5:39 AM on September 26, 2005

Yeah, this is stupid... people acting like it's some revolutionary thing, all they did was restate the basics... And sin/cos are used all the time in calc classes.

Also:

e^(ix)=cos(x)+i*sin(x)
=> cos(x)=(e^(ix)+e^(-ix))/2
=> sin(x)=(e^(ix)-e^(-ix))/(2i)

Is that as simple as it looks? Why did they try to teach us these insane trig identites then? It seems like you could integrate/difrentiate pretty much any trig function like this, what's the deal?
posted by delmoi at 8:55 AM on September 26, 2005

Integrating complex numbers can get a little tricky. It's all well and good as long as everything cancels out and you end up with a Real solution - but if it doesn't, and you end up having to explain branch cuts and Complex path integrals to your calc 1 class, I'm not sure you've made things better. The trig identities (which I hate) do allow you to solve a Real problem without leaving Real numbers, which is nice.
posted by freebird at 9:01 AM on September 26, 2005

This teaching method may work out great for carpenters and the like, but it would really hinder students who ultimately continue on in math, physics, or engineering. The trig functions pop up spontaneously when integrating really simple rational functions, and the ability to jump between cartesian and polar coordinate systems is a necessity. Also, "rational trigonometry" would really suck for dealing with physics/engineering problems involving, say, rotation with constant angular velocity.

So yeah, not terribly impressed. On the other hand, more students should be aware of these relationships, as they can certainly save a lot of work in some situations.
posted by Galvatron at 9:15 AM on September 26, 2005

I figured this out when I was 9, playing around with a protractor and ruler. When I was finally taught trig formally, it was "oh wow, this is such a cooler method of doing it."
posted by PurplePorpoise at 12:21 PM on September 26, 2005