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## Trigonometry Redefined without Sines And Cosines966

Posted by CowboyNeal
from the numbers-and-stuff dept.
Spy der Mann writes "Dr. Norman Wildberger, of the South Wales University, has redefined trigonometry without the use of sines, cosines, or tangents. In his book about Rational Trigonometry (sample PDF chapter), he explains that by replacing distance and angles with new concepts: quadrance, and spread, one can express trigonometric problems with simple algebra and fractional numbers. Is this the beginning of a new era for math?"
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## Trigonometry Redefined without Sines And Cosines

• #### No sines and cosines? (Score:5, Funny)

on Saturday September 17, 2005 @08:33AM (#13584108)
Perhaps Dr. Wildberger is trying to take geometry off on a weird tangent.
• #### Re:No sines and cosines? (Score:3, Insightful)

Actually, it does look like just a tangent of traditional trigonometry. After reading the first chapter, most of his math seems to be the switching forms of the Pythagorean theorem from:

(a^2 + b^2)^(1 / 2) = c

to:

a^2 + b^2 = c^2

With a lot of just applying that paradigm to every aspect of trig. Pretty nifty time saver, but I fear the unique insights from this method may be few.

• #### Re:No sines and cosines? (Score:4, Interesting)

on Saturday September 17, 2005 @08:59AM (#13584243) Homepage Journal
a^2 + b^2 = c^2
That's the way that I learned it and we still had traditional trig.

What did I miss?
• #### Re:No sines and cosines? (Score:3, Insightful)

It wasn't intended to give rise to unique insights. It was intended to simplify the teaching and calculation of geometry.
• #### Re:No sines and cosines? (Score:5, Funny)

on Saturday September 17, 2005 @10:03AM (#13584621)
Well, when Dr. Wilberger explained his great idea to his close circle of friends. They were all in a chord.
• #### Re:No sines and cosines? (Score:5, Funny)

on Saturday September 17, 2005 @10:10AM (#13584675)
After spending several hours trying to explain his theorem to his wife, he determined secant understand it. Ok, I'm stretching....
• #### Re:No sines and cosines? (Score:5, Funny)

<biryokumaru@gmail.com> on Saturday September 17, 2005 @10:26AM (#13584758)
A mathematician was trying to explain something to someone who isn't a mathematician? Of cosecant!
• #### Re:No sines and cosines? (Score:3, Funny)

So, is this collection of puns now a hyperbolic cotangent?

• #### Re:No sines and cosines? (Score:5, Funny)

<<moc.oohay> <ta> <otropoeinre>> on Saturday September 17, 2005 @12:42PM (#13585556) Homepage Journal
He even has the positive testimonial of Barbie, who now claims "math is easy."
• #### Now ... (Score:3, Funny)

on Saturday September 17, 2005 @08:35AM (#13584115) Homepage Journal
If only he could redefine Calculus to use simple algebraic expressions.
• #### Re:Now ... (Score:5, Interesting)

on Saturday September 17, 2005 @08:43AM (#13584161)
Actually, i think this is a perfect point. My major problem when learning integral calculus was my utter loathing for trigonometric identities (and my inability to remember them) required for solving all sorts of weird integrals. I'm curious how this stuff plays with calculus. If it does, maybe this'll make my life easier if i ever go back and attempt calculus again. anyway, reading TFA, hopefully it says something regarding this :)
• #### This reminds me of a test in grad school (Score:5, Interesting)

on Saturday September 17, 2005 @09:28AM (#13584402) Homepage
I was taking a real analysis class in my first semester of grad school. I did a good job on the first question, figured everything out, and got that the answer was the integral of 1/(1 + x^2).

I got no points at all for the question - despite solving the parts relevant to the class - because I didn't know off the top of my head that the integral of that is the arctan function.

I love abstract math but I hate trig.
• #### Re:This reminds me of a test in grad school (Score:3, Informative)

Hi,

If you have some experience in solving integrals of that sort, the substution x = t is pretty standard.

In this case letting x = tan t is very productive. Working through the algebra one finds that (TeX notation)

Just remembering $\tan = \sin/\cos$ and $\cos^2 t + sin^2 t = 1$, on can work out the following:

We have $1/(1+x^2 = 1/(1+\tan^2 t = \cos^2 t$

Also $dx = 1/cos^2 t dt$, therefore

$\int_0^a \frac{1}{1+x^2} = \int_0^{\tan^{-1} a} 1 dt = \tan^{-1} a$

So you don't have to remember the form of the integ
• #### Re:Now ... (Score:5, Interesting)

on Saturday September 17, 2005 @09:33AM (#13584428)
As my Quantum Chemistry professor once said: "Trigonometry is just two formulas: cos^2(x)+sin^2(x)=1 and exp(ix)=cos(x)+i*sin(x). I don't know how they can blow it up into an entire 16 week course"
• #### Re:Now ... (Score:4, Informative)

on Saturday September 17, 2005 @01:18PM (#13585744)
Not to be a pain, but actually you only need exp(ix) = cos(x) + i*sin(x)! Since exp(-ix) = cos(x) - i*sin(x) (just remember sin is odd, cos is even), you can multiply 1 = exp(ix)*exp(-ix) = cos^2(x) + sin^2(x). So the first formula is actually encapsulated in the second, which is ALL of trignometry!
• #### Re:Now ... (Score:5, Interesting)

on Saturday September 17, 2005 @11:53AM (#13585279)
Sadly, you had this problem because those bastards never ever let you in on the secret:

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.
• #### Re:Now ... (Score:5, Informative)

<wongwwy.member@ams@org> on Saturday September 17, 2005 @12:12PM (#13585370) Journal
No, it would make learning Calculus all the more painful. He admits in his first chapter that the transcendental trignometric functions "cannot be understood without a better understanding of calculus". The same can be said in reverse. His "prettification" of geometry, while simplifying trigonometric calculations, makes general geometry and calculus more difficult.

For example, instead of working on a Euclidean affine coordinate system, by using "Quadrance" as he calls it, the coordinates would not be translation invariant, and you will be forced to attach a non-trivial measure to make integrals work out. So while the integrand might be simplified in the trigonometric identities, you will end up, instead of integrating over "dx", over something like "1/sqrt(x) dx", which hardly makes the integral any more appealing.
• #### Re:Now ... (Score:3, Insightful)

I wouldn't have any problems with (yet) an other mathematical notation and method. In any case we use different notations for various rules of physics (tensors, vectors, fourier transformations etc.) depending on the aim and whatever method is easier for the problem. The problem would be teaching high-school level pupils because at that age you usually accept anything you are thought as the norm and then get confused when you are in the university and someone shows something completely different (tensors an
• #### The method doesn't matter, as long as the answer (Score:3, Funny)

on Saturday September 17, 2005 @08:35AM (#13584117)
is 42
• #### Wonderful! (Score:5, Insightful)

on Saturday September 17, 2005 @08:35AM (#13584118) Journal

I did a stint as a Maths teacher, and it was hard enough trying to convince the kids that it was worth learning Trigonometry then. They'll be even more determined to be ignorant if they hear of this.
• #### Don't worry... (Score:3, Insightful)

As a math teacher, it may not be as obvious to you, but as someone who learned trigonometry in school and isn't a math teacher now I can say for certain. When people say they'll never use that in the real world, they're absolutely right.

Now sometimes while slogging through my day, paying bills, shopping, working and things like that I sometimes say "Man I really should calculate the cosine of this electric bill", but as of yet I haven't been harmed by not actually doing that.
• #### Re:Don't worry... (Score:4, Insightful)

on Saturday September 17, 2005 @09:04AM (#13584271) Homepage
I think it kind of depends on what you do for a living. If you work at Wal-Mart, you probably don't need it. If you are and Physicist or Engineer (like my wife and I), you probably do. And thats hard to know in High School
• #### Not just physicists or engineers use trig.... (Score:5, Insightful)

on Saturday September 17, 2005 @09:12AM (#13584312) Homepage
Even kids who go into trades like carpentry would benefit from a knowledge of the fundamentals of trig. Laying out angles for roof rafters, staircase stringers, etc....
• #### agreed (Score:4, Insightful)

on Saturday September 17, 2005 @09:38AM (#13584445)
My dad is a machinist, and that job is very heavy on trig. If you're trying to figure out how to calculate a correct taper or how to calculate threads, you need to know trig.

• #### Re:Don't worry... (Score:5, Insightful)

on Saturday September 17, 2005 @10:09AM (#13584663) Journal
But its pretty easy to know that you only have a slightly greater chance of being a physicist than you do of being a profesional basketball player. You dont see us trying to train our kids to be basketb... Oh shit. Yep were fucked. They will end up at Wal-Mart.
Luckily its a great store for Physici...
Do you need a cart sir?
• #### Re:Don't worry... (Score:5, Insightful)

on Saturday September 17, 2005 @09:06AM (#13584281) Homepage
Everyone complains how trig is not useful, but perhaps its useful because it is hard and is one of the few things left in schools today that actually mentally challenges students.
• #### Re:Don't worry... (Score:5, Insightful)

on Saturday September 17, 2005 @09:10AM (#13584296)
I take it you taught math in elementary school (K to 5th) then, as your point is completely wrong. For a physicist or computer scientist, the principles of trigonometry are invaluable. All those games you might play. All those electronic devices (cell phones, tv, etc) you use on a daily basis. Much of the theory used to devise how they could possibly work was done with *gasp* trigonometry and to a greater extent, calculus.

Simply because you choose a profession does not use it, does not mean it doesn't have value.
• #### Re:Don't worry... (Score:3, Insightful)

There are many people who use trig in the real world all the time. How is a student in high school supposed to be able to make the final decision that they will or won't be one of these people?

A lot of the point of learning math is keeping your options for the future open.

• #### Re:Don't worry... (Score:3, Insightful)

Maybe someone should make a list of professions you rule out doing well at if you don't learn about trig -- I don't think it's just scientists, carpenters, surveyors, engineers, mathematicians, navigators,.. that need trig.

I don't understand why Math gets singled out so badly. How many people need to use the details of history in life, after all, and Literature classes , Speech classes are also of dubious merit, after all -- most people won't be historians, professional writers, speakers, or politici

• #### Re:Don't worry... (Score:3, Funny)

Don't be so sure. Chief Sohcahtoa helped me figure out how long my Christmas lights need to be to fit along my roof line. Thanks Chief!
• #### Re:Don't worry... (Score:4, Insightful)

<elentar@symp a t ico.ca> on Saturday September 17, 2005 @09:44AM (#13584480) Homepage
Don't try game programming--it's all trigonometry. Same goes for most engineering.

This sounds like a variant of trig calculations that you often use in computer algorithms, where it's much faster to calculate the square of something than the root. If you do it right, you can avoid roots completely for comparisons, and only do one at the very end of the calculation for actual lengths and distances. Sine and cosine usually appear as the quotient of lengths of sides of a triangle--you rarely calculate sin(x) or cos(x). The one place where roots are unavoidable is normals, which are just so damn handy. But even there you can sometimes get away without normalizing for comparisons in things like backface culling.
• #### Re:Wonderful! (Score:3, Insightful)

Notice that you hardly ever hear the question of usefulness in the real world in a music or art class.

I think one big problem is that people are given the impression that mathematics has something to do with the real world, and that it's supposed to be "useful". (Well it is, but not for the obvious reasons.)

Mathematics really just consists of a bunch of structures. These structures can be really quite beautiful on their own, and if it's presented the right way, people should see some reason to study mathema
• #### Figures. (Score:5, Funny)

on Saturday September 17, 2005 @08:35AM (#13584119)
He does this the year after I take Algebra II/Trig. Bastard.
• #### The "New" has an initial capital for a reason (Score:3, Informative)

on Saturday September 17, 2005 @08:36AM (#13584123) Homepage
The "New" refers to South Wales, rather than the uni. Might wanna read a bit closer next time.
• #### Wow (Score:4, Interesting)

on Saturday September 17, 2005 @08:36AM (#13584124)
I'm a failing engineering student at Iowa State University- or at least I was.. My failing point: math. I can out code some of the instructors in my classes, but I can't do math or memorize functions and derivitives of trigonometric functions.

If this book pans out, it would ultimately change Calculus for the better and might allow me to pass my classes- I wonder if I figured out how to apply this book to my classes and came up with the correct answers despite having worked the problems with his methods, if I'd still pass the class?
• #### Re:Wow (Score:5, Insightful)

on Saturday September 17, 2005 @08:57AM (#13584230)

If you want to be the kind of engineer that implements other engineer's ideas then, by all means, blow off your math classes. But if you want to be someone who your peers turn to when they need help, do yourself a favor and learn the math.

All of the engineering sciences are founded on math (this is espescially true of computer science). If you can out code your instructors, that means you can probably out math them too. What you are interpreting as an inability to memorize functions, is probably really just disinterest.

This disinterest may stem from a feeling that what you are studying has little utility, it may stem from a personal dislike of an instructor, it may stem from the notion that math geeks are all squares and smell funny.

Whatever the reason, you need to get past it. A thorough understanding of the math behind engineering will make your life MUCH easier on down the road.

• #### Re:Wow (Score:5, Interesting)

on Saturday September 17, 2005 @09:36AM (#13584438) Homepage Journal
Here's an easy way to remember the integrals and derivatives of trigonometric functions.

But first, this small reminder:
sin x (vertical component)
cos x (horizontal component)
tan x = sin x over cos x
sec x = 1 over cos x
csc x = 1 over sin x
cot x = cos x over sin x

-> sctsct

Now we substitute these trig functions with simple symbols:

I = sin x (vertical component)
II = cos x (horizontal component)
III = tan x = sin x over cos x
IV = sec x = 1 over cos x
V = csc x = 1 over sin x
IV = cot x = cos x over sin x

Think of them as one, two, three, four, five, and six. Now it boils down to remembering simple combinations of numbers:

integral{ I } = -II
integral{ II } = I
integral{ III } = ln | III + V |
integral{ IV } = ln | IV + VI |
integral{ V } = ln | III |
integral{ VI } = ln | I |

Once you write down more of these combinations, you'll discover patterns in it and from there on it should be easier than ever to remember trig integrals. And when you know most of the trig integrals, you will know most of the trig derivatives, too! :)

If anyone is interested in some more documentation on this, then by all means contact me. I am in the process of writing these things down. They should be available some time next week on my homepage.
• #### Or you could just sketch the functions (Score:3, Informative)

and figure out the derivatives that way.
• #### Re:Wow (Score:4, Insightful)

<(moc.oohay) (ta) (dlonrasg)> on Saturday September 17, 2005 @09:47AM (#13584507) Homepage Journal

I don't know quite how to put this, so I am just going to say it.

The degree doesn't make you an engineer. The MATH makes you an engineer. The degree is just your univerity vouching that you have completed your math and other engineering studies competently.

.. or did you think you could argue a structurally unsound bridge you designed to be more sympathetic and resist collapsing because the math in college was too hard?

In my opinion, I think the author of this book is a quack and all I had to see was the first paragraph on the first page of his web site where he states that he has dispensed with (geometric) axioms. You cannot do anything in mathematics without axioms. Period. Math is not capable of proving something from nothing.
• #### UNSW .. not South Wales (Score:5, Informative)

on Saturday September 17, 2005 @08:37AM (#13584126)
As per the article .. Dr Wildberger is from UNSW, the University of New South Wales .. in friggin' Australia. South Wales is somewhere all together different. But as always people don't RTFA
• #### Hopefully (Score:3, Insightful)

on Saturday September 17, 2005 @08:37AM (#13584127)
This sounds promising, but I have two educational concerns: 1. Is this just a dumbed-down version of trig? ...and on the opposite end of the spectrum... 2. Would this lead to bombarding students with too much math as the requirement shifts from alg/calc/trig to alg/calc/trig/trig2?
• #### Just Wait... (Score:4, Interesting)

on Saturday September 17, 2005 @08:38AM (#13584133)
Just wait until someone reimplements many software functions that rely on sine, cosine, and tangent, using these new concepts, and get a patent on them.
• #### Better LInk (Score:2, Redundant)

DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry by N J Wildberger [unsw.edu.au] is a link to the advertisment for his upcoming book, which also has a PDF dowbload of the first chapter.

Oops .. just realised that *I* didn't RTF summary .. my bad
• #### Redefinition? (Score:3, Insightful)

<AKrotkov@gmail.com> on Saturday September 17, 2005 @08:41AM (#13584150)
Erm, I actually read the sample chapter, and one thing I don't get: What did he do that is so revolutionary?

He redefined a side of a triangle with a Quadrance - a square of distance. He claims this removes the square root, but guess what? d^2 has the same effect.

He also defined a spread, which is the relationship between two lines. The catch? It is PROPORTIONALLY EQUAL to an angle. 90 degrees is 1, 0 degrees is 0, 45 degrees is 1/2.

I haven't read the full book, but from what I can tell, all this is doing is redefining the constraints of trigonometry, causing nice even numbers to be used.
• #### Re:Redefinition? (Score:2, Insightful)

If you were a programmer that relied on an implimentation that used traditional trig, you would understand why 'redefining' the route to the correct answer to use simple algebraic expressions would be a good thing...precision. I am a computer graphics enthusiast and I dwell in alot of 3d math that involves calculus (mainly all sorts of complex curves). Guess what, that crap all likes trig! If I can define the formula for a three dimensional sphere without trig, thank you, thank you, thank you. I'm gonna g
• #### Re:Redefinition? (Score:5, Insightful)

on Saturday September 17, 2005 @08:53AM (#13584213) Homepage
Spread is NOT proportionally equal to an angle. 30 degrees is 1/4 and 60 degrees is 3/4

spread is the square of the sine of an angle.
• #### Uh... (Score:2)

It develops a complete theory of planar Euclidean geometry over a general field without any reliance on `axioms'.

Uh... that's not just redefining trig, that's totally redefining mathematics and logic. I find that hard to believe. Is it just marketing talk? Or did this guy revolutionize the axiomatic system upon which we built all human knowledge? I find the latter doubtful.

And it shows how to apply this new theory to a wide range of practical problems from engineering, physics, surveying and calculu

• #### Re:Uh... (Score:2)

the axiomatic system upon which we built all human knowledge You think your knowlege of where you parked your car is built on an axiomatic system?
• #### SOHCAHTOA and abstract survery results (Score:5, Insightful)

on Saturday September 17, 2005 @08:44AM (#13584164) Homepage
ahh Sin= Op/Hyp

By as someone who's done some surveying it (which uses angles and distance), its easy to see why people use angles and distance. They are fairly easy to measure, so usefull for plans etc..

Quandrance = distance ^2
• #### Re:SOHCAHTOA and abstract survery results (Score:3, Informative)

The concept of spread is actually pretty straight-forward. Basically, given any two lines L1 and L2 that intersect at a single point O (parrallel lines are too trivial), spread is, informally, a function of their 'shortest quadrance (distance^2) apart'. Formally:
(1) take any point A on the first line L1. Denote qudrance between O and A is Q.
(2) compute the shortest distance between A and the other line L2 and square it for quadrance (call the quadrance R)
(3) spread between L1 and L2 = s(L1,L2)=R/Q

Calcula
• #### Faster calculations ?? (Score:5, Interesting)

on Saturday September 17, 2005 @08:45AM (#13584176)
I am wondering if this could be used to make faster calculations
in raytracers and 3D engines by using integer numbers.
• #### Re:Faster calculations ?? (Score:3, Interesting)

by Anonymous Coward
AFAIK most 3D engines already use tables with values for different angles and extrapolate for faster trigonometric calculations since you don't need that much precision in a game anyway
• #### Yes, for some badly written code (Score:5, Interesting)

on Saturday September 17, 2005 @09:45AM (#13584486) Journal
The guy is a little mad but his points are basically sound.

If you implement a geometrical function whose input is a bunch of lengths and their ratios and whose output is a bunch of lengths and their ratios then you frequently shouldn't need any trig in your function, instead you should be using algebraic functions (+, -, *, /, nth root etc.). But many people find it easy to solve their problem by converting a bunch of stuff to angles using trig and then converting back. In fact, I optimised someone's rendering code recently by literally looking for every line of code where he'd used trig and replacing it, where possible, with the algebraic version. It worked well and speeded up the code massively. In addition I uncovered what you might call bugs. For example at one point this guy did a 180 degree rotation by converting to polar coordinates, adding M_PI to the angle and converting back. Rotating (x,y) by 180 degrees gives (-x,-y). The guy who'd written it was 'trig happy' and should have stopped to thing about the geometrical significance of what he was doing rather than just doing the obvious thing. If this book encourages people to use trig less it might be a good thing. But you don't need to talk about 'quadrance' and 'spread' to use the principles I'm talking about.

But of course there are times when you need trig. I'd hate to see the guy differentiate rotations without trig.

• #### Re:Faster calculations ?? (Score:4, Insightful)

on Saturday September 17, 2005 @11:57AM (#13585296) Homepage
I doubt it. In the end, the numerics are probably the same. Inside the computer, nobody computes "sines" they compute truncations of infinite series. In general this guy's computations will also end up with infinite series that need to be truncated (for example taking the square root at the end). It doesn't really matter, therefore, when it comes to numerical computation. A square root and a sine are very similar if you're a computer.

Furthermore, a lot of what this guy did is kind of a trick. Using 'spreads' may work when given an explicit triangle, but the part he's skimming over is that spreads are missing a REALLY nice property of angles. They don't add. Angles are a very nice parameter for rotation because a rotation of 10 followed by a rotation by 20 is the same as a rotation by 30. This property is implicitly used all over the place in graphics. So, in the end you probably have to use some angle-like measure when doing computer graphics (which is all about transformations, not measurements of unknowns). And in doing so, I'm sure you end up computing sines and cosines to do projections based on those rotations.

In the end, you just can't cheat your way out of the fact that a projection based on a rotation is a transcendental operation that numerically requires computing a truncated infinite series.

• #### Interesting - but not entirlely new (Score:4, Insightful)

on Saturday September 17, 2005 @08:50AM (#13584192) Homepage
The professor seems to have found an interesting way to present trig - however, it should also be noted that he does actually rely on sines as an underlying concept.

I read the article and part of hist concept of spread for an angle is the ratio (in a right triangle) of the side opposite the angle to the hypotenuse of the triangle. As I recall from my high-school geometry classes, though, this is exactly how the sine is calculated. (The cosine is the adjacent side divided by the hypotenuse.)

The interesting thing about his approach, though, is that he defines the concept of spread so that there's no need to refer to the underlying sine concept and the calculations are all (relatively) simple algebra - squaring and addition and subtraction. I would like to read some more and play with it a bit - the fact is I still have bad dreams about those damned trig identity tables, which I never really successfully felt comfortable with!

Interesting.
• #### Great for eighth grade, but ... (Score:5, Insightful)

on Saturday September 17, 2005 @08:56AM (#13584225) Homepage
What happens when kids get to math subjects where trigonometric functions are used for more than just calculating the dimensions of geometric figures? How does this "spread" thing represent angles greater than 180 degrees without redefining what you are measuring, and how does this really make a persons life any easier unless someone tells you the spread as in the textbook chapter? If his explanation is to be taken as any sort of indication of how to measure the spread, then you might as well just walk off the dimensions of what you're measuring because you'll have to do that anyway to calculate it. How will you integrate problems that call for constant rotation using spread? This seems like trading a little bit of pain now for a lot more down the road, and I pray that it won't catch on in the US. If students are really having this much trouble with this subject then it should be introduced earlier and in smaller portions, not ignored. The last thing we need is for someone to take another slice out of the already anaemic math programs in our primary schools.
• #### Re:Great for eighth grade, but ... (Score:3, Interesting)

I don't trust anyone who claims that a proper definition of angle requires the calculus. I wonder if this guy has ever read "Foundations of Geometry" or heard of its author David Hilbert.

This stuff is junk. On page 8: Square roots are to be avoided whenever possible.

Followed by page 16: To convert back to distances, take square roots.

He claims that sines and cosines are hard because the poor student can't calculate them by hand. How many here can extract a square root by hand?
• #### I don't see how this is "easier" (Score:5, Insightful)

on Saturday September 17, 2005 @09:02AM (#13584254)
Imagine if we'd been using "quadrance" and "spread" for years - and then some bright spark suggested calculated using sines and cosines. Which would people see as easier?
• #### Bah!! (Score:3, Funny)

on Saturday September 17, 2005 @09:11AM (#13584310)
Now THESE are some divine proportions. [ariagiovanni.com]
• #### Is this silly? (Score:3, Interesting)

on Saturday September 17, 2005 @09:14AM (#13584321) Homepage
It looks like all that is being done is removing squareroots and negative numbers.

quadrance is the square of the distance
spread is the square of the sin angle

If you want to teach trig this way, it will certainly be easier to learn. But then, your students will not appreciate or understand square roots and negative numbers.

It is highly unlikely to lead to any revolutionary new algorithms. It appears that you are changing an angle to something rational, because all most of the popular sine values have square roots in them. However do not be fooled by this, it is just as hard to calculate spread as it is to calculate the sine of an angle. (except in a few cases).
• #### Galileo got there first (Score:3, Interesting)

on Saturday September 17, 2005 @09:14AM (#13584324)
Well, nearly. The reason that Newton is regarded as the originator of modern kinetics is that he derived the formulae that link acceleration,mass,velocity and time. In fact, Galileo got part of the way there but his unit of "speed" was the square of velocity. This meant that his comments about the relation between acceleration, time and mass were correct but his velocity unit was not useful, because in the real world we most typically want to be able to use the simple relationship between velocity and time. If car speedos were calibrated in metres per second squared, we would not be able easily to work out how long it takes to travel a given distance.

In the same way, as I understand it based on the sites referenced in the story, this guy makes use of an angle replacement that is not useful to most people. If I want to divide a circular cake into n portions, using spread will not be helpful. Although degrees are a unit with no rational basis(there are not 360 days in a year, and 480 might be better-divisible by 5,3 and powers of 2) they are relatively easy to use for circle division, which is all the circle math that most people need to do most of the time.

Unfortunately, mathematics lectures at Cambridge left me with the permanent belief that mathematicians' ideas of what is simple and what is complex merely illustrate the fact that physics and engineering use math, but they use a useful subset (which changes with time) and do not necessarily buy into ideas which mathematicians regard as self-evident.

• #### I looked at one of his first examples (Score:3, Interesting)

on Saturday September 17, 2005 @09:23AM (#13584370) Journal
The one where the solution involves sqrt(7). The fact is, you don't need trig to solve that problem and people shouldn't be using trig to do so. His approach isn't new, it's what a mathematician should do anyway. If there's one thing that is taught wrong it's a tendency to use trig when pythagoras's theorem and similar triangles will do the job anyway. But this guy isn't doing anything new.
• #### What a silly non-sense (Score:3, Insightful)

on Saturday September 17, 2005 @09:47AM (#13584500)
Sorry to spoil the fun, but while his approach is another way of presenting trigonomic geometry that some people might find cute (I don't care for it), this buzz about "establishing new foundations" of geometry is absolute non-sense.

Here is what he does: He replaces the distances by its square, and calls it a squandrance. He replaces the angle by the square of its sine and calls it a spread.

Ok, the relations between the squandrances and spreads of a rectangular triangle are simpler than those between its sides and its angles -- they are just simply obtained from pythagoras' theorem.

However, two way more fundamental relations suddenly become horribly difficult: Say going from town A to C I go from A to B then from B to C. A to B is 5 squandrance, B to C is 3 squandrance, how much squandrance is it from A to C? No, not 8, but 5 + 3 + 2*sqrt(8). The simple addition of distances becomes a square-root function...

Also, say I turn left by 30 degree, and then do that again. Guess what, I turned left by 2*30 = 60 degrees. However, if you were doing this in spreads, you don't want me to tell you the answer. I think there is a reason why sailors have been using angles and not spreads...

As for engineers: Why do they have to learn trigonometry? Not for geometry, you can just do that with coordinates. But because it turns up in waves everywhere, which in turn turn up everywhere. (Aside: the mathematical reason is that sine and cosine are the solutions of one of the most fundamental differential equations.) For example, the voltage of an AC current will depend proportionally on a sine fucntion. The angle you substitute in sine will be proportional to time. The spread you would have to substitute in a rational geometry function instead is NOT proportional to time, but to the square of the sine of time. Too bad my stopwatch measures time in seconds and not in the square of its sine...

Will his approach lead to faster computations? Of course not. Whereever it would help, people of course already knew last year already (and maybe even, gosh, in 2003) to use some trigonometric substitution....

• #### Most of you missing the point. (Score:5, Insightful)

on Saturday September 17, 2005 @09:49AM (#13584521)

I am a high school mathematics teacher and I train students for mathematics competitions. I think most of you are missing the point of Dr. Wildberger.

Dr. Wildberger is not trying to redefine trigonometry, he is simply trying to give it a new perspective and hopefully, the new perspective would allow new insights into new methods of solving trigonometric problems. Protesting that memorizing the trigonometric functions as side adjacent over side opposite, etc., etc., is very easy and intuitive ignores the fact that in analytic geometry, that is not even how the trigonometric functions are defined!

Yes, really! For example, the sine of theta is defined in analysis as the y component of the radial vector from the origin to a point in a circle of unit radius whose arc distance from the x-axis is theta. The cosine of theta is defined similarly but this time taking the x-component. From this two simple definitions, the entire panoply of the trigonometric identities can be usefully derived!

The analytical definition is certainly not intuitive and not easy to memorize for a high school student! The side opposite, side adjacent trick is just that, a trick that is useful sometimes and certainly useful enough for high school mathematics but it is not a very useful definition as far as analysis is concerned.

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! Not to mention the fact that most students think there is magic involved in the computation of the trigonometric functions because the method of computation is not in their textbooks. It is only when one studies the calculus that the methods for computing the trigonometric functions are explained!

Dr. Wildberger has an idea that he thinks will make trigonometry more intuitive and I hope he is really onto something here. It would certainly help me with my students. I have read only the downloadable first chapter of the book and the idea is intriguing. Waving off Wildberger's new ideas without reading the entire book and without understanding the mathematics of trigonometry is just tragic.

In times like this I always remember the architect (I forgot the name, help me out here please) who refused to accept an architecture medal because the society that was giving out the medal invited Prince Charles to hand out the medal. That architect said, "I refuse to accept a medal from a person who believes that our grandfathers already know everything there is to know about how to build buildings and that there is nothing we can ever add to that knowledge anymore."

Just my two cents.

• #### No, I think you miss the point (Score:3, Insightful)

The Wildberger version is harder to understand because most school students will never understand mathematical theory, they will understand only things that relate to real world examples. And squaring distance and angle is not a concept that is easy to relate to the real world. I taught math for some years before finding that engineering paid a lot better for less stress, and while the more gifted pupils would understand this stuff, they were also the ones who did not find sines and cosines hard. For the ma
• #### Very nice. Makes sense to a game programmer (Score:5, Informative)

on Saturday September 17, 2005 @10:50AM (#13584908) Homepage
Most of the relationships Wildberger explains are well known to those of us who write physics engines, or the more geometrical parts of game engines. Trig functions are too expensive to use in inner loops, and their corner cases are annoying. If at all possible, everything is done with linear operations on vectors, matrix multiplies, and quaternions. These operations not only go fast, they parallelize; all 16 multiplies of a 4x4 matrix multiply can be done simultaneously, and every modern graphics card has the 16 multipliers necessary to do that.

Wildberger has put a cleaner theory underneath the kind of geometry game engine developers use. This may turn out to be useful.

Lately I've been doing robot motion planning, which has too much unnecessary trig in it. With enough work, it's often possible to derive a trig-free solution to some of the key problems. Better ways to think about trig-free solutions will help.

• #### But if you graduate at Dr. Wilberger's group... (Score:3, Funny)

on Saturday September 17, 2005 @11:36AM (#13585190)
you still get a degree!

Bert
• #### Trig is not hard, it's just taught REALLY badly (Score:3, Informative)

on Saturday September 17, 2005 @11:40AM (#13585212)
Trig should be about a 1 to 2 week topic in school. If instead of having students memorize endless identities you simply teach them 1 (Eulers equation) and show them how to easily derive the rest then it becomes pretty trivial.

Euler's equation:

e^(i*x) = cos(x) + i*sin(x)

Need a double angle formula? No problem.

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

So you can clearly see that

cos(2*x) = (cos(x))^2 - (sin(x))^2
sin(2*x) = 2*sin(x)*cos(x)

All of the trig identities fall out as simply through simple algebraic games when you start with Euler's equation. It also makes looking at the calculus of trig pretty trivial, greatly simplifies the study of waves, etc. I cannot for the life of me understand why everyone doesn't teach it this way.
• #### What a completely silly idea. (Score:3, Interesting)

<wongwwy.member@ams@org> on Saturday September 17, 2005 @12:07PM (#13585338) Journal

It is not revolutionary, nor redefined. What it does it to rewrite trigonometry in similar thing as the good old "SOH-CAH-TOA" method taught in American high schools.

Wildberger's sole insight are the following:

• Instead of using the linear norm, he chooses to use the equivalent quadratic norm for distances, thus removing the squares from Pythagorean theorem. (So, for a right triangle, his version would be BASE + HEIGHT = HYPOTENUSE).
• Instead of using angles and calculating sines and cosines from it, he uses the concept of Spread, which is essentially just the sine of the angle squared!!
Well, one immediately sees a problem with the second point when trying to do something more than traditional planar Euclidean geometry: an obtuse angle will have the same spread as one other acute angle, and they share spreads with two other angles greater than pi radians!

His argument is foundamentally flawed even in the first chapter when he "explains" the traditional method of measuring angles. He claims it as the following (paraphrasing):

Take two lines, the measurement of the angle is taken by drawing a unit circle about the intersection point. For each line, choose a point in which the line intersects the unit circle. Take the arc length between the two points, and that gives the angle.
so far so good, but he goes on to argue that
For each line there are two choices of intersection points with the circle, resulting in a total of 8 different pairs with 4 different arc-length measurements.
That is complete bullshit. When I was in middle school in Taiwan, we were taught that 1) we start with two rays instead of two lines. The angle between rays A and B would be given by taking the arc length of on the unit circle by going counter clockwise from A to B. So, semantically, the angle between A and B, and the angle between B and A, would sum up to a nice 2pi. While colloquially, we often prefer to deal with the smaller of the two measurements.

There are obvious problems with the Quadrance and Spread concept when applied to real life. For one thing, you can't just add Quadrances: say I know that points A B and C are on a line, in that order. Say I also know the distance AB and BC, then the distance AC = AB + BC. But if I measure using the Quadrance, sqrt(AC) = sqrt(AB) + sqrt(BC) would hold. He is just passing the buck in his "redefinition". While making things simpler for trigonometry, he is making things more difficult for other bits of Euclidean geometry.

Similar, while traditionally angles can just sum, his definition using a spread cannot sum angles.

In summary, his "getting rid of sines" is just replacing it with its valuation, which is hardly novel: it is something taught standard in American classrooms (remember SAHCOHTOA?).

• #### "Decimal number plane"?? (Score:3, Interesting)

on Saturday September 17, 2005 @12:59PM (#13585639) Homepage
What the hell is that? I started reading the first chapter. OK, maybe there's something mildly interesting here; some calculations could be simpler expressed in these terms. But alarm bells went off when I read "decimal number plane" (let alone everything about how this will revolutionize mathematics).

He seems to mean real plane, but with a pejorative connotation of needing decimal numbers to do ordinary trigonometry.

Google the phrase (in quotes); you get exactly one hit - this book.

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