Trigonometry Redefined without Sines And Cosines

No sines and cosines? by Joey+Patterson · 2005-09-17 01:33 · Score: 5, Funny

Perhaps Dr. Wildberger is trying to take geometry off on a weird tangent.

Re:No sines and cosines? by biryokumaru · 2005-09-17 01:41 · 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.

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Re:No sines and cosines? by SilverspurG · 2005-09-17 01:59 · Score: 3, Interesting

a^2 + b^2 = c^2
That's the way that I learned it and we still had traditional trig.

What did I miss?

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Re:No sines and cosines? by smallpaul · 2005-09-17 02:37 · Score: 2, 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? by Gorobei · 2005-09-17 02:40 · Score: 2, Insightful

I doubt there are any unique insights from his approach: he's basically made angle the fundamental element of trig, rather than the more usual distance.

That said, it might be an interesting way to teach/explore trig. By doing away with the trig functions (which are just the distance->angle mappings,) he gets to solve many simple problems with just algebra and a final square root. Because the sqrt is explicit, this approach might give students a better mental model for trig: as things stand, most students just treat sin, cos, etc, as black boxes, and apply the rules (SOHCAHTOA) by rote.

I'd like to see how his method stacks up when applied to, say, an entire beginner level book or classic text, on geometry/trig.
Re:No sines and cosines? by multipartmixed · 2005-09-17 02:43 · Score: 2, Funny

My teachers would accept answers like "sqrt(c)" (except written with the square-root symbol).

Rumour has it that one year a freshman ran out of time on a trig exam, stuck on the first question, trying to write out the square root of two..

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Re:No sines and cosines? by Darth_Burrito · 2005-09-17 03:03 · Score: 5, Funny

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? by Darth_Burrito · 2005-09-17 03:10 · Score: 4, Funny

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? by biryokumaru · 2005-09-17 03:26 · Score: 4, Funny

A mathematician was trying to explain something to someone who isn't a mathematician? Of cosecant!

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Re:No sines and cosines? by Associate · 2005-09-17 04:41 · Score: 5, Insightful

Those black boxes are the reason that while I was relatively good at math, I sucked at trig, which screwed me when I got to calculus. I had always thought that when learning math, I could follow the steps to a solution which lead to an understanding as to why it worked. Black boxes, as you described it, do not do this.

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Re:No sines and cosines? by Anonymous Coward · 2005-09-17 05:36 · Score: 2, Funny

It's irrational :P
Re:No sines and cosines? by shokk · 2005-09-17 05:42 · Score: 4, Funny

He even has the positive testimonial of Barbie, who now claims "math is easy."

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Re:No sines and cosines? by techno-vampire · 2005-09-17 06:15 · Score: 5, Insightful

I wasn't taught trig functions as black boxes. We learned right from the start that they're the ratios of the various sides. Once you understand that, it's easy to know which function to use to find which side or angle, and why. Identities were just s easy: they're just formulas that don't depend on the angle; they're right for any angle, so you can use them to simplify equations. Trig was fun, and I was good at it, but that might be because my teacher understood how to explain it instead of simply demanding rote memorization.

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Re:No sines and cosines? by emandres · 2005-09-17 06:42 · Score: 2, Informative

This is an interesting enough concept, but the math involved with it would require a bit more algebra than I knew when I learned the trig basics. Also, this doesn't seem like it would have much practical application in calculus. Anyone who's ever taken calculus beyond just the basics can tell you that it is a pain in the butt integrating and deriving rational functions. Unless his replacements for sine and cosine, etc, are all related in the way they are in classical trig, it would be a nightmare trying to do the simplest of integrations, like proving the sine is the antiderivative of cosine.

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Re:No sines and cosines? by abb3w · 2005-09-17 10:05 · Score: 2, Funny

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

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Now ... by LordKaT · 2005-09-17 01:35 · Score: 3, Funny

If only he could redefine Calculus to use simple algebraic expressions.

Re:Now ... by NoTheory · 2005-09-17 01:43 · Score: 4, Interesting

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 :)

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Re:Now ... by miskatonic+alumnus · 2005-09-17 02:33 · Score: 4, Interesting

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 ... by Viv · 2005-09-17 04:53 · Score: 4, Interesting

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 ... by omega_cubed · 2005-09-17 05:12 · Score: 4, Informative

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.

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Re:Now ... by bennigoetz · 2005-09-17 06:18 · Score: 4, Informative

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 ... by M1FCJ · 2005-09-17 07:04 · 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 anybody?).
Re:Now ... by don.pratt · 2005-09-17 08:56 · Score: 2, Funny

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 have a truly marvelous demonstration of this proposition which this margin is too small to contain.

The method doesn't matter, as long as the answer by PtrToNull · 2005-09-17 01:35 · Score: 3, Funny

is 42

Wonderful! by h4rm0ny · 2005-09-17 01:35 · Score: 5, Insightful

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.

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Re:Wonderful! by cgibbard · 2005-09-17 05:54 · 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 mathematics without any reference to application.

The problem is that, in highschools, it is usually presented as a jumbled mess of formulas with almost no logical stucture to it at all.

There are huge gaps in the reasoning, partly owing to the fact that calculus is left entirely to the end, and then largely mistreated. You can't talk about angles without first talking about limits, and you can't really talk about limits until you understand what the real numbers are (hint: if you were confused about the 0.9999... = 1 thing, you've probably never been given a proper definition of the real numbers).

Angles need some notion of arc length, which needs at least the concept of a limit superior. (If not an integral.) The book in the article shows how to accomplish the tasks normally associated with trigonometry without needing the concept of an angle (or really anything from calculus or analysis).

If you look at the things that students have trouble with, it's usually the curriculum's fault for not explaining things in a reasonable logical order.

One of the things many people have trouble with in highschool is the whole issue surrounding the logarithm and exponentiation with a positive real exponent. The reason why they struggle is that these things get defined circularly. Nobody ever really tells you what the expression 2^(sqrt(2)) or 5^pi is supposed to represent. You need to know things about limits and convergence of series in order to define a^b where a is real, and b > 0 is real.

I was lucky, and found things to read on my own which described enough of mathematics to me to get me interested, and then went to university for pure mathematics.

The reason why mathematics should be taught in highschool is that people should gain some concept of logic, which is useful no matter where you're headed, and by proving propositions and theorems, one eventually gains an incredible grasp of logic. This isn't currently done though.

Mathematics is basically presented as an awful illogical mess where at best, the students are taught to solve some very specific problems in a mechanical, unthinking fashion, and at worst, their self-esteem is damaged and they come away thinking that they are bad at something which they've never been exposed to. I've seen some very bright people who thought that they were terrible at math, and for this reason avoided going into fields of study that they'd otherwise have been interested in.

I hope we can eventually do something about this because, as a student of mathematics, I can say that the present state of affairs at the elementary and highschool level is terrible, and while I can easily see ways in which it could be made better, actually carrying it out is another thing altogether.
Re:Wonderful! by sigmoid_balance · 2005-09-17 12:10 · Score: 2, Insightful

I lived in Romania. I learnt in Romania. I still live in Romania, but that's another story :)
We now have a new model of teaching math, which concentrates mostly on "computing" things; every exercise asks you "blah, blah, a=6, b=8, blah blah blah, x=?". Geometry, trigonometry, algebra, analysis, everything. We call this "evolving to the way the western society does teaching".
When I started really learning math, by this I mean the 5th grade, the exercises were like "Hypothesis: Given A and B _prove_ that C holds". Simple things, things which solved _a whole class_ of exercises with numbers, which later developed into more complex things, which were built with these bricks.
When you put things like this the student has to think of a way to prove C, maybe even be original about it. Maybe prove a few lemmas before proving that C holds. An exercise like this will have a two page solution in which you will never see a number, possibly (I'm exagerating a little, but you all get the ideea). When you find numbers in an exercise you'll be happy to get out of it the easy way: you have solved the problem before, you just filled the dotted spaces, trivial.
Also as an example, when we were shown the formula A^2 + B^2 = C^2 (the Pythagora theorem), we were shown the prof for this and also prof for the reciprocal theorem. When we were told that cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b), in the 9th grade, we were also given the demonstration.
For every theorem I saw during my highschool, if the reciprocal theorem holded, the reverse implication was true, I was shown the prof for that too.
I like math, I think math is easy, or at least not harder than other things that are more difficult than crossing a street.
You might think I was very smart at that time compared to the majority of my generation to be able to understand all this at that age, you are probably right :) (I'm also very modest). But really, all my colegues who found all this much more difficult than me, and who pursued other kind s of carrers which are not related at all with math have now a very well formed way of thinking. Math taught us to think.
We were ofcourse lucky to have gifted math teachers, who could teach us all this the right way, but my final point is math is good for your brain, it allows you to develop good thinking, makes you learn how to think. Even people who study liberal arts (yes, i mean you two liberal-arts-students-reading-slashdot), should have a good understanding of basic math.
As a final word: Math is like sex ... err ... no it's not like it ... math is ... err ... may the ... err ... no that was not it ... err ... "This is a good day for science!" ... yes, this is it, or at least close to it.

Figures. by Musteval · 2005-09-17 01:35 · Score: 5, Funny

He does this the year after I take Algebra II/Trig. Bastard.

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The "New" has an initial capital for a reason by Bewbewbew · 2005-09-17 01:36 · Score: 3, Informative

The "New" refers to South Wales, rather than the uni. Might wanna read a bit closer next time.

Wow by Loconut1389 · 2005-09-17 01:36 · Score: 3, Interesting

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 by lobsterGun · 2005-09-17 01:57 · Score: 5, Insightful

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 by WilliamSChips · 2005-09-17 02:25 · Score: 2, Interesting

You're half-right. Much of programming doesn't need much math. But certain fields require a lot. Robotics, for example.

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Re:Wow by Pinball+Wizard · 2005-09-17 02:30 · Score: 4, Insightful

Well you certainly aren't working in animation or writing simulations, or writing AI programs, or code for robots, or doing any kind of graphics conversion, or audio programming or making any kind of games with your "programming"(I'll stop here, but I could go on and on). I would guess with your attitude toward math you're really not a programmer, you probably just tie stuff together that other people have written with your own code or scripts. You use libraries rather than write them. Not trying to insult what you do, but there's a lot more to programming than that, and it does take math.

And you are wrong about algorithms. Algorithms is math. No ifs, ands or buts about it.

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Re:Wow by chris_eineke · 2005-09-17 02:36 · Score: 5, Interesting

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.

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Re:Wow by Tony+Hoyle · 2005-09-17 02:46 · Score: 2, Interesting

I'm software development manager, and I didn't get there by not knowing how to program. I've also lost count of the number of libraries I've written. When I first start there *was* no code that other people had written - no internet to get it... you always wrote from scratch.

None (well, very little) of this needed maths.

Algorithms are *not* maths. Why should they be? Anyone can derive something like a bubble sort from first principles without the use of a calculator. A binary search is intuitively obvious - people do something like it all the time in things like interviews (the game of 20 questions as it's known). I could go on... OTOH it's rare to actually work at that level these days - the STL, Java libs, etc. provide all the primitives then you just build on top of them.. there's nothing wrong with this - going back to the days when everything was written was scratch just aint fun.
Re:Wow by Asprin · 2005-09-17 02:47 · Score: 3, Insightful

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.

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Re:Wow by kamapuaa · 2005-09-17 02:55 · Score: 2, Insightful

Sure, you don't need to understand math all that well to implement quicksort, but you most certainly need to understand math to have come up with it!
You could apply this to pretty much any field with just as much stretching. The music scale works on logarithmic principles, and don't get started on how complex the calculations can be for what makes a good instrument, or good acoustics. That doesn't mean you average cello player need to know anything about sines and cosines.

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Re:Wow by the+morgawr · 2005-09-17 03:13 · Score: 3, Interesting

Algorithms by definition ARE math. They are not numeric based math, but they absolutly are math. Math is fundamentallly about patterns. Algorithms are imperative math statements, equations are declarative. Just because it's a different type of math doesn't mean that it's not math.
The only reason people don't realize this more is because most of the really hard stuff is already worked out for them. If you were stuck coding in assembler with no libraries to help you out, you'd realize how much math there is under the hood.

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Re:Wow by Hydrogenoid · 2005-09-17 03:23 · Score: 2, Insightful

Some of them sucked.. some of them were really good (IMO).

And that is all you can get without using maths to prove that they are good or bad, be it in the average, best or worst case.

UNSW .. not South Wales by OzPeter · 2005-09-17 01:37 · Score: 4, Informative

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

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Re:UNSW .. not South Wales by zaguar · 2005-09-17 03:04 · Score: 2, Funny

You must be new here.

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Hopefully by JasonEngel · 2005-09-17 01:37 · Score: 3, Insightful

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... by DataPath · 2005-09-17 01:38 · Score: 4, Interesting

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.

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Redefinition? by AndreiK · 2005-09-17 01:41 · Score: 3, Insightful

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? by DarkPixel · 2005-09-17 01:52 · 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 go read this book when it comes out.
Re:Redefinition? by sameerd · 2005-09-17 01:53 · Score: 5, Insightful

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.
Re:Redefinition? by Anonymous Coward · 2005-09-17 02:20 · Score: 2, Insightful

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.

No, the spread isn't directly proportional to the angle. It actually turns out to be the square of the sine of the angle -- which just looks proportional if all you look at are 0, 45 and 90 degrees.

It still isn't revolutionary, though. You're just working directly with the sines and cosines, since you have nice algebraic relationships between them, and never looking at the actual angles, which would take you into the transcendental domain.

I'm sure mathematicians and physicists do this sort of thing all the time. Probably nobody thought to write a whole book about it. Guys, say hello to the next Stephen Wolfram.

SOHCAHTOA and abstract survery results by acomj · 2005-09-17 01:44 · Score: 4, Insightful

ahh Sin= Op/Hyp
Cos = Adj/Hyp
Tan = Op/adjacent.

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..

Replacing angles and distance with abstract quadrance and spread is exchanging one difficult thing (tan,cos,sin) for another (quad, spread)

Quandrance = distance ^2
Spread hard to see.

Re:SOHCAHTOA and abstract survery results by controlguy · 2005-09-17 03:16 · 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

Calculation of (1) and (3) is trivial. Calculation of (2) isn't so bad either (if you have a coordinate system -- but you can always add one). I believe that it basically involves a vector dot-product for a projection and then an application of the Pyth. Thm. using quadances.

The beauty is that you can do this by hand! In classical trigonometry, you practically need a calculator to handle angles and you'll likely end up with an irrational number somewhere that you'll approximate to a rational one. In a world of rational numbers, quadrance and spread give you rational numbers back! Now THAT's accuracy. In fact, you get rationals of polynomials with rational coefficients.

Basically, we've been spoiled by the advent of calculus and computers. Classical trigonometry is hard. The mesurement of an angle actually requires the computation of limits, and our modern calculations of COS, SIN, ... use, I believe, Taylor series expansions.

For purposes of surveying (though IANA Surveyor so I'm sorry if this sounds ignorant), a machine that measures spread instead of angle and a calculator that inputs distances (and converts to quadrances) is the biggest change. As two lines become more separated, spread increases just like angle, though not at the same rate (probably at a rate of something like cos or sin).

Of course you can express all of it using SINs and COSs, but that's not the point. The real question for us in the engineering discplines is how it will effect our use of complex numbers. What we have now is fairly convenient, but I wonder what this has to offer? Unfortunately, they didn't provide the PDF for *that* chapter.

Faster calculations ?? by AeiwiMaster · 2005-09-17 01:45 · Score: 5, Interesting

I am wondering if this could be used to make faster calculations
in raytracers and 3D engines by using integer numbers.

Re:Faster calculations ?? by Anonymous Coward · 2005-09-17 01:57 · Score: 3, Interesting

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
Re:Faster calculations ?? by birge · 2005-09-17 04:57 · Score: 3, Insightful

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.

Re:huh? by HateBreeder · 2005-09-17 01:48 · Score: 4, Insightful

You're wrong.

It's not that trigonmetric functions are hard to learn, it's that they rely on transcendal function like sine/cosine which are calculated numerically (as a taylor power series expansion, for example) thus are only approximations to the true values (an accurate number would require the calculation of an infinite series, which isn't practical in given time/space).

The article clearly states that: "Advanced mathematical knowledge, such as linear algebra, number theory and group
theory, is generally not needed." (to use this method)

I think that having a percise, simple (polynomials, rational fractions) alternative to current methods in eucleadian trigonometry, is very welcome.

Wouldn't it be nice to be able to calculate angles and distances without having to use a calculator (for sine/cosine calculations)?

--
Sigs are for the weak.

Interesting - but not entirlely new by caffeined · 2005-09-17 01:50 · Score: 4, Insightful

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.

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Sigh. My id isn't prime. 2 2 2 2 2 3 5 313

Don't worry... by tgd · 2005-09-17 01:50 · 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... by anderm7 · 2005-09-17 02:04 · Score: 4, Insightful

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
Re:Don't worry... by Dr_LHA · 2005-09-17 02:06 · Score: 5, Insightful

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... by PocketPick · 2005-09-17 02:10 · Score: 4, Insightful

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... by DarkPixel · 2005-09-17 02:16 · Score: 2, Insightful

Of course in high school everyone is so certain what they will become proffessionaly... like maybe an engineer? Oh crap, I don't get math...oh well, no more engineers in the world.

People need to stop dissing math in their k-12 education as "not something I'd use in real life". That is so not true. Learning that math opens you up to opportunities otherwise unavailable. Kinda like reading, that's useful right? Ok so reading and math are on different levels, but I believe I sorta hinted at my point.
Re:Don't worry... by arsenick · 2005-09-17 02:24 · Score: 2, Insightful

Before devaluating elementary education in such an ignorant way, you should stop and think about the people who developed such things as electricity, cars, or your computer. They had to build on the knowledge of their predecessors, and they had to work for it. And it probably was not so obvious to them why they should learn trigonometry at first.

Without trigonometry, my friend, you'd still be thinking the Earth is flat and we that we live in the center of the Universe.

Some of us are actually glad that we have progressed since cave men and do our best to further improve society. Education is the starting point.
Re:Don't worry... by scrondle · 2005-09-17 02:25 · Score: 2, Interesting

I'm sorry, but that is just wrong. I think it is the most practical branch of Mathematics. I used it when I was working as a metal fabricator, and I use it now that I am writing software for a living. I think that makes it pretty universal. Contemplate drawing a map without trig and I think you will get my point.
Re:Don't worry... by sketerpot · 2005-09-17 02:27 · Score: 2, 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... by eweu · 2005-09-17 02:32 · 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... by Thangodin · 2005-09-17 02:44 · Score: 3, Insightful

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:Don't worry... by RAMMS+EIN · 2005-09-17 02:47 · Score: 4, Informative

``Your high school required every student to take Trig?''

Yes. But then, I live in the Netherlands. Our system is different from the US; instead of (basically) lumping everyone in the same school and making sure they all pass, we send people to different types of high school, based on how they perform on a test at the end of primary school. I went to the highest types of high school, where you get at least 3 years of math IIRC; I took math for the full six years of the program. In the other other types of high school, you get less math because (1) they last shorter, and (2) they tend to focuse more on practical issues than on theoretical ones.

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Please correct me if I got my facts wrong.
Re:Don't worry... by mdwh2 · 2005-09-17 02:48 · Score: 2, Informative

Yes, but simply because there are professions which use it doesn't mean it has to be taught to everyone. I agree that school has to provide a certain common base, but I don't really think trigonometry belongs in there. A class teaching how to automate common computing tasks would have been more useful to more people, I imagine.

True - part of the problem is that (at least here in the UK, I don't know how the US works) Maths is compulsory (until 16), along with English and Science. With optional subjects, you can presume people taking them may want to work in those areas, so it is important to teach accordingly.

With compulsory subjects, I agree that the compulsory bits should be only those which everyone needs in everyday life. In maths, I'd say that things like understanding statistics are more important than trigonometry (consider how often statistics are given in the news and so on, and how many people misunderstand them).

So ideally you'd have "core maths/english/science" then a separate set of classes instead for those who choose to take all of that subject.

But here's the problem: I suspect that most schools won't have the resources to teach two sets of those subjects; it may be simpler just to do things as they are now. (Plus as someone else points out, you may not know what you want to do when you are 14 - or they may not realise just how many jobs may require an application of maths - so it's good to teach it anyway)
Re:Don't worry... by Mac+Degger · 2005-09-17 02:50 · Score: 5, Insightful

No. Absolutely not. People need a basic understanding of this stuff, because it is sop important to the things which make modern society work. People need to know enough to be critical of obviously dumb assumptions, at the very least. You need to know that your contractor is screwing you over by quoting you for more than twice the square-footage than you actually have; and it's amazing how many people can't even handle Pythagoras.

As an aside: I'm always amazed how many people who do sciences and other technical stuff are always interested in many things, like music, politics, aesthetics, social structure...but hardly any political science or sociology student has even a passing interest in the sciences. I'm starting to believe that the latter are just to stupid to realise how much of an impact those things have on their life.

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-- Waht? Tehr's a preveiw buottn?
Re:Don't worry... by mysidia · 2005-09-17 03:08 · 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 politicians. Don't even get me started about the professional merits of Art classes for non-artists.

Math is so generally useful, that I think people are attacking a subject for which there is no rationale merit to attack. I can only speculate this is due to a perceived subjective difficulty of the subject.

Yet it all doesn't matter if there is limited professional scope, and all the material is still very important to be taught well. The purpose of elementary schooling may be misunderstood -- it is not to prepare one for a particular professional but to prepare one for life, which can include many things, despite people having specific plans.

Certainly, the thing that will define what a person's plan will do in the future should not be excluded by something like the difficulty of the kind of math done in math classes. Once learned, a difficult subject could be easy and pleasurable -- if nothing more, having knowledge of trig, places students in a kind of elite: just having the knowledge may be advantageous.

Even people who do not ultimately or intend to choose a technical career may need to talk with scientists and engineers. They may regret it if due to a lack of even high-school knowledge, they cannot be conversant enough on a subject to discuss anything interesting.

By not teaching things like calculus, trig, chemistry, or biology early on, we would rob our youth of a basic knowledge pool -- our future scientists and engineers might never have discovered their favorite subject. Future scientists, etc, could accomplish more in life, get going faster, by learning the basics (which anyone should be able to understand) early on.

Just because a subject's hard to learn or painful at first and therefore encourages some learners to complain or be taken aback by the subject, doesn't mean it's of less value or knowledge will not later be useful. Particularly when surprising things happen in life.

Yeah, most of us may not be scientists and engineers, and most of us don't get stranded on desert islands either. Who says we always get a choice of what knowledge we will need in life? Consider things like mountain climbers... etc... it is rather possible that having or not having technical knowledge of mathematics or physics or not becomes a life-or-death matter. You just never know if some basic tidbit may happen to be extremely helpful or not.

The apparent difficulty of a subject for some, or lack of effective presentation is no reason to obscure the basics or stop classes from existing -- it's reason to find better ways of getting people to come to learn the importance of these subjects.
Re:Don't worry... by dilvish_the_damned · 2005-09-17 03:09 · Score: 5, Insightful

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?

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I think you underestimate just how much I just dont care.
Re:Don't worry... by Bastian · 2005-09-17 04:31 · Score: 2, Insightful

Also, whatever happened to being well-rounded intelligent beings? Since when did high school become the place you go to learn a trade?

That actually happened about the time when high school was created. The masters of determining cirriculum were standing in the balance - they could create a model of school that encourages kids to think critically, and to focus on the process of thought and reasoning.

Or they could take the "student as shoe, randomly-assembled array of facts and figures as foot, us as shoehorn" approach and force a bunch of crap into kids brains on the swallow-and-regurgitate model. As a kicker, they could make the model one that would encourage independent acts of swallow-and-regurgitate by rewarding it with good grades for minimal mental flip-flops. And they could put a cherry on top by discouraging independent thought by making critical thinkers who try to form their own opinions have to defend their work much more vigorously in order to get good marks (i.e., making "not what the teacher thinks" a synonym for "incorrect").

Guess which one I think models our primary education system?

(And no, I don't blame teachers. The problem is completely systemic, and I see the teachers I've gotten to know as victims of this brain-crushing system, too.)

Great for eighth grade, but ... by levin · 2005-09-17 01:56 · Score: 5, Insightful

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.

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`which fortune`

Re:Great for eighth grade, but ... by miskatonic+alumnus · 2005-09-17 03:04 · Score: 2, 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" by Curmudgeonlyoldbloke · 2005-09-17 02:02 · Score: 4, Insightful

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?

Re:I don't see how this is "easier" by random_me · 2005-09-17 07:04 · Score: 2, Interesting

> Imagine if we'd been using "quadrance" and "spread" for years - and then
> some bright spark suggested calculated using sines and cosines.

It would be an amazing breakthrough, because there are some very important things which are simpler and easier using sines and cosines. Read some of the other comments about the effect of rational geometry to calculus. Sines and cosines show up all over physics and more specialized descriptions of the real world (chemistry, thermodynamics, electrical engineering, etc).

Many people have been asking the question (and I haven't seen anybody posting an answer) about what is really easier to do using quadrance and spread that we don't already use some similar form for?

Re:Why are there 360 degrees? by Skirwan · 2005-09-17 02:07 · Score: 2, Funny

There are 360 degrees in a circle because there are 365 (point whatever) days in a year. The ancient Greeks were more primitive than we are today; lacking computers, they couldn't manage a simple off-by-one error, and had to fall back on the less sophisticated off-by-five-and-a-long-decimal error.

Re:Units? by TheRaven64 · 2005-09-17 02:08 · Score: 4, Insightful

If you'd R'd TFC then you would know that spread is a unitless quantity. It is a ratio between two quadrances (lengths squared), and as long as the quadrances are homogenous with respect to their units then they cancel out.

I don't usually advocate this kind of behaviour, but the chapter is actually well worth reading. Quadrance is a neat hack - use the square of distance instead of distance to eliminate some nasty square roots, but spread is a much more interesting concept. The notion of spread removes the dependence upon circles to define relative direction, which removes a lot of complexity from trig.

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I am TheRaven on Soylent News

Bah!! by doi · 2005-09-17 02:11 · Score: 2, Funny

Now THESE are some divine proportions.

--
A man's reach must exceed his grasp, or what's an erection for?

Not just physicists or engineers use trig.... by Ellis+D.+Tripp · 2005-09-17 02:12 · Score: 4, Insightful

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....

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Re:Not just physicists or engineers use trig.... by miskatonic+alumnus · 2005-09-17 03:38 · Score: 3, Insightful

Good point, but the people in those professions can save time and $ by simply buying a device to do any of those calculations for them. The only person who then needs to "know the math" is the one who builds the device.

Number one: The people using the device still have to know the math. Suppose you want to calculate compound interest using the formula:

[P(1+r/n)^(nt)]-P

Several times I have seen college students fail to produce the correct answer even armed with a textbook, this formula and a calculator. Why? Because they don't understand the math.

Number two: Calculators lie virtually always. Often they produce answers that are "good enough". However, without an understanding of maths, "good enough" typically means "whatever the calculator reported as an answer". Example: what is the sum

10000000 + 0.00000001 ?

The correct answer is 10000000.00000001
The calculator's answer is 10000000.
Re:Not just physicists or engineers use trig.... by mrbnsn · 2005-09-17 04:17 · Score: 3, Informative

If you would RTF Sample Chapter, you would see that this is exactly what Wildberger has done: redefined trignometry in terms of "rise/run" ratio ("spread") and the pythagorean theorem ("quadrance").

So your complaint basically boils down to this: "carpenters don't need to know trignometry, they only need to know Rational Trignometry".
Re:Not just physicists or engineers use trig.... by ciggieposeur · 2005-09-17 07:40 · Score: 3, Insightful

10000000 + 0.00000001 ?

The correct answer is 10000000.00000001

No, the correct answer is 10000000. Each term has only one significant figure, so after truncating to the correct precision you get the calculator's displayed answer. Although many calculators will have the inaccurate figure (10000000.00000001) rounded to the nearest base-2 floating point in memory and a long-enough fixed-point format will display it.

Now, if you had said 10000000.00000000 + 0.00000001, then the correct answer would indeed be 10000000.00000001.
Re:Not just physicists or engineers use trig.... by alekd · 2005-09-17 08:41 · Score: 2, Insightful

Example: what is the sum

10000000 + 0.00000001 ?

The correct answer is 10000000.00000001
The calculator's answer is 10000000.
A more interesting example would probably have been 0.00000001 + 10000000 - 10000000 = 0 and not 0.00000001 given that you evaluate from left to right. It is a phenomenon of floating point arithmetics know as catastrophic cancellation.
Re:Not just physicists or engineers use trig.... by petermgreen · 2005-09-17 10:00 · Score: 2, Insightful

the correct answer at an abstract mathematical level assuming all numbers given are exact values would be 10000000.00000001.

Sometimes you can take precision is implied but you have to be very carefull with doing so especially with integers after all is 100 a number given to 1 SF or a number given to 3 SF where two of the digits happened to be zero? you can't tell!

When using a calculator or computer you have to understand that the response won't be an exact answer for most functions and you need to know if the loss of precision is significant or not.

to take one possible example suppose for a simulation you decided to represent the position of your boat and something it is towing as relative to its home base using a pair of numbers.

Mathematically the reference point is arbitary and so this seems fine. However you then decide to represent those numbers as whatever floating point type is conviniant and suddenly the accuracy decreases as you move away from the reference until it eventually gets bad enough that your simulation breaks in some way.

Nievely written algorithms can lose a huge ammount of precision way beyond the obvious loss from the data type limits when converted from mathematical perfection to computer data types.

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Re:Not just physicists or engineers use trig.... by miskatonic+alumnus · 2005-09-17 16:08 · Score: 2, Insightful

No, the correct answer to the problem as stated is still 10000000.00000001. If these numbers were measurements then I would concede your point.

Is this silly? by sameerd · 2005-09-17 02:14 · Score: 3, Interesting

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).

Re:Why are there 360 degrees? by pg133 · 2005-09-17 02:14 · Score: 2, Insightful

With the power of the internet, behold, an answer to your question Babylonian Mathematics

The base 60 number system of the Babylonians was successful enough to have worked its way through time to appear in our present day modern world. We still have 60 minutes in an hour, 60 seconds in a minute, 360 degrees in a circle and 60 minutes in a degree. Even our 24 hour clock is a legacy from the ancient Babylonians.

Galileo got there first by panurge · 2005-09-17 02:14 · Score: 2, Interesting

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.

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Panurge has posted for the last time. Thanks for the positive moderations.

I looked at one of his first examples by exp(pi*sqrt(163)) · 2005-09-17 02:23 · Score: 2, Interesting

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.

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This reminds me of a test in grad school by zzyzx · 2005-09-17 02:28 · Score: 5, Interesting

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 by HuguesT · 2005-09-17 02:57 · Score: 2, 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 integral but you do have to remember how to do a variable substitution in an integral, though, as well as some classical tricks.

agreed by i41Overlord · 2005-09-17 02:38 · Score: 3, Insightful

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.

Yes, for some badly written code by exp(pi*sqrt(163)) · 2005-09-17 02:45 · Score: 4, Interesting

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.

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Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.

What a silly non-sense by greppling · 2005-09-17 02:47 · Score: 3, Insightful

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. by yeOldeSkeptic · 2005-09-17 02:49 · Score: 5, Insightful

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.

You've by Anonymous Coward · 2005-09-17 02:53 · Score: 2, Insightful

confused the area of mathematics with a subset of mathematics called calculus.

Try proving that a sorting algorithm is O(log n) without using any mathematics. Then you'll understand why mathematics is important.

Re:Why are there 360 degrees? by damiam · 2005-09-17 02:58 · Score: 2, Informative

Because 360 is divisible by a lot of numbers, making it easier to work with than say, 359. It's really pretty arbitrary, which is why mathematicians use radians for most serious purposes.

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It's hard to be religious when certain people are never incinerated by bolts of lightning.

Re:Well, not exactly by moonbender · 2005-09-17 03:23 · Score: 2, Informative

Look again. The circle is there, but only to show a similarity to the previous examples. The radius of the circle is irrelevant, and he only uses one point on the circle - in other words, the circle is totally unused and you could use any point on any of the lines.

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Switch back to Slashdot's D1 system.

Very nice. Makes sense to a game programmer by Animats · 2005-09-17 03:50 · Score: 4, Informative

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.

No, I think you miss the point by panurge · 2005-09-17 03:51 · Score: 2, 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 majority who will not become mathematicians and physicists, abstracting mathematical ideas away from real world experience will not be beneficial.

In fact, you give away your attitude a bit by mentioning Prince Charles. I'm a republican myself but I think you misrepresent his views. He appears to believe that people should live in human scale buildings designed to give a sense of community, and not live in buildings designed to return the maximum income to the shareholders in developers (who don't have to live in the buildings that bring in that income, and wouldn't want to.)
He also appears to believe that many architects produce stuff intended to glorify themselves and their clients regardless of their fit with the landscape and the environment. This is a neat parallel to the idea that mathematics teaching for the majority should reflect concepts that tie back to the real world so they will be useful, and that people who propose radical changes are often more interested in their own fame and reputation than the real benefits.

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Panurge has posted for the last time. Thanks for the positive moderations.

Re:Yeah, I looked it up a while back by LaCosaNostradamus · 2005-09-17 04:21 · Score: 2, Insightful

What's the point of a standard if it's not pervasive and useful? Most standards are useful, and just by being standards, they are pervasive. But being widely accepted also means incredible costs for invoking changes across most sectors of society on the basis of marginal increases in performance.

There are proposals to change time, but like changes to the English alphabet, the benefits have to outweigh the almost impossibly large costs of transforming. Look at the English/Metric systems. The USA still has not changed. Metric's benefits still have not outweighed the costs of changing all rulers, indicators, speedometers, odometers, signs, etc. -- and as well, the internal rulers in people's minds that have used feet and miles for centuries.

P.S. The number 60 has these low factors: 2 3 4 5 6. That means that if you use the number 60 for measuring, it's easy to divide whatever you're doing into 2 to 6 parts, and each part is another integer. The number 10 only has factors 2 and 5. Arguably, the number 12 (having factors of 2 3 4 6) is more useful than 10. 10's usefulness is that it matches the base (10) that we use ... but only having 2 factors makes it insufficient for other uses.

P.P.S. If there was some real political will behind it, the USA might be able to change all typing keyboards to Dvorak from Qwerty. But that would be a social "Apollo Project" in scale, hence it's never going to be done. Keyboards will probably change format when the entire concept of a keyboard changes, like if pervasive voice recognition or neural connections arise. But then the change will be invoked as the standard becomes nonstandard, and the keyboard fades away.

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[You have a stable society when some nut guns down a schoolyard and the law doesn't change.]

But if you graduate at Dr. Wilberger's group... by kanweg · 2005-09-17 04:36 · Score: 3, Funny

you still get a degree!

Bert

Trig is not hard, it's just taught REALLY badly by hagbard5235 · 2005-09-17 04:40 · Score: 2, Informative

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.

quack? by selfdiscipline · 2005-09-17 04:40 · Score: 2, Interesting

If you read the first page of his site, you probably noticed that he put the word axioms in quotes.
Math is all about discarding old "axioms" and coming up with new axioms. You just have to realize that as axioms age, they often become "axioms". Get it?

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Incite and flee.

What a completely silly idea. by omega_cubed · 2005-09-17 05:07 · Score: 2, Interesting

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?).

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Engineers also speak PDE, only in a different dialect.

Or you could just sketch the functions by Dioscorea · 2005-09-17 05:13 · Score: 2, Informative

and figure out the derivatives that way.

"Decimal number plane"?? by Bob+Hearn · 2005-09-17 05:59 · Score: 2, Interesting

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.

Parent is factually incorrect by Jesus+2.0 · 2005-09-17 07:55 · Score: 4, Informative

Parent may be "4, Interesting", but nonetheless is factually incorrect.

He didn't replace distance with angles, nor is one of the two "the fundamental element of trig" - they are together the fundemental elements of (standard) trigonometry.

What he actually did was that he replaced distance with distance squared and angles with sine squared.

Distance-squared and dx/ds, dy/ds by BrianMarshall · 2005-09-17 08:57 · Score: 2, Interesting

It is hard (for me) to say whether his approach will provide any unique insights, but it reminds me of something...

I have done programming involving coordinates and trig from time to time - originally, stuff like finding where a line is clipped by a polygon. I can remember and do SohCahToa, but that was close to the limit of my knowledge of trig.

The big problems that I found, while trying to write the code, were positive versus negative angles, infinite-angle of vertical lines, and having to calculate a lot of square-roots.

I found that two principles were a great help...

Like the man says/implies, if distance-squared works as well as distance, use it; you avoid a square-root calculation.
Express angles as a pair of numbers dx/ds and dy/ds (change in x and y as you move along the line).

The second point eliminated a lot of if-statements and similar but not quite identical code (if both angles are positive..., if angle A is positive and angle B is negative..., etc.)

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"When the going gets weird, the weird turn pro" -- HST

Re:Distance-squared and dx/ds, dy/ds by Woodham · 2005-09-17 21:09 · Score: 2, Funny

I can remember and do SohCahToa, but that was close to the limit of my knowledge of trig.
Sex On Hard Concrete Always Hurts, Try Other Arrangements.
That's how my maths teacher said to remember it.
It worked really well for some reason...

Re:Read the Article by Dashing+Leech · 2005-09-17 10:10 · Score: 4, Interesting

"As noted by many posters, spread is just a function of angle. "

Ironically, you've actually harmed, not helped, anyone who hadn't read the article. The main point of this work is that distance AND angle are the wrong things. The quadrance (square of distance) and spread (effectively, the square of the sine of the angle -- though help points out 'angle' is a handwaving concept to begin with) should be the fundamental elements and makes the trigonometry meaningful and easy. That quadrance and spread are (independently) functions of distance and angle is trivially obvious, but it is exactly the difference that is his point that makes the math so much simpler.

Perhaps a more specific example for those programmers out there. No more need for lookup tables or Taylor expansion for trig functions unless a human at some point needs an angle value at which case it can be converted in the final step before output. (Internally, programs would never need them.)

Insurance claims by harvey+the+nerd · 2005-09-17 10:13 · Score: 2, Interesting

You really need trig for good high school physics. On different insurance claims for house damage and for a car wreck, I have needed hs physics when someone (claims adjuster or other party) was trying to screw with me.

Once I showed people the nature of their statement/position, I said, bring all the lawyers you want, my friends are engineers...

End of discussion and bs.

Worth exploring for vector graphics programs by Temeraire · 2005-09-17 11:21 · Score: 2, Interesting

Ask any vector graphics program (Adobe Illustrator, Corel Draw, etc, etc) to generate an outline around some text and you will rapidly see the limitations of conventional trigonometry. Increase the width of the outline and/or the complexity of the text and sooner or later the maths will blow up.
A few years ago my software house needed a subprogram to create paths offset any chosen distance from another 2D path. (Necessary for machining in the sign-making industry.) I fondly imagined this was half a day's work for a clever visiting student.
Alas, no, it turned out to be a 3-month coding nightmare. Finding the precise intersection of two nearly parallel vectors (expressed as lines, circle arcs, or Bezier curves) is surprisingly difficult, within the limits of precision and time set by computers. You end up dealing with special case after special case.
In ignorantly fumbling towards a better way of expressing the calculations, I got as far abandoning angles and using quadratures. If only Rational Trigonometry had been around at the time .....!

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