Trigonometry with a twist?
I’d love to hear from someone who knows more about this than me: a mathematician from New South Wales (Australia!) claims to have developed a new trigonometry, i.e., a new way of dealing with angles and triangles.
Apparetly, in Norman Wildberger’s formulation, you don’t worry about angles and distances, and instead work in “spread” (which is between 0 for parallel lines and 1 for right angles) and “quadrance” which is just distance squared. He claims that this produces equations which can be solved “exactly” rather than the approximations that go with sine and cos that you might be familiar with from school.
To me, this sounds awfully fishy. (I’m a skeptical sort of guy, aren’t I?) The whole 0 to 1 thing sounds a bit like mapping the angle to the sine of the angle (spread = sin(theta), say) and I wonder if using distance squared (his “quadrance”) isn’t just a convenient way of using Pythagoras’s theorem (base^2 + side^2 = hypotenuse^2 - what you use to find the long edge of a right angled triangle! In his quadrances, it sounds like the sum of the base and the side of the triangle give you the hypotenuse straight out (rather than having to square and square root things.
Furthermore, even if his equations could somehow give “exact” answers (which I doubt - I’m sure there’s a theorem or something here!) having 10 decimals places is more than enough for any real purpose (e.g., engineering) while mathematicians would just leave sin(theta) as…sin(theta)! (And no, that’s not a factorial sign for the geeky mathematicans out there…)
I’m not going to buy his book based on this story, although I’m puzzled that he got it published if this is all there is to it. Perhaps it really just simplifies enough stuff that he thinks it’s a justifiable suggestion? I really can’t see us changing the way we deal with angles etc now, and, ultimately, it’s just a different way of looking at the same thing - there’s nothing wrong with maths now, and I suspect (I’m almost certain, in fact!) there’s nothing we can’t do now that we could do with his formulation. So anyone with any thoughts would be most welcome to comment!
Dave Barry Says:
I’ve read a little bit of the online chapter.
The author makes a good case that his method is better for solving arbitrary triangles, because you naturally end up with answers in terms of radicals. It is possible to get exact solutions using the usual method, but you end up having to work out horrible things like sin(pi/5) or something like that (I’m not sure if sin(pi/5) can be expressed in terms of radicals, but some similar expressions can be).
He makes a reasonable case that spread is more fundamental than angle, but he also says (I think) that students are turned off by geometry because of the vagueness of the usual definition of angle, which would be avoided if everyone used spread. I think he drastically overstates his case here, and that the average student isn’t going to like any sort of geometry. Angles are very intuitive, they are easy to add (as Dave Bacon says above), and I don’t think vagueness of their definition in high school is why many people don’t like geometry.
Courtney Mewton Says:
sin(pi/5) with radicals?
sin(pi/5) = sqrt{5} / sqrt{ 10 + 2*sqrt{5} }
Here is something else. sin(2pi/5) / sin(pi/5) gives you the golden ratio.
Grandite Says:
Joel raises an incredibly good question, and it is right to be skeptical of such claims. However, I’ve read most of the book by now and I believe that Wildberger’s approach does have merit.
Firstly, let’s consider the current system of geometry. Our basis for geometry was created perhaps as far back as 4000BC by ancient Babylonians, who decided to divide the circle into 360 degrees because they thought the year was 360 days (probably). In the years that followed, geometry just grew up out of this. Now, I’m not sure if people reading this are familiar with the concept of a ‘field’, but a few examples of a field include the set of Real Numbers, or the set of Complex numbers, or modulo 5, or modulo 7 or modulo 11, etc.
That may not have made sense to you, but basically, our current geometry was tailored to work over the real field (the set of all numbers you know, 5, -3, 0, sqrt(7), pi, etc). Geometry was seen to be needed to work only over the reals - after all, why else would you want to study geometry.
Mathematics is progressing and we see now that restricting geometry to the Reals is somewhat arbitrary and there are many many cases today where we do geometry over other fields. However, the concept of angle does not have a counterpart in many fields and geometry grinds to a halt. Wildberger’s Geometry works over any field.
This means that over the Reals, Wildberger’s Geometry and the current system of geometry are pretty much on par, but over other fields (like modulo 7,modulo 11, etc), Wildberger’s geometry is by far superior - superior in the sense that it can do things the current system of angle and distance cannot even come close to doing.
It will probably take a great debate amongst professional mathematicians to see if his view of geometry is sufficiently better than the current system to warrant such a large reform, but I believe that in the years to come, Wildberger’s new outlook on Geometry will be hailed as a milestone in mathematics.
Greg P Says:
As far as I can tell from a brief view, is that this guy has simply taken geometric forms and imagined them on a graph, in which case you can use the various formulae for graphs to make calculations.
Trigonometry is really just a pragmatic way of dealing with angles and distances. Sine, cosine, etc. are just tabular values of ratios using a unit length of 1.
Well, it’s kind of silly and simple, from my understanding. One can, of course, deal with the variables he defines, and they have some nice properties. However you lose other nice properties along the way. For instance the reason we like angles is that rotations about the same access yield formulas with just simply addition. But in this formulation, adding angles is a very complicated procedure. One could say at this point: “fine but why do we care about these differences.” But it is exactly the parts of trig which are important for resolving vectors which one loses in this formulation. And to me, since all I do is resolve vectors aka quantum theory, this seems like way to severe a price to pay.