Talk:Rational trigonometry

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[edit] Doubt

I am not so certain about this new trigonometry. It is in its calculations letting you treat a circle like a square. The reassignments of basic components also makes future mathematics a real pain to accomplish. How would onw go about taking the integral of the quadrance. It is even possible?

What are the rules to deal with the acute and obtuse spreads that he aludes to in Chapter 1 (e.g. there are two destinct solutions since the spread doesn't contain quadrant information) --142.176.130.187 11:16, 29 September 2005 (UTC)


I was tempted to put a flag up for an accuracy warning, but it's not quite wrong enough. I am no expert on the subject, so I hope someone who knows a bit more will come in. Neildogg 03:30, 18 September 2005 (UTC)

Nothing actually could be wrong here. These are just definitions of new terms upon which rational trigonometry could be built. I believe that this article is quite correct as it just describes an idea that stroke a mathematician elsewhere and references the page where further information may be obtained. As for me, I looked through the first chapter of the book and found nothing quite obviously delirious that would make deletion favorable. --ACrush217.23.131.98 09:52, 18 September 2005 (UTC)

[edit] Relationship between angle and spread is oversimplified / somewhat misrepresented in current text.

>For all intents and purposes, angle and spread >are the same thing in terms of perception, but >quite different in terms of the underlying >mathematics.

So far as I can figure out what exactly this means, I'm not sure that it's correct.

I will say, and I think someone should point out, that while increasing an angle (up to 90 degrees) increases the corresponding spread, spread is *not* proportional to angle. So in the sense that they are not proportional, they are not "the same thing in terms of perception".

For that matter, a spread describes a relationship between two lines, whereas an angle describes a relationship between two rays emanating from a common point. A spread doesn't specify an angle as specifically as an angle does; for example, the angle 89 degrees has the same spread as the angle 91 degrees has.

--C. Niswander (19 September 2005)

Addendum: I did some clarifying and cleaning up in this article, but it could still use additional improvements.

Whatever it's called, spread is a function of angle. Rational and conventional trigonometry appear to have a connection rather like Cartesian and polar coordinates: we're talking about the same geometrical relationships, but describing them in terms of different variables. In short, rational geometry is analytical geometry using length^2 and sin^2(angle) as fundamental variables rather than length and angle - a transformation chosen because it disposes of square roots and trig functions. 195.92.67.75 19:21, 19 September 2005 (UTC)
Yes, spread is a function of angle. In fact, I think that your sentences are suitable for the actual article. --C. Niswander (20 September 2005)
I tend to disagree. Having attending several of Norman Wildeberger's talks, the rationale behind rational trigonometry is that the concept of an angle belongs to a circle (ie, Euler's formula), and that the concept of spread is far more natural for a triangle (c.f. Thales' theorem). Angles and distance also break down in fields other than the real numbers, whereas spread and quadrance do not - in these cases, spread is *not* a function of angle. Personally, I don't think it will overthrow trigonometry as we know it, but it may lead to some inovation in algebraic geometry. Dmaher 03:43, 5 April 2006 (UTC)

I suspect it could eventually overthrow applications of trigonometry to things like navigation, land surveying, and those aspects of astronomy that use two- and three-dimensional geometry simply because those model physical space in the obvious way. But I suspect it cannot touch things like trigonometric functions in Fourier series and the like. Michael Hardy 19:49, 11 April 2006 (UTC)

According to Wildberger, this is intentional:
"The trig functions sinθ and cosθ still have a role to play in the study of circular or harmonic motion, but there the knowledge needed is rather minimal. Indeed for the study of circular motion the trigonometric functions are best understood in terms of the (complex) exponential function." [1]
and:
"[Rational trigonometry] cleanly separates the physical subject of circular motion and the mathematical subject of trigonometry. For the former, the trigonometric functions are useful, for the latter they are—or should be—largely irrelevant." [2]
--Piet Delport 20:13, 11 April 2006 (UTC)

[edit] Basic laws

I have attended Wildberger's book launch and have bought his book and obviously I will be writing of what I understand of his work. I have quickly put his five basic laws up in the hopes that it will be useful for others; the proofs are included in the book but this will have to wait until I have read his book more throughly. Feel free to ask me any questions on this topic and I will try my hardest to find out the answer keeping in mind that I am not a mathematician. Alanl 15:07, 20 September 2005 (UTC)

It's worth getting one in; I'm a trifle rusty on the subject, but I notice that (at least) four of the five laws are just transformations of standard trig formulae. For instance, take the law of cosines:
c^2 = a^2 + b^2 - 2ab \cos C . \;
rearrange:
a^2 + b^2 - c^2 = 2ab \cos C . \;
square it:
(a^2 + b^2 - c^2)^2 = 4a^2b^2 (\cos C)^2 . \;
and there you have the Cross law
(Q1 + Q2 − Q3)2 = 4Q1Q2c3
Or take the condition for collinearity. Three points are collinear if the area they enclose is zero. By Heron's formula for area:
\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\ = 0
square it:
{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\ = 0
substitute s, the semiperimeter, s=\frac{a+b+c}{2}, multiply out:
− a4 + 2a2b2 + 2a2c2 − b4 + 2b2c2 − c4 = 0
add 2a4 + 2b4 + 2c4 to both sides:
a4 + b4 + c4 + 2a2b2 + 2a2c2 + 2b2c2 = 2a4 + 2b4 + 2c4
factorise:
(a2 + b2 + c2)2 = 2(a4 + b4 + c4)
and there's your triple quad formula:
(Q_{1} + Q_{2} + Q_{3})^2 = 2(Q_{1}^{2} + Q_{2}^{2} + Q_{3}^{2})
So far, then, it looks an interesting idea, but I wouldn't overstate its difference from conventional geometry. Tearlach 20:17, 20 September 2005 (UTC)
For the record: It must be kept in mind that the point of rational trigonometry is not to be fundamentally different than conventional trigonometry, but to be a more elegant, accurate and general representation of the same underlying mathematical idea(s). In other words, characterizing rational trigonometry as "just a transformation" of classical trigonometry is (distantly) like characterizing continued fractions as just a transformation of plain fractions: it's entirely true, but not what the distinction is about.
Some of the aphorisms displayed by the ticker on Wildberger's website seem to indicate what he's trying to achieve with rational trigonometry:
In order of discovery:
Proofs, theorems, definitions, notation.
In order of importance:
Notation, definitions, theorems, proofs.
And:
A good notation is worth a hundred theorems.
--Piet Delport 11:18, 5 April 2006 (UTC)

[edit] Diagrams

I was reading through this article. It make sense. But it would have been easier for me to follow had there been more diagrams. Val42 16:54, 4 February 2006 (UTC)

[edit] "Adding" quadrance

I don't know if this is useful or anything, but given two connected collinear line segments with quadrances Q1 and Q2, you can obtain the quadrance of the combined line segment with the following formula: Q_3 = Q_1 + Q_2 + 2\sqrt{Q_1Q_2} Ubern00b 22:49, 5 April 2006 (UTC)

[edit] The Controversy

This is a very unsatisfactory article. The book is in the general vein of somebody's latest proposal for a phonetic alphabet. As such it seems to merit an article of some sort but the ideas can be presented in a paragraph, and the only other points of interest would be the sort of reception it has found in various communities, with appropriate external links. I'm not suggesting deletion. I think the book has attracted enough attention and comment to deserve a brief and informative mention. The author's proposal is controversial and intended to be such, so some effort is needed to strike a proper balance. The fact that it is interchangeable with the standard theory is somewhat beside the point, as my example of a phonetic alphabet is intended to suggest. As a historical note, the movement from Indian Sine to modern sine was motivated by analogous though less radical considerations. And for that matter degrees vs. radians is capable of stirring up emotions in some circles. Abu Amaal 18:08, 11 April 2006 (UTC)

I don't think rational trigonometry can be dismissed that easily. For example, rational trigonometry claims to separate the mathematics of trigonometry and circular motion from each other more cleanly than conventional trigonomotry, and also generalize across arbitrary fields (among other things). I can't personally verify those claims, but they would make rational trigonometry more different than just an alternate "phonetic alphabet", as far as i understand it. --Piet Delport 19:32, 11 April 2006 (UTC)
I don't understand. How is this related to a phonetic alphabet? Did I miss something?--88.101.76.122 (talk) 15:51, 2 February 2008 (UTC)

I disagree with Abu Amaal. The author is not merely proposing new conventions to replace old ones. He is proposing that certain aspects of the subject, when separated from certain other aspects, can be presented in a way that is simpler and also can be applied when the field of scalars is any field other than the reals. It is not interchangeable with standard theory, since it applies to other fields of scalars just as neatly as to the reals, and also since it has been separated from some parts of standard theory that are needed elsewhere—for example, in the theory of Fourier series; in other words, the greater simplicity comes at a price. Michael Hardy 19:38, 11 April 2006 (UTC)

None of the examples I gave are "interchangeable" as you construe the term. Be that as it may - I wish to discuss the structure of this page and not the merits of the theory it describes. So, again: the reader coming to this wiki page should be informed (a) that the theory is controversial (b) what, briefly, the proposal is (c) what the major pros and cons are currently felt to be and (d) where fuller discussions may be found, on both sides of the issue. (One might also point toward the enormous literature in the foundations of geometry which since Hilbert has analyzed the field theoretic content of a wide variety of geometric notions and the axiomatic significance of various constructions and theorems. But this may not be an essential point in this context.) Wikipedia has no need to take a position on this matter, but does need to reflect the existing range of views. Abu Amaal 23:35, 11 April 2006 (UTC)
I don't think the theory can be controversial: it's just mathematics, and i'm not aware of any mathematicians that consider it flawed. The controversy, as far as i understand it, is about the author's opinions on things like mathematics education; a discussion of which probably belongs in an article about the author. Or, failing that, in a separate article about his book. --Piet Delport 08:31, 12 April 2006 (UTC)

The previous comment further illustrates the need for a sensible discussion on the main page. What this commenter is unaware of is no doubt something that many are unaware of. And the use of interchangeability as a supporting argument is a nice balance to the denial of interchangeability. Both have merit, and I think I have managed to avoid stating my own views, a policy I would recommend to others. Now that I have been disagreed with from both sides, after having said very little, with a denial of controversality tossed in to liven things up further, perhaps we can return to the problem of putting together a useful article. The theory is the book, and has been in existence since September of 2005. It is more or less a current event. There are forums where the merits can be discussed. We just need to produce one article, with links.

One might want to review Guidelines_for_controversial_articles. I may initiate this myself but it would be better if someone who is more interested in the subject, or more committed to Wikipedia, would take it in hand. Abu Amaal 16:04, 12 April 2006 (UTC)

I agree with Piet and Michael. I don't think there's much wrong with the article, except that it stresses the application to 2D geometry (which is understandable as that has been the focus of all the promotional and preview material) and needs a little work on the wider implications. I suggest you lighten up. Starting a mediation procedure, which is meant to be a nuclear option when all else has failed (see Wikipedia:Mediation Cabal/Cases/2006-04-12 rational trigonometry) is sheer overkill at this stage, and suggests a misunderstanding of Wikipedia conventions. Tearlach 19:20, 12 April 2006 (UTC)
Fine. That was Pepsidrinka's suggestion, as I understood it, and I requested that any communications about that be addressed privately to me. I am unimpressed with the quality of the advice and the results of following it. But see my talk page, and if anybody can figure out where the misunderstaning lies, add a comment there. I was looking for ideas to make this interchange more profitable for all concerned. Along the way I stuck an expert tag on the page. I still don't know if that was an appropriate thing to do here.

Now returning to the point under discussion: the subject matter is highly controversial, but one cannot assume this is known. The page will probably be visited mainly by high school and college students. For them some understanding of the nature of this controversy will be an important piece of information. Michael in particular has identified clearly some arguments that are typically made on one side or the other. There are others. My point is that the page needs to develop this aspect to be useful for its primary audience. This is the point I tried to make in the third sentence of my initial posting.

Michael as far as I can see has taken no position on the point. Piet has denied that the theory is controversial (and appears to be denying that a logically sound theory can be controversial). But I see that Piet does acknowledge the existence of a controversy, so while we disagree about the correct characterization of that controversy we can still discuss our response to it. In dealing with such controversies I suppose one links to places representative of the various points of view, and one gives some information about those points of view. This is the proposal: (1) document, efficiently, the controversy surrounding this theory; (2) do so on this page.

I have changed the title of this section. I hope you like the new title. I have written a great deal here and the section is getting very long. Is this a problem?

Abu Amaal 21:24, 12 April 2006 (UTC)

I don't think the subject matter of this page is controversial, nor that it would be if it were more widely known. Nor do I think secondary-school pupils or students at any level would be done any disservice by this material, even though it could probably be better written by someone more thoroughly familiar with the material. There may be potential for controversy over the question of whether this ought to replace the more traditional approach to trigonometry in the secondary-school or lower-division undergraduate curriculum. I don't think this approach is sufficiently widely known at this time for such a controversy to have started. But the page doesn't deal with the curriculum, but only with the mathematical topic, so I don't see why it would be controversial. There's a trade-off: this approach makes some applications of trigonometry (those that deal directly with metric geometry) much simpler, and makes others impossible (those that would be applied to Fourier series and the like). It also extends trigonometry into geometry over finite fields and other fields of scalars. So what to do in the elementary curriculum could become controversial, but the mathematics that this page deals with is not. Michael Hardy 22:59, 12 April 2006 (UTC)


All right. Now that we have established that we do in fact disagree about whether the subject is controversial at this time (which is more of a question for a journalist than a mathematician) we can wait until further developments cast a clearer light on that, or until other participants in the controversy wander by this page and make their own views known. Meanwhile, if you are interested, look at the public discussions that have taken up the topic and keep your eyes open for strongly worded dissent - but only if you are interested, as it is a tedious business. I take it that nobody cares much one way or another about the tag. I feel that this page does a disservice to the uninformed reader by leaving all of these matters to the talk page. Others feel as strongly that it should be left as is. For the moment, you win, since I am not going to edit it and nobody else involved in the page sees any need to do so. No doubt this discussion will be revived in due course if the subject does not fade entirely from view, and perhaps this section will serve as a point of departure.
Addenda: (1) We have had trigonometry over finite fields since Witt, who remarks that Dickson pointed out the desirability of it but seemed to think it was undoable. (2) However, this has not impinged on the role of the circular functions, particularly in the complex case where they lead to the complex exponential on the one hand and also serve as a sort of genus zero counterpart to the theory of abelian functions (Gauss, Disquisitiones, to begin with). You are aware that Fourier series are just one nice way of exploiting the complex exponential function, and that the modern view of mathematics gives this particular function a highly privileged role. (3) Historically, the battle to combine the theory of circular functions with classical trigonometry at an elementary level in education came late to the U.S., and was controversial, and was actively championed by mathematicians. Allendoerfer and Oakley is an early attempt to show how this might be done. (4) I think you will find that mathematicians divide broadly into two camps: (A) those who don't give a hoot what the nonmathematicians do with their mathematics and (B) those who would like the approaches to the subject which they have found useful to be widely disseminated. Of course, there a few hundred other issues one could bring to bear here (for example, one could just say "Thomas Kuhn" or "Lakatos" and then go on for ... 30 pages I suppose). There is probably no significant field of mathematics which has won acceptance merely by being correct. (Oh my, now I'm making massive generalizations, and I had really best close.) Abu Amaal 03:53, 13 April 2006 (UTC)

[edit] References

Could you tell me where to find the public discussions to which you refer? Michael Hardy 21:24, 13 April 2006 (UTC)
See my talk page for a response (briefly: Google). But there is also a review by one of Wildberger's colleagues to appear in the Intelligencer, which should be seen by quite a few mathematicians. The publisher is Wild Egg Books, run by Norman Wildberger, and with one book published to date (according to my reading of the site). As they say: 'Wild Egg Books is delighted to offer the controversial new book DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry by N J Wildberger'. I suggest people who are aware of reviews and other extended comments may wish to add to this section as they come across them (and also to the main page presumably, but the threshold here is much lower). Abu Amaal 01:35, 14 April 2006 (UTC)
The Drexel Mathforum site is relevant. This thread is a short one initiated by Wildberger. I imagine further discussion will accumulate on that site. Abu Amaal 19:40, 14 April 2006 (UTC)

[edit] Proposal: Divine Proportions article

Are there any objections, or alternative suggestions, to starting a Divine Proportions: Rational Trigonometry to Universal Geometry article, to discuss the book, and the more controversial views presented therein by Wildberger? --Piet Delport 03:06, 14 April 2006 (UTC)

I somewhat prefer keeping the discussion brief and on this page. But your proposal may be more broadly acceptable at this time. I would suggest in that case making the title short: Divine Proportions and putting a link (otheruses4 I believe) to Divine Proportion at the top of the page. I'm a little uneasy about the clash with a very different topic of major interest (if my short title is adopted) but it seems to me it can be handled cleanly. Abu Amaal 14:31, 14 April 2006 (UTC)
Agreed. Rational trigonometry, Wildberger, and his book are so intermingled as topics, it seems unnecessary to split them (it's not like Stephen Wolfram and A New Kind of Science, where there's a huge amount of reportage and controversy, and the author and book have separate notability). I recommend creating redirects pointing to Rational trigonometry. Tearlach 14:53, 14 April 2006 (UTC)

[edit] Proposal: reshape - on second thought

I'm tempted to put my hand to this article and try recasting it. I'm thinking more in terms of shape than content, though I expect there would be consequences at the level of content too. Broadly speaking, this article has two pieces, namely, an introduction which has a lot of content in itself, in terms of setting context and so on (but which I think requires a close reading to pull it all out), and a second piece which gives some necessary items - definitions, in particular - and a variety of formulas. I have a clearer sense of why the first part looks the way it does than I have for the second part. Anyway I would keep this general shape but no doubt tilt things more toward the front end. While one can always revert afterwards, if people are attached to the status quo it would be good to know. Abu Amaal 00:24, 16 April 2006 (UTC)

Well, after looking at some links I see that your part 2 is very firmly anchored in place and is certainly not going to get reworked by me after all. I'm surprised by all of those other pages. I don't see what they are doing on Wikipedia - though in a way I do. I mean, it's clear why one might want to make pages like that somewhere, and it's clear that somebody did some real work putting them in. But I wouldn't have thought they would be on Wikipedia. It seems to me the line between a mathematics page and a vanity page is getting quite blurred here. The reasons for this are partly intrinsic: one has a unique source for this approach to the subject. So I see why things look the way they do. I don't agree with it, but I think I understand more or less why it was done.

So part 2 is pretty safe from me. I'm not sure you really want to repeat the definition of quadrance at the end, there's a misprint in the spread section (1 radian), and the discussion of periodicity strikes me as a bit strange since in this theory spread is the independent variable and angle is banished from consideration; maybe the increased ambiguity relative to usual angle measurements is what is being pointed to here.

At this point the vestiges of my initial proposal are:

  • would anybody object to putting some of the introductory part into a section covered by the table of contents?
  • If I would like to fiddle with the introductory part, would you like to hear about it on this talk page first or just see it as a proposed edit? Abu Amaal

[edit] controversy?

I have now learned that Norman Wildberger thinks that what is controversial is not the content of the mathematics itself. Not surprising, of course. Michael Hardy 02:13, 16 April 2006 (UTC)

Do you mean "not surprising that Wildberger thinks that", implying that he's mistaken? --Piet Delport 23:25, 16 April 2006 (UTC)
I meant (of course) it's not surprising that Wildberger thinks that, implying that he's obviously right (see my comments above!). Michael Hardy 01:09, 17 April 2006 (UTC)
I may add a section on the philosophical and pedagogical points Wilderberg is trying to make. Two points in particular give a rationale for rational geometry - his lack of belief in "non-computable decimal numbers" and his belief that the concept of "angle" is weak. Nick Connolly 19:43, 23 October 2007 (UTC)

[edit] Anybody else read it?

With much discussion on Rational trigonometry, I was wondering how many have actually read ``Divine Proportions"? Best regards, Dmaher 10:21, 1 June 2006 (UTC)

I've read some parts of it. Not the whole thing yet. Michael Hardy 21:50, 1 June 2006 (UTC)
Nearly finished it. Had a longish conversation with Wildberger a few weeks ago also.Nick Connolly 20:04, 23 October 2007 (UTC)

[edit] Notability

What evidence is there that "rational trigonometry" is notable enough for the WP? As far as I can tell, it is being promoted by one not-well-known mathematician in one self-published book, with an Amazon sales rank of 900,000. The WP guideline is: "a minimum standard for any given topic is that it has been the subject of multiple non-trivial published works, where the source is independent of the topic itself". This topic has been the subject of only one published work (self-published to boot) by the promoter of the concept (that is, not independent). The only references to it in Google Scholar are by Wildberger himself; there are no outside citations. It may be a brilliant contribution to mathematics, or mathematics education, but until it is taken up by the profession at large (or some significant subgroup), it doesn't belong in WP, surely not in three articles! --Macrakis 18:55, 19 October 2006 (UTC)

Google Scholar actually gives two citations:
  • Vladimir V. Kisil's paper "Elliptic, Parabolic and Hyperbolic Analytic Function Theory–1: Geometry of Invariants" [3]
  • David G. Poole's short essay "The Impossibility of Trisecting an Angle with Straightedge and Compass: An Approach Using Rational Trigonometry" [4]
There is also:
Taken together with the popular interest in the subject (coverage in technical news articles, web journals), i believe the minimum notability criteria are met. --Piet Delport 13:19, 20 October 2006 (UTC)
We could quibble about whether Kisil's incidental mention, Poole's paper (unpublished?), and one book review make it 'notable', but I suppose if there is in fact significant interest in it in the news media, etc., it's worth including. --Macrakis 15:16, 20 October 2006 (UTC)
Agreed; i only mentioned the citations to set the record straight, not as a real argument for notability (by that measure alone, zillions of obscure, specialized academic works would be notable). --Piet Delport 15:25, 20 October 2006 (UTC)

To me it seems notable simply on the grounds that the questions answered by Wildberger's book are notable. Michael Hardy 23:11, 14 November 2006 (UTC)