Talk:Divine Proportions: Rational Trigonometry to Universal Geometry

Doubt

I find this current article dubious in way of it essentially advertising a book from the UNSW University with all citations centered around a seemingly single source. I am unsure of the quality of any particularity new insights gained from this reformulations of trigonometry that may be necessarily inherited from such an approach nor the technical merit of thing being on Wikipedia as this currently stands. This article does not convey any particularly new knowledge, rather more a reflection of a minor in-print book, Wikipedia is not a minor book review service. —Preceding unsigned comment added by 149.171.184.73 (talk) 09:44, 3 May 2011 (UTC)

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 one 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)

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)

Wildberger provides no motivation for separating out circular motion and others have done finite field exact rational calculation for most other parts of mathematics. Linear algebra libraries exist for this and the literature provides many references:

"We have used rational arithmetic in the implementation of: the matrix inverse using both Gauss-Seidel iteration and LU Decomposition (Golub and van Loan, 1999; Jennings and McKeown, 1992); exact calculation of the rank of a matrix; generalized matrix inverses for rank deficient, including rectangular matrices (Ben-Israel and Greville, 1974; Nashed, 1976); and an iterative matrix inverse using the conjugate gradient method (Golub and van Loan, 1999; Jennings and McKeown, 1992). In each case, the implementation involves only elementary matrix transformations which are carried out exactly in rational arithmetic... We show here how rational arithmetic can be used to compute indefinitely many digits of the DFT. In the longer version of this paper (Anderson and Sweby, 1999), this result is extended to the eigensystem of a symmetric matrix." (pg. 62 of Godunov Methods: Theory and Applications)

Non-linearities arising from rational points must be qualified with the fact that iterated rotations can be used and that any angle can be described by lengths alone, allowing any equispaced measurement. An irrational length of a circular makes it impossible to produce and can only rationally be divided in 4 by well-known theorems. Modular groups can handle rational rotations and rational trig functions (with inverse) are easily constructed with transrational arithmetic.

--Matthew Cory June 4th, 2017 — Preceding unsigned comment added by 173.64.61.197 (talk) 15:24, 4 June 2017 (UTC)

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:

$(a^{2}+b^{2}-c^{2})^{2}=4a^{2}b^{2}(\cos C)^{2}.\;$

and there you have the Cross law

$(Q_{1}+Q_{2}-Q_{3})^{2}=4Q_{1}Q_{2}c_{3}$

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:

$-a^{4}+2a^{2}b^{2}+2a^{2}c^{2}-b^{4}+2b^{2}c^{2}-c^{4}=0$

add $2a^{4}+2b^{4}+2c^{4}$ to both sides:

$a^{4}+b^{4}+c^{4}+2a^{2}b^{2}+2a^{2}c^{2}+2b^{2}c^{2}=2a^{4}+2b^{4}+2c^{4}$

factorise:

$(a^{2}+b^{2}+c^{2})^{2}=2(a^{4}+b^{4}+c^{4})$

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)

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)

"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: $Q_{3}=Q_{1}+Q_{2}+2{\sqrt {Q_{1}Q_{2}}}$ Ubern00b 22:49, 5 April 2006 (UTC)

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)

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)

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)

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

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)

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)

Yeah, it's awful and not worth your time. Qerty123 (talk) 12:59, 4 November 2013 (UTC)

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:

• James Franklin's review[5] (appearing in The Mathematical Intelligencer)

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)

so here we are 7 years later. Wildberger's youtube channel has over a million hits. a google search "site:.edu norman wildberger" turns up thousands of hits, and not all of them are just book reviews or his grad student's papers. Then there's an article in New Scientist. im pretty confident that it is notable now. Decora (talk) 05:39, 16 September 2013 (UTC)

What is this article about?

It's called rational trigonometry. Is it about a particular book or the mathematics in it? If it's the latter, it can include some references to earlier similar work, and if it is the former it should include something about the reactions to the book. As it stands, it seems entirely based on one source. Shreevatsa (talk) 00:57, 1 January 2009 (UTC)

Furthermore, the article Norman J. Wildberger appears to derive its notability entirely from its subject's introduction of the notion of rational trigonometry. Per WP:ONEEVENT, it seems likely that the latter article should be deleted and its relevant content merged here. siℓℓy rabbit (talk) 07:06, 1 January 2009 (UTC)

I implemented the merge. —David Eppstein (talk) 03:13, 17 March 2009 (UTC)

Universal Geometry?

Okay folks, I haven't yet read the book, but I've been enthralled since the moment I heard about the concept of rational geometry. Not only does this theory fit better with my intuition about real numbers, it accomplishes its primary mission. It truly is easier to grasp than traditional trigonometry; I remember my struggles from those days. If I was asked to teach trigonometry, this technique would be my first choice.

What would like to see the article do a better job of, however, is explaining the connection this work has to Universal Geometry, as the book title suggests. My sense is that it relates to the abstraction of the notions of quadrance and spread to other geometric domains, but this is at best implicit. 70.247.160.102 (talk) 19:48, 20 February 2010 (UTC)

Wildberger should have a biographical page

Consider:

• Philip DeFranco -- whose videos I often enjoy.
• Fred Figglehorn / Lucas Cruikshank -- I used to laugh at Fred, but these days his oeuvre seems stale.
• Jared Lee Loughner -- looks like a One Eventer to me.
• Jenna Marbles -- who is sometimes clever and funny -- oh, wait, she is up for deletion, but with the following notice: "If you can address this concern by improving, copyediting, sourcing, renaming or merging the page, please edit this page and do so. You may remove this message if you improve the article or otherwise object to deletion for any reason." Wildberger gets no such courtesy. He is gone -- poof -- with no explanation on the page, only the speechless redirect.

Or consider the following mathematicians:

• Robin M. Williams
• K. S. Amur
• Tatomir Anđelić
• George Anderson (mathematician)
• Ben Andrews

For perspective, look at Wildberger's research page. He is not just all about rational trigonometry.

Are certain editors guarding the gates here -- against repeated attempts to put this man in, as I did yesterday, only to have that deleted overnight? I have nothing to do with Mr. Wildberger, and only read about him yesterday via a tweet from a mathematics professor. I looked up his name and found this odd redirect.

Incidentally, if you want to question my impartiality in some other way, know that I dislike creationism, and that I agree that so far Wildberger is a one-trick pony in terms of the public eye, or among mathematicians. But so is Jared Loughner -- or is he actually famous for more than just that shooting spree? And get this: Kim Kardashian "is known for a sex tape with her former boyfriend Ray J as well as her E! reality series that she shares with her family, Keeping Up with the Kardashians." Now, if Norman Wildberger could possibly make a sex tape and put that up on Youtube along with his lectures, he'd stand a better chance of getting a page in WP. Dratman (talk) 23:18, 18 September 2011 (UTC)

Despite all the wikidrama which comes up from time to time, WP:N isn't all that complicated, and can be roughly summarized as "if you think the article should be kept, find more reliable sources independent of the subject". Most of the above arguments seem like WP:OTHERSTUFF. Last time I looked for sources on this, I found more for rational trigonometry than for Wildberger (not unusual for academic topics), and I added a few to the article. Kingdon (talk) 14:49, 24 September 2011 (UTC)

A decade later, I think you could certainly come up with enough material in third-party sources to write a defensible Wikipedia article about Wildberger himself, independent from his book or specific topics. Plenty of the chatter about him is based on self promotion and artificially hyped controversy, but there are also enough independent "reliable sources" to fill out a short encyclopedic biography, if someone is willing to put in the work. –jacobolus (t) 22:54, 7 May 2023 (UTC)

Pathological cases

If the underlying field has a square root of -1, then the line L with that slope is pathological. (For example, in F_13, this would be a line x+5y=constant or x-5y=constant). Any two points on L have quadrance 0, but are not identical. The spread between L and any other line is infinite. What does Wildberger say about this case?

The Pythagorean Theorem section states:

The lines A₁A₃ (of quadrance Q₁) and A₂A₃ (of quadrance Q₂) are perpendicular (their spread is 1) if and only if:

$Q_{1}+Q_{2}=Q_{3}.\,$

but the proof only goes in one direction. In fact the converse seems to be false: in a field of characteristic 2, all triangles are Pythagorean (Q_1 + Q_2 = Q_3), but not all spreads are 1. Does the "iff" apply only when the characteristic is not 2? Joule36e5 (talk) 11:11, 26 July 2012 (UTC)

Aha, a peek inside Google Books reveals (p 121) "all theorems in [Universal geometry] are necessarily valid over a general field, excluding characteristic two." So, evidently char 2 is indeed a pathological case. I would suppose that the concept still makes sense in such a field, but the theorems need to be qualified as to whether they're valid in a char 2 field. (On p 96 there's also some mention of char 13 or 17, but I can't view the context; perhaps this applies only to some specific exercise.) Joule36e5 (talk) 11:30, 26 July 2012 (UTC)

Too many links pointing here

Please, someone check if all links pointing here are indeed necessary. I have come here from Hyperbolic geometry, and spent a lot of (unhappy) time trying to understand this "theory" with no results. So I relied to Mathscinet and Zentrlablatt, which, at least, confirmed that I am not the only one who does not understand! As I have added to the page, there is a secondary source asserting that it is not clear what [the author's geometry] is. The review (by Victor V. Pambuccian for Mathscinet) is in fact even cruder When using venerable names, such as "hyperbolic" and "geometry", one expects to find an axiom system... and a representation theorem for that axiom system, describing all models of it.... None of it is happening here, so we have no idea what precisely "universal hyperbolic geometry" is. And Anyone wanting to actually learn "universal" metric geometry should, instead of this paper, study Bachmann's book [Aufbau der Geometrie aus dem Spiegelungsbegriff, zweite ergänzte Auflage, Springer, Berlin, 1973;], a paper by Struve and Struve [H. Struve and R. Struve, Z. Math. Logik Grundlag. Math. 34 (1988), no. 1, 79-88] as well as their other paper [J. Geom. 98 (2010), no. 1-2, 151-170], R. Lingenberg's [Metric planes and metric vector spaces, John Wiley & Sons, New York, 1979], and R. Frank's [Geom. Dedicata 16 (1984), no. 2, 157-165; (which is what I am going to do, rather than staying here).

By the way, I believe that the page should be kept, provided it is clearly explicitly stated that, so far, professional reviewers have found the theory unclear. Indeed, Wildberger has also written a lot of other books and papers on important journals.--78.15.196.22 (talk) 17:03, 6 September 2013 (UTC)

PS: to be clear, I do not mean that Wildberger's book contains errors, actually I do not think so. It simply appears that it contains just only a part (the simplest part) of trigonometry, and that it is really far from getting some kind of "Universal Geometry", whatever this means. This is also confirmed by Franklin's review, when he says "It is true that there is a need to retain the “circular” or “harmonic” functions to deal with circular motion, Fourier analysis and the like...". How this can be inserted into Wildberger framework I really do not understand. It looks like someone wanting to "simplify" the theory of the algebraic sums of integers just by considering only sums of positive numbers... Of course the resulting theory is simpler ;) but I call this a restriction, not a simplification!--78.15.196.22 (talk) 17:33, 6 September 2013 (UTC)

Deletion

I think this article should be deleted, the only sources explaining how this theory works comes from Wildberger's book which surely cannot be a third party source. There are no other reasonable sources on the topic because the theory isn't actually used by anyone else, and thus is not notable enough for a wikipedia page. — Preceding unsigned comment added by BBF3456789 (talk • contribs) 01:27, 12 November 2013 (UTC)

I agree. Started the process. If anyone disagrees, post here first before removing the tag. SohCahToaBruz (talk) 12:31, 13 November 2013 (UTC)

• I have removed the proposed deletion tag because that procedure is intended for articles where deletion is uncontroversial. It is not the appropriate procedure for an article like this one with an extended editing history, a prior related deletion discussion that saw material merged into this article (see Wikipedia:Articles for deletion/Norman J. Wildberger), and editors who have expressed support for the article in the past. Instead, it can be nominated for deletion as set forth at Wikipedia:Articles for deletion. --Arxiloxos (talk) 18:43, 13 November 2013 (UTC)

I do not agree with the proposal to delete or consideration for deletion. However the title of the page is Rational trigonometry, and yet some or all biographical information has been merged.. Someone was redirected here when searching for universal and or hyperbolic Geometry which is another subject Professor Wildberger has lectured on extensively. In that context RT is the introduction to his Universal Hyperbolic Geometry..

If there is a section on trigonometry then I suggest this page could usefully be merged with that. Unless and until other mathematicians are willing to cite or review the material in the book and or in his lectures and define yet another subject boundary, I feel the above discussions shows the work is significant enough to warrant a section on the trigonometry page.

Someone has already usefully pointed out the connections to the circular functions. The notion of quadrance requires contextualising as it is a proposal to reintroduce ancient Greek thinking, which is necessarily restricted to the Greek counting system and developments therefrom . In that regard the reference in the trigonometry page I propose, could be linked to math foundational issues or a page relating to that topic..

The Universal Hyprbolic Geometry link should be qualified to a you tube reference in the hyperbolic geometry page.

While I am a great admirer of Prof Wildberger's work and insights, as someone pointed out the page in Wikipedia is the matter under discussion. The material in the page is a fair summary and could even be made more concise. The book is available as is an extensive YouTube lecture series for anyone washing to find out moreJehovajah (talk) 19:32, 30 November 2013 (UTC)

• I disagree to delete. In my opinion, rational trigonometry is notable and mathematically sound. Wildberger makes a very good point that it is computationally accurate in contrast with other approaches. --Asterixf2 (talk) 10:00, 17 May 2016 (UTC)

This page is very misleading. It should be revised into an article about "Divine Proportions" or N.J. Wildberger, but it doesn't warrant it's own math page. As it is currently written, this page makes a fringe math theory seem like it's actually a useful, promising, perhaps even revolutionary theory. For example, the article says "Rational trigonometry makes nearly all problems solvable with only addition, subtraction, multiplication or division, as trigonometric functions (of angle) are purposefully avoided in favour of trigonometric ratios in quadratic form." This is a huge claim. But what's the source? It's an article that Wildberger wrote for an undergraduate math magazine that aims to "[be a journal that is an] accessible forum for practitioners, students, educators, and enthusiasts of mathematics, dedicated to exploring the folklore, characters, and current happenings in mathematical culture." The article is peer-reviewed, but it's a peer-reviewed feature article in a journal that is properly categorized as general interest rather than math research. The claim that "Rational trigonometry makes nearly all problems solvable..." is too general and it reads like propaganda for a fringe theory; someone who did not check the citations might think that Rational Geometry was a special new technique that can solve nearly all problems, when it is actually the case that "nearly all [Geometry and Trigonometry] problems" cannot be solved using Rational Trigonometry, as Rational Trigonometry gives such an impoverished coordinate system that you can't do more or less anything that the Greeks were able to do with a compass and a straight edge. What is this article doing here? This topic is one man's pet theory, described in a book that he published himself, that has been around for well over a decade yet has made no purchase either in secondary education, mathematical research, or anywhere except for YouTube and it's comments section. Until there are some third-party sources other than Wildberger (or book reviews of Wildberger!) who make use of the theory and show that it is indeed a useful, interesting, and productive area of study, this article should be deleted or turned into a page on Wildberger's Rational Trigonometry book. Wikipedia must do a better job weeding out articles like this one, or else it's credibility will suffer. Matheducator22 (talk) 05:00, 1 May 2019 (UTC)

Wildberger against infinity

My impression is that Wildberger's main interest is not so much to simplify trigonometry as to eliminate infinity from mathematics. The page should make this clear. Tkuvho (talk) 12:07, 25 November 2013 (UTC)

Babylonian precedent

The Plimpton 322 tablet has recently been claimed to be a trigonomic table using the rational approach:

https://phys.org/news/2017-08-mathematical-mystery-ancient-babylonian-clay.html 207.224.80.52 (talk) 03:24, 25 August 2017 (UTC)

Erroneous conflation of "exact values" with rational numbers

The page, as it is currently written, says "The laws of rational trigonometry, being algebraic and 'exact-valued'..." This is not correct. This phrase implies that rational numbers are "exact" and that irrational numbers are not. There is nothing "inexact" about irrational numbers or real numbers in general. For example, the exact solutions of $x^{2}-x-1=0$ are the golden ratio and the silver ratio, two irrational numbers, for example. There is nothing approximate about them. Any solutions other than these two would be, at best, approximations to the actual solution. If you use a number system that does not permit irrational numbers, then the equation $x^{2}-x-1=0$ has no exact solutions, just as the equation $x^{2}+1=0$ has no solutions if you limit your number system to the real numbers. Matheducator22 (talk) 05:49, 1 May 2019 (UTC)

@Matheducator22: WP:SOFIXIT . –Deacon Vorbis (carbon • videos) 12:32, 1 May 2019 (UTC)

Full quote: "The laws of rational trigonometry, being algebraic and 'exact-valued', introduce subtleties into the solutions of problems, such as the non-additivity of quadrances of collinear points (in the case of the triple quad formula) or the spreads of concurrent lines (in the case of the triple spread formula) absent from the classical subject, where linearity is incorporated into distance and circular measure of angles, albeit 'transcendental' techniques, necessitating approximation in results."

Response: I agree the use of the term 'exact-valued' is inaccurate. But if you take that out what remains is correct. It is expressing the idea that the algebra is finite but not linear as a result whereas if using power series to approach (say) 'arctan(x_1)' giving linearised expressions the algebra becomes an infinite polynomial - equivalent to transcendentalism. (You pays your money and you take your choice.) I have taken out the offending term.

Paul White 07:16, 5 May 2019 (UTC)

And if you approximate rational numbers as decimal or binary or floating point then the algebra becomes just as infinite. There is nothing special about rationality here. —David Eppstein (talk) 08:09, 5 May 2019 (UTC)

Rewrite

There has been some discussion on Talk:Euclidean distance about whether we should try again to delete this article (the last attempt was in 2013), and how to deal with the inevitable "keep because it has references" arguments. Instead, I have WP:BOLDly completely rewritten the article to be about the book rather than about the more-or-less-nonexistent-except-for-Wildberger field of study, based on reliable secondary sources (published reviews of the book) rather than primary sources (the book itself). —David Eppstein (talk) 18:25, 30 December 2020 (UTC)

Impressively quick labor! When I have more time I will see if there are things that might be salvaged from the earlier article.

A point that the reviews I have seen did not make forcefully enough (or at all in some cases) is that even accepting the idea that quadrance should be the basic quantity, at some point one will want the quadrance-preserving symmetries, which are the same as the classical isometry group, and therefore it is inevitable that the one-parameter subgroups fixing a point (rotations) or line (translations) will be central to the story. But classical distance and angle measure are set up precisely to be compatible with (ie, additive with respect to) the parameterization of those subgroups and this goes haywire in Wildberger's approach. Wilderberger's stuff is more something that could, hypothetically, work in a computer application where data is already presented as squared distances and squared sines, and one operates on that directly using his formulas. Whether anyone has tried and found it beneficial, I don't know. 73.89.25.252 (talk) 19:02, 30 December 2020 (UTC)

That's a good point, but I think not one we can make in the article unless we can find sources that say it. —David Eppstein (talk) 19:33, 30 December 2020 (UTC)

One of the reviews hosted at Wildberger's web site calls it a geometry without rotations (or very similar words), and some of the other reviews talk about solution of triangles versus cyclometric functions (the implication being that it's a geometry and trigonometry of surveying, as in relations between coordinates of points at fixed locations, which is another way of saying the dynamic rotations are missing).73.89.25.252 (talk) 03:47, 31 December 2020 (UTC)

I think this was a good move. XOR'easter (talk) 02:06, 31 December 2020 (UTC)

Undue weight on self-publication

The article emphasizes the self-published nature of the book, which I think goes overboard in this case, and leads to UNDUE and BLP. The implication of self-published, together with all the other criticism included in the article is that the book is a vanity publication, crackpotry or something that would not have passed review at an academic press or a peer-reviewd journal. But that is not true here; the mathematics is undisputedly correct and competent, and any controversy or skeptical reviews concern the claims about advantages of Wildberger's approach compared to the traditional ones.

There are a quite a few mathematicians who have become publishers and sellers of their own books in the age of LaTeX and journal boycotts. Nobody criticized Michael Spivak's books on differential geometry, or Thurston's notes on 3-manifolds, for being essentially self-published. I don't think Wildberger's book has to be of that quality for him to not be cast as the operator of a vanity press. 73.89.25.252 (talk) 18:57, 31 December 2020 (UTC)

It's only because I've been typically putting who wrote a book and how it was published into the lead for lack of a better place to put it. In this case it happens to be self-published. I don't think it ended up being important to how it was received — most self-published works and even many conventionally published books don't get this level of published book reviews — but it should be mentioned somewhere. Compare Adventures Among the Toroids, another article I wrote about a self-published book (where again I had no intention of implying crankery but self-published went into the lead for lack of a better place to put it). —David Eppstein (talk) 19:43, 31 December 2020 (UTC)

Wiswell is actually Wisewell, but...

Cited is a review by Laura Wiswell, who is unknown to the Internet and appears to be Laura Wisewell, then a mathematician at Edinburgh, with the name misspelled in the source. Since Ms. Wisewell has died it is hard to confirm this, and the document is listed under the misspelling, things could be left as they are with a note in the reference, but if there is sufficient confidence in the determination of the true author it would make sense to correct it throughout the article. What sayeth thou, WP? 73.89.25.252 (talk) 21:08, 1 January 2021 (UTC)

Thanks for catching this. It seems clear that they are the same person. I think we should just silently correct the typo. —David Eppstein (talk) 21:27, 1 January 2021 (UTC)

Changed now in the text, with a silent note on the misspelling in the reference, and the coding of the citations unchanged (to retain the name actually used in the journal). 73.89.25.252 (talk) 22:03, 1 January 2021 (UTC)

Math in a math article?

Is there a reason why this article has changed from an article about mathematics to a criticism of his book? It was originally called "Rational Trigonometry" because it was about the math in his book, not about Wildberger or how he feels about the choice axiom. Rational trigonometry is interesting and isn't wrong. Because its math. Any possibility of changing it back to actually having math? Maybe moving all the stuff from people who didn't like his book to the controversy section? 73.162.96.217 (talk) 01:56, 7 January 2023 (UTC)

With multiple published reviews, the book itself meets Wikipedia's notability standards, regardless of whether we have a separate article on rational trigonometry. But with little attention to Wildberger's theories beyond criticism of the book, the notability for the theories themselves is much less clear. So instead of not having an article on rational trigonometry at all, we have included a redirect to the closest related notable topic. See past discussions at Wikipedia:Articles for deletion/Spread polynomials and Wikipedia:Articles for deletion/Rational trigonometry. Or, for that matter, read the book reviews to get a clearer idea why the theories might not have made much impact. —David Eppstein (talk) 02:07, 7 January 2023 (UTC)

It seems quite extreme to entirely cut any discussion of the content of the book including all of the figures and formulas, and does a disservice to readers. It would be helpful to add back one or two sections clearly describing the basic ideas, including an concrete example or two and some figures. It doesn't need to reprise the full content of the book. Currently readers coming to this page (or e.g. redirected from rational trigonometry) who are curious about this topic will be left mystified. –jacobolus (t) 17:30, 7 May 2023 (UTC)

With essentially no work on it beyond Wildberger's, rational trigonometry as formulated by Wildberger is not a notable subject. The "overview" section of this article is intended to describe the overall flavor and content of this subject. I don't think there's any salvageable content in the old version that would be helpful to readers and could be properly sourced to independent secondary sources rather than to Wildberger's primary work. Yes, it looks "mathy": it has big piles of formulas and illustrations. Looking mathy is not the same as being informative. —David Eppstein (talk) 18:49, 7 May 2023 (UTC)

essentially no work on it beyond Wildberger's – this is too harsh. The book has been cited like 150 times in Google Scholar's citation index, most often by Wilderger himself or his students, but also by a pretty wide range of other people, with applications to various technical fields. It would take some significant work to read through the list and accurately characterize the influence of the book so far, but your characterization is not fair. I think it would be useful to briefly describe the top few abstractions Wildberger uses and compare the main few formulas of trigonometry in Wildberger's book vs. the traditional expressions. Otherwise readers coming to this page to figure out what Wildberger's system is supposed to be about are left in the dark. This article currently mostly consists of the gut feelings and speculation of reviewers writing immediately after the book's publication. While those are somewhat relevant, they aren't going to answer the most basic questions readers are likely to have. –jacobolus (t) 20:52, 7 May 2023 (UTC)

(Note: I'm not planning to work on this myself in the foreseeable future, among other reasons because the Wikipedia articles about "traditional" approaches such as angle, triangle, trigonometry, history of trigonometry, solution of triangles, law of cosines, law of tangents, Menelaus's theorem, inscribed angle, central angle, spherical geometry, spherical trigonometry, great circle, spherical triangle, stereographic projection, mathematical instrument, etc. in my opinion need a lot of help.) –jacobolus (t) 21:12, 7 May 2023 (UTC)

My own feeling is that your phrasing "formulas of trigonometry in Wildberger's book vs. the traditional expressions" already makes a mistake, in presupposing that the formulas in Wildberger's book were not already in regular use. For instance, the simplification of doing distance comparisons using squared distance instead of distance long predates Wildberger's giving it a different name. —David Eppstein (talk) 21:26, 7 May 2023 (UTC)

That's fine enough. It's a fair criticism that Wildberger invents weird new names for existing ideas and doesn't sufficiently credit past work. But a reader coming here is still likely to be curious what his book actually says. For example, Glen Van Brummelen's recent book about the history of trigonometry mentions Wildberger (in a throwaway line near the end). I can easily imagine a reader coming to Wikipedia to figure out what he's talking about. –jacobolus (t) 21:56, 7 May 2023 (UTC)

Where I am coming from is that I appreciate people at least trying to rework and revisit the foundations of elementary subjects, even if they aren't always as successful as their authors might hope. Even though I don't think e.g. gyrovector space or rational trigonometry or tau (mathematical constant) is going to take the world by storm, there's at least some merit to these proposals, and Wikipedia should still try to neutrally describe what they are and what their relation is to mainstream methods. Sometimes, as in the example of geometric algebra, what was once a tiny niche can through sufficient work and clear enough concrete advantages start to see significant practical adoption.

To go back to Wildberger's "spread" and "quadrance" as a practical example, the Google S2 library for spherical geometry uses squared chord length between points on the unit sphere (4 times spread, a.k.a. "spherical quadrance") as one of its primary representations of spherical distances. This was arrived at entirely independently of Wildberger, but the library authors had overlapping reasons for adopting such a representation: it can be worked with using rational arithmetic instead of transcendental functions. If the authors of the library used spread per se and did a full systematic study of it as a representation, the way Wildberger has started to do, they might find clearer or more legible expressions or explanations for their work or additional places that their code could be extended or applied. –jacobolus (t) 22:15, 7 May 2023 (UTC)

Covering the use of polynomial and rational methods in geometry, such as the use squared Euclidean distance and dot products, in articles related to those topics, with proper historical documentation of those topics, is obviously unproblematic. For instance, I included material on squared distances in the Euclidean distance good article. But that doesn't mean we should cover them in Wildberger's terms. If Wildberger had merely written a survey of such methods, properly crediting past results, we probably wouldn't be here. But instead, any attempt to cover this topic in Wildberger's terms is inherently a continuation of Wildberger's own promotionalism, where he pretends that all of these standard methods are new and original to him. Your Google example is a good instance of this: you say yourself that it was arrived at without reference to Wildberger and yet you want to use it as an example of this material. It is not an example of this material. It is just a continuation of the standard use of squared Euclidean distance to simplify geometric calculations that has long been used in many contexts.

If there is anything novel about Wildberger's approach to this, it is the dogmatic hyperfinitist view that denies the existence of regular pentagons because they cannot be expressed with rational coordinates. But that is not a point of view that leads to greater practicality, and it is not a point of view that comes across to readers by showing them paragraph after paragraph of symbolic derivation of equations said to be a reformulation of the Pythagorean theorem. —David Eppstein (talk) 22:58, 7 May 2023 (UTC)

you want to use it as an example – just to be clear, I am not suggesting Google's project should be discussed in this article. My claim is that (a) these methods have some clear benefit in various contexts, and (b) are worthy of systematic study taking them on their own terms instead of just treating them as a funny one-off trick in an otherwise angle-measure-dominated world. –jacobolus (t) 23:33, 7 May 2023 (UTC)

That's a worthy goal but not one that can be accomplished in the context of a book that deliberately divorces itself from the systematic study of the uses people have made of these methods, instead pretending that they are some new kind of geometry. —David Eppstein (talk) 00:36, 8 May 2023 (UTC)

paragraph after paragraph of symbolic derivation – I am also not advocating this! I think a clear summary can be made of the core content of this book with 2–3 figures and a few paragraphs, with identities shown (e.g. in a table) side by side with the "standard" trigonometric identities, if you like with commentary about where similar ideas had been developed/applied previously or independently. We don't need to include Wildberger's made up new names beyond "quadrance" and "spread". There are independent sources reprinting these formulas if you want to cite something other than Wildberger's book.

dogmatic hyperfinitist view – I don't think there's much point in belaboring this part, even though Wildberger himself certainly does. Beyond motivating him, it's largely tangential to the actual content of his work. –jacobolus (t) 23:41, 7 May 2023 (UTC)

Are you trying to explain how the book relates to known material or are you trying to explain these computational methods to people who might want to use them? If the former, I think we are better served by sourced material explaining the similarities of the book's material to known methods, rather than by deceptive tables that put the book's material only up against related but non-rational formulas as if to say that the rationality is novel. If the latter, I think this material would be more helpful elsewhere, not in the context of the book, using sources with conventional terminology. An example of such a source (usable in a context about computing with distances but not in a context about the book) would be the paragraph devoted to how and why one should avoid square roots for the distances in Euclidean minimum spanning trees, in Yao, Andrew Chi Chih (1982), "On constructing minimum spanning trees in k-dimensional spaces and related problems", SIAM Journal on Computing, 11 (4): 721–736, doi:10.1137/0211059, MR 0677663. (Note the date!) —David Eppstein (talk) 00:45, 8 May 2023 (UTC)

@David Eppstein – In my opinion we should try to answer the main questions article readers are likely to have, which include both (1) what is this and how does it work, and also (2) who uses this and how does this relate to other mathematical works. Since Wildberger has done a lot of promotion to a wide audience and pitches his book to laypeople, it would be helpful to aim at the ~high school level if we can.

I will take a look at your link when I get the chance. Another source with squared distance:

Kendig, Keith (2000), "Is a 2000-Year-Old Formula Still Keeping Some Secrets?", American Mathematical Monthly, 107 (5): 402–415, JSTOR 2695295

This one talks about "separation-squared" as a concept and overlaps a little bit with Wildberger's book. –jacobolus (t) 19:47, 8 May 2023 (UTC)

@Jacobolus: I do not appear to have subscription access to the source you added (in the wrong format, violating WP:CITEVAR: should have been a list-defined reference in Citation Style 2, not an inline-defined reference in Citation Style 1), "Mind-bending mathematics: Why infinity has to go". It appears to be an article about finitism, rather than about Wildberger or this book. Since you are using this to claim that Wildberger is a finitist and that his finitist views motivated this book, can you please provide me the quotes in this article that state that he is a finitist (not merely quoting him about finitism, but clearly stating that he is himself a finitist, so that we don't run afoul of WP:BLP by attaching an unsourced epithet to him) and that his finitist views informed the book (so that we don't run afoul of WP:SYN, my reason for already reverting your edit once)? I don't want to revert your edit a second time without discussion but I am not convinced that this source is adequate for what you are using it to source. —David Eppstein (talk) 19:49, 7 May 2023 (UTC)

But if infinity is such an essential part of mathematics, the language we use to describe the world, how can we hope to get rid of it? Wildberger has been trying to figure that out, spurred on by what he sees as infinity’s disruptive influence on his own subject. “Modern mathematics has some serious logical weaknesses that are associated in one way or another with infinite sets or real numbers,” he says. ¶ For the past decade, he has been working on a new, infinity-free version of trigonometry and Euclidean geometry. In standard trigonometry, the infinite is ever-present. Angles are defined by reference to the circumference of a circle and thus to an infinite string of digits, the irrational number pi. Mathematical functions such as sines and cosines that relate angles to the ratios of two line lengths are defined by infinite numbers of terms and can usually be calculated only approximately. Wildberger’s “rational geometry” aims to avoid these infinities, replacing angles, for example, with a “spread” defined not by reference to a circle, but as a rational output extracted from mathematical vectors representing two lines in space. ¶ [...] While Wildberger’s work is concerned with doing away with actual infinity as a real object used in mathematical manipulations, [...] –jacobolus (t) 20:15, 7 May 2023 (UTC)

Here is another source calling Wildberger a finitist and crediting that as the motivation for his work: Schramm, Thomas (2019), "Divine Proportions Norman Wildbergers andere–rationale Trigonometrie", Proc. 15. Workshop Mathematik in ingenieurwissenschaftlichen Studiengängen (PDF) (in German), p. 53–62. –jacobolus (t) 20:20, 7 May 2023 (UTC)

If you consider Wildberger himself a credible source about his own motivations, he has directly claimed this repeatedly, including in peer-reviewed papers, his book, self-published papers, interviews, YouTube videos, etc. (Though I am not sure if he calls himself a "finitist" per se.) –jacobolus (t) 20:34, 7 May 2023 (UTC)

"Pythagoras' theorem proof (rational trigonometry)" listed at Redirects for discussion

The redirect Pythagoras' theorem proof (rational trigonometry) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 April 21 § Pythagoras' theorem proof (rational trigonometry) until a consensus is reached. 1234qwer1234qwer4 19:33, 21 April 2023 (UTC)

"Coturn (rational trigonometry)" listed at Redirects for discussion

The redirect Coturn (rational trigonometry) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 April 21 § Coturn (rational trigonometry) until a consensus is reached. 1234qwer1234qwer4 19:34, 21 April 2023 (UTC)

Franklin review is being mischaracterized

Ping @David Eppstein – Franklin has a review in which he says, essentially, 3D geometry is often and successfully modeled using vectors, and Wildberger's "rational trigonometry" is implicit in the vector identities commonly used. This is evidence that Wildberger may be right to try to replace "traditional trigonometry" (i.e. lots of transcendental functions and opaque calculations) with simple algebraic relationships. To directly quote:

Secondly, a careful examination of 3D vector geometry will reveal that a certain amount of Wildberger's philosophy is implicit in it already, suggesting that he is on the right track at a more basic level. What makes geometry with vectors so successful is that all the information about lengths and angles is contained in the scalar product of vectors, which is algebraically very simple. The student soon learns that the way to approach typical problems, say on the closest distance between two non-intersecting lines, is to stay with vectors and their scalar products as long as possible and only extract any needed lengths and angles at the last moment. Wildberger simply goes one step further: he recommends we do the same in two dimensions, and suggests that we hardly ever have any real need for lengths and angles in any case.

This article currently entirely mischaracterizes that review, and uses it to imply that Franklin is criticizing Wildberger for lack of novelty. It's an unfair and underhanded interpolation which should not be allowed here per WP:SYN. Wikipedia should not be putting words into reviewers' mouths and using those to advance Wikipedians' personal agendas. ––jacobolus (t) 23:28, 7 May 2023 (UTC)

This is not evidence that "Wildberger may be right to try to replace traditional trigonometry" because that is an inaccurate description of what Wildberger did, which is actually to formalize the simplified calculations that many people had already been using for much longer (non-problematic) and then file off all the serial numbers and scribble his own name all over them instead (problematic).

It was not intended to imply that Franklin is criticizing Wildberger. It was not intended as criticism at all, except maybe by implication. The intent was to inform readers of how a concept from the book relates to more standard formulations. More specifically, it was intended to note the close relation between one of Wildberger's two central concepts, "spread", and standard dot product based calculations, just as the other material on squared Euclidean distance was intended to clarify the relation between the other of Wildberger's central concepts, "quadrance", and long-standard calculations. By making it much more vague, you completely lost that point, so that nowhere in your version of the article does one learn that spread is not a particularly novel concept. —David Eppstein (talk) 00:08, 8 May 2023 (UTC)

The article saying James Franklin points out that for spaces of three or more dimensions, modeled conventionally using linear algebra, the use of spread by Divine Proportionsis not very different from standard methods involving dot products in place of trigonometric functions. Is not supportable from the source. First of all, Franklin says nothing about >3 dimensional space, nor does he mention "linear algebra". But most importantly he does not say Wildberger's method is "not very different"; what he says directly is that Wildberger's approach "goes one step further". Whatever you think Franklin's review should have said, the summary suggested in these sentences located in a section called "Critical reception" is a flagrant mischaracterization. –jacobolus (t) 00:19, 8 May 2023 (UTC)

What he says directly is that you can do the same things with dot products. You are misinterpreting the "goes one step further", which is not a claim that the use of spread is different from the use of dot products. It is a claim that people didn't use dot products in 2d and that Wildberger is novel in extending the use of dot products from 3d to 2d. I believe that the idea that people didn't use dot products in 2d is obviously false, so I think we're better off not trying to repeat that part of the claim. —David Eppstein (talk) 00:26, 8 May 2023 (UTC)

Spread is obviously not the same as the use of (just) dot products per se: The spread between vectors is the squared magnitude of the cross (or wedge) product divided by the squares of the vectors. Arguing that wedge products are underused in 2D seems fair enough to me; I have read at least a couple of expository papers in computer graphics arguing such a point.

That's all somewhat off the main point though, which is that this source does not adequately support the claim that Wildberger's method is "not very different". The source does not say anything remotely similar to that. –jacobolus (t) 03:58, 8 May 2023 (UTC)

It says the same idea is implicit in the philosophy that all students "soon learn", that "the way to approach typical problems ... is to stay with vectors and their scalar products". Do you have a less-blunt way of saying that the only difference is that spread is 1 − squared dot product (of normalized vectors) instead of dot product itself? —David Eppstein (talk) 05:01, 8 May 2023 (UTC)

In my opinion Wildberger's "rational trigonometry" concept/approach is (for better or worse) significantly different from the usual vector methods, and not just because he makes up new words for everything. I personally find Wildberger's approach to be somewhat unintuitive, inflexible, and parochial compared to vector expressions I would typically use, but it is in my opinion substantially novel as a whole system (I don't think I have seen any previous source which developed anything particularly comparable), even if the individual parts involved are nothing fundamentally new. YMMV. –jacobolus (t) 05:55, 8 May 2023 (UTC)

Works citing Wildberger on rational trigonometry and related topics

I'm parking some more citations here, in roughly chronological order, in case they're useful to anyone trying to work on this article. (All of the citations in the current article are reviews published shortly after the book first came out.)

I tried to exclude anything either (a) directly by Wildberger or his students, or (b) that only mentioned his work incidentally (but I did include some articles that merely adopted the name "quadrance" for squared distance). Obviously most of these are not fit for inclusion in this article. But sifting through may help if anyone wants to write sections about influence, applications, etc. Feel to modify the below list if there's anything I missed. There are a lot of masters theses and conference papers here, but also a nontrivial number of journal papers. –jacobolus (t) 03:45, 8 May 2023 (UTC)

• Almeida, João Pequito (2007), Rational Trigonometry Applied to Robotics (PDF) (MSc thesis), Universidade Tecnica de Lisboa.
• Kosheleva, Olga (2008), "Rational trigonometry: computational viewpoint." (PDF), Geombinatorics, 1 (1): 18–25.
• Ida, Tetsuo; Marin, Mircea; Takahashi, Hidekazu; Ghoura, Fadoua (2008), "Computational origami construction as constraint solving and rewriting" (PDF), Electronic Notes in Theoretical Computer Science, 216: 31–44, doi:10.1016/j.entcs.2008.06.032.
• Camargos, C. N.; Mendonça, C. A.; Duarte, S. M. (2009), "Da imagem visual do rosto humano: simetria, textura e padrão", Saúde e Sociedade (in Brazilian Portuguese), 18 (3): 395–410, doi:10.1590/s0104-12902009000300005.
• Kurz, Sascha (2009), "Integral point sets over finite fields" (PDF), The Australasian Journal of Combinatorics, 43: 3–29, arXiv:0804.1289.
• Aarts, Jan (2010), "From a rational angle" (PDF), Nieuw Archief voor Wiskunde, 11 (3): 209–211.
• Shparlinski, Igor E (2010), "On point sets in vector spaces over finite fields that determine only acute angle triangles" (PDF), Bulletin of the Australian Mathematical Society, 81 (1): 114–120, doi:10.1017/S0004972709000719.
• Fuhrmann, Thomas; Navratil, Gerhard (2013), "Ausgleichungsrechnung mit Gröbnerbasen" (PDF), Zeitschrift für Geodäsie, Geoinformation und Landmanagement (in German), 138: 399–404.
• Silva, Luiz José da (2013), Trigonometria Racional: Uma Nova Abordagem para o Ensino de Trigonometria (PDF) (MA dissertation) (in Brazilian Portuguese), Universidade Federal da Bahia.
• Vinh, Le Anh (2013), "The number of occurrences of a fixed spread among n directions in vector spaces over finite fields", Graphs and Combinatorics, 29: 1943–1949, arXiv:0809.4214, doi:10.1007/s00373-012-1242-3.
• Martins, Maria Cristina Fernandes (2014), Trigonometria passado, presente e futuro (MA dissertation) (in Portuguese), Universidade da Beira Interior.
• Gunn, Charles (2014), Rational trigonometry via projective geometric algebra, arXiv:1401.2371.
• Resa, Jorge; Cortes, Domingo; Navarro, David (2015), "A fast algorithm for coordinate rotation without using transcendental functions", Proceedings of the International Conference on Scientific Computing (CSC), Athens, 2015, WorldComp, p. 42.
• Li, Xiao-Bo; Burkowski, Forbes J.; Wolkowicz, Henry (2016), "Protein structure normal mode analysis on the positive semidefinite matrix manifold" (PDF), 8th International Conference on Bioinformatics and Computational Biology (BICOB 2016).
• Horváth, Ákos G. (2016), "Constructive curves in non-Euclidean planes", Proc. 19th Scientific-Professional Colloquium on Geometry and Graphics, arXiv:1610.00473.
• Falvo D’Urso Labate, G.; Baino, F.; Terzini, M.; Audenino, A.; Vitale-Brovarone, C.; Segers, P.; Catapano, G. (2016), "Bone structural similarity score: a multiparametric tool to match properties of biomimetic bone substitutes with their target tissues", Journal of Applied Biomaterials & Functional Materials, 14 (3): e277–e289, doi:10.5301/jabfm.5000283.
• Brakhage, Karl-Heinz; Niemeyer, Alice C.; Plesken, Wilhelm; Strzelczyk, Ansgar (2017), "Simplicial Surfaces Controlled by One Triangle" (PDF), Journal for Geometry and Graphics, 21 (2): 141–152.
• Juhász, Máté L. (2018), "A Universal Linear Algebraic Model for Conformal Geometries", Journal of Geometry, 109: 1–18, arXiv:1603.06863, doi:10.1007/s00022-018-0451-1.
• Lund, Ben; Pham, Thang (2018), "Distinct spreads in vector spaces over finite fields", Discrete Applied Mathematics, 239: 154–158, arXiv:1611.05768, doi:10.1016/j.dam.2017.12.027.
• Cabieces García, Pablo (2018), Aplicación de la trigonometría racional en la náutica [Application of rational trigonometry to nautical studies] (PDF) (MEng thesis) (in sp), Universidad de Cantabria{{citation}}: CS1 maint: unrecognized language (link).
• Li, Zijia; Schicho, Josef; Schröcker, Hans-Peter (2019), "The geometry of quadratic quaternion polynomials in Euclidean and non-Euclidean planes", Proceedings of the 18th International Conference on Geometry and Graphics 2018 (ICGG 2018), Milan, Italy, Springer, pp. 298–309, arXiv:1805.03539, doi:10.1007/978-3-319-95588-9_24.
• Schramm, Thomas (2019), "Divine Proportions Norman Wildbergers andere–rationale Trigonometrie" (PDF), Proc. 15. Workshop Mathematik in ingenieurwissenschaftlichen Studiengängen (in German), p. 53–62.
• Martínez Peralta, Rogelio; Zamora Gómez, Erik; Sossa Azuela, Juan Humberto (2019), "Efficient FPGA hardware implementation for robot manipulator kinematic modeling using rational trigonometry", IEEE Latin America Transactions, 17 (9): 1524–1536, doi:10.1109/TLA.2019.8931147.
• Bongardt, Bertold (2019), "On circle intersections by means of distance geometry", Advances in Mechanism and Machine Science: Proceedings of the 15th IFToMM World Congress on Mechanism and Machine Science, vol. 15, Springer, pp. 187–196, doi:10.1007/978-3-030-20131-9_19.
• Scharler, D. F.; Schrocker, H. P. (2020), "Motion factorization and bond theory in hyperbolic kinematics", in Alkhaldi, Ali Hussain; Alaoui, Mohammed Kbiri; Khamsi, Mohamed Amine (eds.), New Trends in Analysis and Geometry, Cambridge Scholars Publishing, pp. 193–218, ISBN 1527545636.
• Cifrián Pérez, Diego (2020), Aplicación de la trigonometría racional a la navegación astronómica [Application of rational trigonometry to celestial navigation] (PDF) (MEng thesis) (in sp), Universidad de Cantabria{{citation}}: CS1 maint: unrecognized language (link).
• Ida, Tetsuo (2020), "Verification of Origami Geometry", An Introduction to Computational Origami, Springer, pp. 113–144, doi:10.1007/978-3-319-59189-6_5.
• Kusaka, Yasuyuki; Fukuda, Nobuko (2020), "Decomposition of pattern distortions by the spread polynomial model in roll-to-sheet reverse offset printing", Journal of Micromechanics and Microengineering, 30 (9): 095007, doi:10.1088/1361-6439/ab999c.
• Schramm, Thomas (2021), "Rational Trigonometry Using Maple", Maple in Mathematics Education and Research: 4th Maple Conference, Waterloo, Ontario, Canada, November 2–6, 2020, Springer, pp. 365–375, doi:10.1007/978-3-030-81698-8_24.
• Scharler, Daniel F.; Schröcker, Hans-Peter (2021), "An Algorithm for the Factorization of Split Quaternion Polynomials", Advances in Applied Clifford Algebras, 31: 1–23, doi:10.1007/s00006-021-01133-8.
• Martínez, Rogelio; Zamora, Erik; Sossa, Humberto; Arce, Fernando; Soriano, Luis Arturo (2022), "Orientation Modeling Using Quaternions and Rational Trigonometry", Machines, 10 (9): 749.
• Bender, Chase; Chakrabarti, Debraj (2023), "Polarization identities", Linear Algebra and its Applications, 665: 211–252, arXiv:2204.12014, doi:10.1016/j.laa.2023.02.006.
• Evers, Manfred (2023), Geometry on real projective Cayley-Klein spaces, arXiv:2301.04024.

---

You know, for all of the effort you are spending on pushing this as some kind of breakthrough in how to do geometric calculations more simply, if you want a book on the same type of topic with 1/10 the hype, 3x the citations, and I think significantly more novelty, you might try instead Stolfi's Oriented Projective Geometry. We have an article on it, oriented projective geometry, but it's in significant need of improvement. —David Eppstein (talk) 06:55, 8 May 2023 (UTC)

I am not pushing this as a "breakthrough". But I also don't think it can be dismissed off-hand as trivial, even if Wildberger rubs you the wrong way for his polemical self promotion. Also, Stolfi's book is nice, though personally I think still not the best representation for my own uses. I like Stolfi, who is a Wikipedian, e.g. User:Jorge_Stolfi/DoW/Vogonization. –jacobolus (t) 13:20, 8 May 2023 (UTC)

"Novel in presenting squared distance"

@Jacobolus: In your latest rewrite attempt, you wrote into the lead that this book is "novel in presenting squared distance" as a method for simplified calculations, despite direct evidence I already showed you (a paper from 1982!) that this idea was long known and even standard in the computational geometry community, directly in the context of simplifying calculations.

There is an entire section about squared distance in the article Euclidean distance, detailing applications including computational geometry, statistics (much older), distance geometry, and optimization. It does not mention Wildberger, not out of any animus against his book, but because I literally could not think of any documented contribution he had made to that topic beyond making up a new neologism for it.

Can you clearly state what exactly you think is novel in Wildberger's use of squared distance? What does he say about it that is not already elsewhere in the literature, beyond a made-up name? Is it merely that he speculates on its pedagogical value for schoolchildren? —David Eppstein (talk) 22:21, 8 May 2023 (UTC)

Are you serious? I'm starting to think you've never looked at the book this article is about, but only a few negative reviews.

The paper you linked is a technical research paper about solving a tricky computational geometry problem via a tricky algorithm proven using a bunch of inequalities. It is not remotely comparable to a trigonometry textbook, nor does it adopt Wildberger's proposed concepts as fundamental. It doesn't present any method for solving triangles, it doesn't include any of the basic formulas from Wildberger's book, and it certainly doesn't propose eliminating the use of angle measure as a concept: indeed it explicitly defines the angle measure between vectors on page 728, and its formulas include arcsine, arccosine, and plentiful square roots. No layperson or high school student is even going to be able to make sense of it.

It uses the squared distance in the course of solving a problem (specifically, this single sentence: "In particular, $d_{2}({\tilde {x}},{\tilde {y}})$ may be replaced by ${\textstyle d_{2}({\tilde {x}},{\tilde {y}})^{2}=\sum (x_{i}-y_{i})^{2}}$ everywhere to produce a valid algorithm without square root operations.) but so do hundreds (thousands? more?) of other sources going back to antiquity.

Nobody ever claimed that mentioning the squared distance is a new idea: Elements is full of squared distances, and Wildberger himself credits most of the relevant formulas to Pythagoras, Euclid, and Archimedes, all of whom lived >2000 years ago. We might further mention Diophantus, Brahmagupta, Fermat, Descartes, Euler, Chasles, Cayley, Clifford, etc. etc. etc.

The book we are trying to describe here is clearly not a cutting edge pure mathematics research paper, nor does it pretend to be. It's a reformulation of a basic subject studied by teenage students worldwide. The novel part is the choice of nonstandard foundational representations while completely eliminating the standard representations from consideration.

I've skimmed dozens if not hundreds of elementary geometry and trigonometry textbooks in the course of researching other topics I am interested in, and for better or worse I have not ever seen anything else like Wildberger's book. (Indeed, the novelty here is precisely the problem most mathematicians have with this book!)

Your summary ("logically equivalent" etc.) is very misleading to readers, especially nontechnical readers who will almost certainly misinterpret that jargon. This article's goal should be to describe the content of the book rather passive aggressively slag Wildberger because he is separately controversial. –jacobolus (t) 23:58, 8 May 2023 (UTC)

You have gone on at great length, but failed to answer my question. What can we do with squared distances after reading Divine Proportions that we would not have thought to do with them previously? To put it another way, is there anything at all that can be included as content, in Euclidean distance § Squared Euclidean distance, sourced from Divine Proportions, other than the trivial issue of Wildberger's neologism for it?

If we want to present it as a significant advance, we have to be able to identify some way in which it was an advance. "I haven't seen anything like this" is not an advance, especially when you have seen things like that but are aggressively dismissing them because they come attached to other material describing actual advances in how to compute things. —David Eppstein (talk) 00:50, 9 May 2023 (UTC)

What you can do that you could not do previously is solve a triangle using a nonstandard assortment of formulas involving only rational arithmetic, or teach a high school student or carpenter to do the same (if you like, even in a finite field). If you like I guess you can also write nonstandard algebraic proofs of many basic results in Euclidean geometry, though that part is not all that interesting IMO.

The difference is a change of perspective rather than any fundamental change to the geometry. But changes of perspective have their own value.

"Aggressively dismissing"? The paper is extremely far removed from this book (and also, in the idea of using squared distances, not itself novel, but just another independent reflection of centuries-old ideas). I honestly don't see how you think it is notably "alike", to the point I am half convinced you are just trolling me.

I would guess if you really hunt in the past literature you could probably find something reasonably similar to most of the individual formulas in Wildberger's book, especially if you are willing to squint pretty hard at them.

(For example, I bet if you skim through thousands of old geometry and trigonometry papers from the 18th–20th century you can find somewhere the statement that

$d_{1}+d_{2}+d_{3}=0\implies (d_{1}^{2}+d_{2}^{2}+d_{3}^{2})^{2}=2{\bigl (}(d_{1}^{2})^{2}+(d_{2}^{2})^{2}+(d_{3}^{2})^{2}{\bigr )}$

which is what Wildberger calls the "triple quad formula" for "quadrances" between colinear points, and I bet you can find somewhere the statement that

${\begin{alignedat}{3}&\theta _{1}+\theta _{2}+\theta _{3}=\pi \implies \\&\quad (\sin ^{2}\theta _{1}+\sin ^{2}\theta _{2}+\sin ^{2}\theta _{3})^{2}={}&&2{\bigl (}(\sin ^{2}\theta _{1})^{2}+(\sin ^{2}\theta _{2})^{2}+(\sin ^{2}\theta _{3})^{2}{\bigr )}\\&&&+4\sin ^{2}\theta _{1}\,\sin ^{2}\theta _{2}\,\sin ^{2}\theta _{3}\end{alignedat}}$

which Wildberger calls the "triple spread formula" for the angles of a triangle. Though I don't know off-hand where to look for these.)

But I don't think you will find them collected together, written down directly in terms of squared distance and squared sine, or treated as a coherent system for solving basic trigonometry problems. –jacobolus (t) 02:08, 9 May 2023 (UTC)

You can "solve a triangle" in the sense that you can compute the quantities Wildberger describes in the way he describes it. Why you would want to compute those quantities instead of others is a different question. For instance, why should I prefer the square of the distance to the fourth power? Both are polynomials. Both are not really the distance but behave like it. Why should I call computing one of them "solving" anything?

You keep saying "teach a high school student" but teach them what? To plug numbers into formulas? How are they going to relate that to things they know intuitively like how far apart things are? This is the weakest aspect of the book because the book itself is useless for that level of teaching (as the reviewers explicitly say), and merely tells other people how they should teach without any testing of whether it might actually be a pedagogical improvement (as the reviewers explicitly say).

Re The paper is extremely far removed from this book: The paper is part of a big literature on geometric computing, specific only in its demonstration that this use of this idea was well known. The book is on geometric computing, but ignores the entire history of geometric computing.

Where to look for the things you describe is the general area of algebraic elimination theory, Gröbner basis theory, and the like, for general methods for turning formulas involving roots into formulas involving only polynomials. You will have to look hard for the specific formulas you mention because that area is not specific to distances and angles and might only give those formulas incidentally as examples. —David Eppstein (talk) 03:29, 9 May 2023 (UTC)

Whether the specific computations Wildberger recommends are more useful, meaningful, convenient or easier to teach or learn compared to the standard computations is something that reviewers seem to be of mixed opinion about: some are cautiously optimistic, while others think any simplification in one place is likely to pop back up as extra complexity somewhere else, or think that any marginal benefits are outweighed by significant switching costs. But that's an entirely different question from whether the approach is novel. I think it clearly is, even though both squared distance and squared sine are clearly far from new. Novel is also not the same as "some kind of breakthrough".

Why would you want to know the squared distances and squared sines rather than some other related quantity? My answer would be that in general you probably wouldn't. Note that I am not personally advocating using this as a practical system or teaching it to classrooms full of ordinary students. Then again, you could certainly say the same about angle measures. If angle measures were not already the primary abstraction used to represent relative orientations, rotations, points on the sphere, etc., I would advocate against adopting them as a standard format in most contexts. Angle measures are very obnoxious to deal with and switching to vector methods saves a lot of trouble.

If you asked Wildberger my guess is that he would start polemically complaining about claimed logical contradictions at the foundation of the real numbers, but if you kept asking beyond that he might say this lets you do your calculations straight-forwardly with pen and paper with no need for mathematical tools beyond the quadratic formula. If you kept probing further he might point out that if you start with points and lines in the Euclidean coordinate space with coordinates in some field, then the resulting "quadrances" and "spreads" will all be elements of the same field.

For more on this point, see:

Yiu, Paul (2001), "Heronian Triangles Are Lattice Triangles", American Mathematical Monthly, 108 (3): 261–263, doi:10.1080/00029890.2001.11919751

This feature might hypothetically be of interest to someone like the author of the VZome software program who does all of his calculations in the field of rational numbers extended by the golden ratio. It has been of some research interest to a few other mathematicians working with finite fields. –jacobolus (t) 04:11, 9 May 2023 (UTC)

"Pythagoras's theorem proof (rational trigonometry)" listed at Redirects for discussion

The redirect Pythagoras's theorem proof (rational trigonometry) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 May 9 § Pythagoras's theorem proof (rational trigonometry) until a consensus is reached. Jay 💬 05:00, 9 May 2023 (UTC)

"Pythagorean theorem proof (rational trigonometry)" listed at Redirects for discussion

The redirect Pythagorean theorem proof (rational trigonometry) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 May 9 § Pythagorean theorem proof (rational trigonometry) until a consensus is reached. Jay 💬 05:01, 9 May 2023 (UTC)