Divine Proportions 192
Divine Proportions - Rational Trigonometry to Universal Geometry | |
author | Norman John Wildberger |
pages | 300 |
publisher | Wild Egg Pty Ltd |
rating | 2 |
reviewer | David Halprin |
ISBN | |
summary | Wilberger presents an ultimately disappointing vision of a new descriptive system for geometry. |
There are various ways to approach Norman's so-called "Rational Trigonometry" and/or "Universal Geometry." I have examined it from various perspectives and it does not live up to Norman's claims, whichever standpoint, that I have taken.
DEFINITIONS
Firstly, the definitions, given in the Introduction:-
quadrance = (distance)2 = (x2 2 - x1 2) + (y2 2 - y1 2)
spread = (sin(angle))2 = sin2A
N.B.When one has an equation to solve, (say it is a quadratic), one expects two solutions and deals with them accordingly. If, however, in order to solve an equation, that has a square root sign within it, then one has to square both sides of the equation at some time and this doubles the number of solutions. These extra solutions are regarded as inadmissible, despite their potential interest and possible geometric interpretation. (See worked example later.)
Here is a point of view which suffices to reject this book on its own merit, whether or not there are any other objections, although many other readers will already know of many other disapprovals to mine.
Let's consider someone proposing new variables in some geometric enterprise. This happened in Plane Geometry (for instance), post Descartes, when some bright sparks came up with Polar Coordinates, Pedal Coordinates, Contrapedal Coordinates, Bipolar Coordinates, Parabolic Coordinates, Elliptic Coordinates, Tangential Polar Coordinates, Cesaro Intrinsic Coordinates, Whewell Intrinsic Coordinates and Euler Intrinsic Coordinates, etc.
There are three essential steps to any such proposal:
- The defining of these coordinates — either in words, with a geometrical description, or in clear mathematical symbology.
- The relationship of these new coordinates with some other planar coordinate system. This amounts to a mathematical statement of a coordinate transformation. (e.g. From Cartesian to Polar and/or Polar to Cartesian.) Once this is so done, then one can transform any previously-found equations to the new symbology, and hence arrive at a new taxonomy for plane curves, or a new way of stating the conditions for two lines to be parallel, perpendicular or concurrent, or for points to be collinear or not, etc.
- The demonstration how this new system can be a better system for certain types of problems, perhaps with some limitations in special cases, but not denying their right to be subsumed into mathematical texts, curricula, etc.(e.g. Curve-sketching made easier for plane curves, which are expressed in the new coordinate system, if it is to be preferred in selected examples.) Other pre-existing coordinate systems have shortcuts to finding such things as asymptotes, cusps, asymptotic circles, poles, points of inflection, maxima and minima etc., so the reader would expect to see similar findings by Wildberger.
This third step, in my humble opinion, is where Norman comes undone, and then some!
viz.1) Wildberger cites many plane curves and their concomitant equations in his new coordinate system, in Appendix A, (pages 279-286), but his diagrams have been drawn using software that is dependent on standard polar equations, which are then converted by the software to Cartesian form for plotting. In no way is his "Rational Polar Equation" suitable for being implemented by the software employed. Certainly, any programmer worth his salt could devise a not-so-easy and/or complicated routine to transform Rational Polar Equations back to the regular form, but that is no pat-on-the-back for Wildberger, rather it shows the counter-intuitive and flawed reason for using that coordinate framework.
viz.2) Wildberger's five laws are merely standard trigonometrical identities disguised by his new symbology, showing no advantage over the original forms. See table in Appendix.
He cites a triangle problem in his first chapter on page 14. He then gives a so-called "Classical Solution" in 5 equation lines, using a trig. table via a calculator, for part of this method.
Then, in the next page, he gives his so-called "Rational Solution", which requires three diagrams and 8 or 9 equation lines, and this is a flawed solution, to which he seems oblivious, and does not own to it therefore.Anyone with a modicum of mathematical sense, who tackles this triangle problem, knows the following:-
The usual properties of arithmetic with respect to commutativity, associativity and distributivity also apply equally to common algebra.
When one has an equation to solve (say it is a quadratic), one expects two solutions and deals with them accordingly. However, in order to solve an equation that has a square root sign within it, one has to square both sides of the equation at some time, and this doubles the number of solutions. These extra solutions are regarded as inadmissible, despite their potential interest and possible geometric interpretation.
Viz. The worked example for the rational method for the triangle on page 15 accepts the inadmissible solution as though it is acceptable, whereas the better solution method is the classical method used properly, without recourse to trig tables, and in only four equation lines.
PROBLEM
A triangle ABC has sides a = 5, b = 4 and c = 6.
A st. line from C to AB, (length d), cuts AB at D,
where angle BCD = 45 degrees. What is the length d = CD?
MY SOLUTION
cos B = 3/4 sin B = 7/4, BDC = 180 - (45 + B)
sin BDC = sin (45 + B) = sin 45.cos B + cos 45.sin B
sin(45 + B) = (3/4 + 7/4)/2 = (3 + 7)/(42)
d = 5 sin B/sin BDC = 57/4 x (42)/(3 + 7) x (3 - 7)/(3 - 7)
= 52(37 - 7)/2 = 3.313693059
So, in this first instance, Rational Geometry does NOT provide anything worthwhile, contrary to Norman's hype.
In chapter two, Norman introduces a dissertation on Fields, as though this is an important factor for understanding and using Rational Geometry, despite the fact that up to a student's age of 17, schools don't find it necessary to introduce into his/her brain any Field lessons together with geometry and trigonometry.
Don't forget that his advocacy is to replace classical geometry and trigonometry, (especially lines and angles), at school level. He doesn't suggest retaining it and using his methods as a adjunct and/or complement, especially since some of those guys and gals will become architects, surveyors etc. etc.
Were the academic institutions which set college and university curricula, to take Wildberger at his word, by eliminating regular trigonometry and geometry and replacing it with his concepts, it would be the downfall of current mathematical knowledge and standards for years to come. What's more, the damage would take years from which to recover; an almost irreparable predicament in education.
c.f. Cuisenaire of yesteryear.
However, you don't have to read between the lines to see on page 21 that Wildberger excludes 'characteristic two fields.' Although I am not versed in Field Theory, I opine that such an exclusion does not apply to classical geometry and/or trigonometry, otherwise he would have said so. So, he is already implicitly confessing, to a failure of Rational Geometry in the global sense.
I have to confess that I look upon his sojourn into Field Theory as a diversion in the same sense that a prestidigitator (magician), in his field of legerdemain (sleight of hand), distracts the audience members, thereby lessening their attention on what's really going on.
Wildberger then goes into proportions using the a:b = c:d symbology, as though it has more merit than the usual a/b = c/d, like we have in the Sine Rule, say. Warum? Wherefore?
On page 9, he states, without proof, the equation for the spread between two lines. From standard trig, one can easily calculate the angle between two lines, and when one squares the sine of that angle one has his equation without recourse to rational geometry. Now if one subtracts this expression from 1, one obtains the square of the cosine of the angle between these two lines. Naturally if one starts with these two terms and adds them one can see why they sum to unity, which he states on page 27 as Fibonacci's Identity.
A rose by any other name is still a rose, I believe; Pythagarose?
Then Wildberger presents variants of this, all of which are obtained with simple college algebra and are further diversions. Then he waffles on about the possibility of a denominator being zero and its implications. WOW.
(See table in Appendix).
Then, we have linear equations and their solutions using determinants as though it is a revelation. WOW WOW!
At this point, why not reinvent the wheel?
Remember, this book is not aimed at secondary students; such a lower level of presentation is promised in an intended future publication. So, why does he tell us `cognoscenti' so much that, obviously, we would know before picking up his book?
Is he just filling up the pages, due to lack of the Step 3 material, so we are drooling to obtain an implied revelation or other especially informative disclosure?
N.B. We mustn't hold our breath, so as to avoid cyanosis!
So now, on page 31, we have Polynomial Functions and Zeros. Wildberger examines an example in F19, but does not explain why on earth that has any significance in curve sketching. After all, we expect our graph to be plottable in a Cartesian Framework in the usual field of numbers, which we, and our computer plotting software, always use by default.
Page 32 teaches us how to solve quadratic equations by completing the square. This is so deep, that I hope the reader's gray matter can cope, especially since he/she is, presumably, at tertiary level!
Now to chapter 3 starting on page 35: Cartesian Coordinate geometry. On page 40, he makes a special reference to the conditions of perpendicularity of two lines. This is easily calculated since the product of their gradients must be -1. However, he stresses "that this is the single most important definition in all geometry, it colours the entire subject." Then he follows this up by naming this "blue geometry."
So mind-boggling WOW WOW WOW! He then promises that other colours will appear. I can hardly wait. I hope the new colours match the colour scheme in my study.
Summarily, there has been nothing from Step 3 to illustrate a finding in Rational Geometry, that gives it an edge, at least. He is just making statements, that are already well-known in geometry and trigonometry, and he is an associate professor in mathematics, who should be able to do a lot better than that. I opine that he doffed his professorial hat and replaced it with a dunce's hat in order to write such pretentious garbage.
One must address one's audience, or write to one's intended readership, at a consistently-appropriate level. In matters of a so-called "New Mathematics," he must demonstrate actual advantages, and not attempt to hoodwink us, as he did in the earlier problem on Pg.14 and its badly worked out, so-called "Classical Solution".
If one searches the web, there appears to be no academic interest in "Rational Geometry" by the diasporic mathematical fraternity.
Especially, I had hoped to find that his fellow mathematicians at UNSW would have had something worthwhile to say, and thereby prove me to be an innumerate imbecile for daring to criticise "Divine Proportions."
Alas and alack, niente, gar nichts, zilch. Woe is me. Es tut mit leid.
CONCLUSION
In its present format, a better title would be:-
"LE GRAND PURPORTISSIMENT"
This book, overall, is a misrepresentation of the facts. It purports to be what it is not. The promotional literature on the author's web site is descriptive, but more of the author's dream for a mathematical breakthrough than an actual innovation.
If finances were no concern, I would suggest a complete re-presentation of all his original findings under a new title, that states, in effect, that this is a new coordinate framework, that, from time to time, has occasional advantage over the Cartesian Coordinate system, comparable to the other planar frameworks, stated on the first page of this review.
So mote it be. Amen.
APPENDIX
|
RATIONAL TRIGONOMETRY LAWS |
ANALOGOUS LAWS IN TRIGONOMETRY |
1. |
Triple Quad Formula for collinearity of three points |
Triangular area degenerated to zero. |
2. |
Pythagoras' Theorem for right triangles |
Pythagoras' Theorem |
3. |
Spread Law for any triangle |
Sine Rule |
4. |
Cross law for any triangle |
Cosine Rule |
5. |
Triple Spread Formula for any triangle (Quadrea) |
16 x (Area)2 |
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geesh (Score:5, Insightful)
"I have to confess that I look upon his sojourn into Field Theory as a diversion in the same sense that a prestidigitator (magician), in his field of legerdemain (sleight of hand), distracts the audience members, thereby lessening their attention on what's really going on."
yes, thanks for providing an explanation for your $10 college words, otherwise we plebs might not have understood you.
Also, what's up with the German and French from out of nowhere? I'm all for using them when there is no easy english equivalent, but what the hell, "Alas and alack, niente, gar nichts, zilch. Woe is me. Es tut mit leid." Those are just extra words.
Re:Too bad (Score:5, Insightful)
It also isn't science.
My idiosyncratic take (Score:5, Insightful)
El Sucko (Score:5, Insightful)
I have an M.A. in Mathematics. I've read some of the "Rational Trigonometry" online before, and yes, it is pretty oddball and has its weakness and can be criticized.
But this review is borederline psychotic. It is poorly written, full of ad hominem attacks, lots of made-up grammar and word usage, wierd random abbreviations... it's scatterbrained, repetitive, and unnecessarily hostile.
There is a critical review to be written about "Rational Trigonometry", but this isn't it. I may not like our current government, but I'm still not going to listen to some incoherent homeless guy raving about it on the street.
Re:geesh (Score:5, Insightful)
Frankly they both bored the shit out of me after about 5 seconds. Why is it that math is always rendered this way? I've met interesting and articulate mathematicians before, so I know they exist...Are they not allowed to write textbooks? Or at least write reviews about textbooks?
I was pushed into a near-hatred of math by hordes of pretentious math prodigys that had zero use for any student who didn't start off with what they felt was obvious knowledge. The text book talks down to you, the professor talks down to you, and god forbid you ask for a practical example!
I'm not a math genius, but I'm damn good at practical math. The only way I managed to pass calculus the first time was because I happened to be taking it at the same time as a physics course, and I could figure it out where I could see an application in physics. For calc II I shopped around, trying to find a decent book with dismal results. Ended up dropping the class, and shopping for a decent professor the next semester.
Math is cool, but goddamn, the way it's taught is awful and jackasses like this reviewer and the joker who wrote the book he's reviewing are a prime reason why.
Re:geesh (Score:3, Insightful)
The symptoms you describe exist in every field, from math to literary critisism to welding to surfing.
A New Kind of Science (was Re:Too bad) (Score:4, Insightful)
Wolfram performs an over-analysis of a very narrow subset of cellular automata while claiming to have invented the field, that 'mainstream science' refuses to look at this incredible discovery, and that his 'new kind of science' based on recursion and cellular automata will change the world, although he has no idea how.
It reads like something written after reading Godel, Escher, Bach, smoking pot, and thinking, "I'm thinking about thinking. Now I'm thinking about thinking about thinking. Now I'm....whoa, I wonder what that looks like on graph paper?"
From the reviewer's not-so-clear description, it appears this book falls into a similar category.
OK, stepping way out on a limb here... (Score:3, Insightful)
A few up-front things:
IANAMathematician;
I appreciate the reviewer's efforts to thoroughly discuss the reviewer's point of view;
I don't mind acknowledging that I'm not as smart as the vast population of Slashdot, but I like math even though I'm not top-notch;
I love to learn stuff, and like to read Slashdot articles/comments that are out of my field, and way over my head;
With the above said...
I don't mind looking up unfamiliar terms that appear in an article or in a review (I like learning) - when the words are concerned with the subject matter at hand. I do mind when I read something that attempts to completely fill up my "new word of the day" calendar (for the next millennium). Why? Because I'm interested in understanding the subject and the review, not in how many new non-topic-related words and phrases that can be crammed into a paragraph.
Lastly, a good review, IMVHO, is one that does not chastise, scold, or belittle the matter of review.
Re:geesh (Score:4, Insightful)
Pretentious, yes, but not Geek. Geeks strive for well-defined, unambiguous terms, rational organization of subject matter, and language that accomplishes exactly as much as is necessary, and no more. Geek writing is efficient.
The OP's analysis is excellent, but frought with writing that goes beyond pretentious. It's just bad. Disorganized, rambling, semi-coherent and full of useless jumbles of letters that communicate nothing.
How about a more qualified reviewer? (Score:4, Insightful)
Wildberger may be a little "out there" (alright, he's completely nuts), but this point is not one you can fault him for. There are a LOT of results which exclude fields of characteristic two. It's not a big deal. In fact, it's commendable that Wildberger has explored the ramifications of his framework in any fields with non-zero characteristic, as the "normal" pedestrian conceptualizations of geometry don't apply.
It would have been nice if /. could have posted a review by somebody who is actually qualified to critique the book. And no, I am not such a person, but I know a couple people who are.
Re:El Sucko (Score:3, Insightful)
Not to mention imprecise. In two instances the reviewer says
in order to solve an equation that has a square root sign within it, one has to square both sides of the equation at some time, and this doubles the number of solutions.
which is not true in all cases. Two examples are
\sqrt(x) = x, which has two solutions before and after squaring both sides, and
\sqrt(x)=10, which has one solution before and after.
Re:geesh (Score:3, Insightful)
.
If you want to see how a REAL scientist writes, without sound pretentious, but yet writing clearly without unnecessary obfuscation, check out anything by Richard Feynmann for instance.
Re:geesh (Score:4, Insightful)
As to your prior experiences, articles like these are part of the reason why mathematicians are distrustful of people that don't find a way to prove themselves. It's an easy field to claim that you've come up with a result, and sometimes it can be a very technical logical fallacy that defeats your efforts. I just wasted a half hour of my time looking up this guy's name for any signs of credibility and reading through the comments.
In experimental fields, even if someone isn't very good, at least they can be used as a technician or research goon. In math, if you're not bright enough to come up with results, you're a non-starter. I know an undergrad who spent four years struggling through basic undergrad classes with the goal of grad school, and then got to his senior year and none of them would take him. It would almost have been a service if someone had been more blunt earlier on.
Of course, I'm not really talking about the calculus sequence, linear algebra I, that kind of thing. Those are more for engineers and scientists. But there you have to bear in mind that to math majors it's the equivalent of Humanities_Course 101, and I dunno about you, but I've taken my share of shitty-ass 101 courses. It's usually because it's foisted off on the newest professor that can't get out of it, they in turn foist off a lot of the work on the TAs, and it's not interesting for anyone's research. It's not a great situation, but then again there are exceptions. I went to a small, teaching-focused school, and my math professors were very personable and great teachers. They loved student research because they got so few who were motivated. I spent some time at a research school, and had a lot more opportunities, but the professors were a lot less accessible and not as good at teaching. It's a trade-off and something worth thinking about before you settle on a school.
Re:A very odd mathematician (Score:3, Insightful)
I try to keep abreast of the current absolutely correct, final theories of everything.
Maths As A Science (Score:3, Insightful)
Because there are two types of mathematics practiced in the world today. Mathematics that follows the scientific method, and mathematics that does not follow the scientific method. The latter is regarded as a more laudable endevour.
Mathematics that follows the scientific method is the kind most geeks are familiar with, and which most engineers and physicists use. Under this type, basic properties are defined from the ground up, with examples, and theorems and proofs are given more concrete relations to basic numbers and geometry. In this reigieme, mathematics is, like the other sciences, an exploration, examination and classification of the universe, albiet in the case of mathematics a more abstract portion of the universe. Here mathematics is by default falsifable, as all our properties and theorems can be subjected to direct experiment by means of calculation of basic numbers and geometric measurements.
Mathematics that does not follow the scientific method is somewhat different. Instead of exploring the properties of basic numbers and geometry, proponents of this method instead propose structures that may or may not exist, defining them through axioms and other definitions. Examples are few and far between as the objects in question may or may not exist "in the real world", and even if they do exist, any concrete example would neccesarily restrict itself to only one minute subset of all possible manifestations of the object.
Here, mathematics is not falsifiable, as experiments to test the validity of properties are pointless, because the axioms restrict the objects we consider to only those with certain properties. Experiments to test the validity of theorems are also largely impossible or unfeasable, as most of the objects under consideration have never been constructed or explored, and indeed there is no guarantee that anyone can ever be able to construct them. In general, falsifiabilty is only really guaranteed when mathematics can be ultimately reduced to basic elements which we candirectly observe and manipulate, such as real numbers, finite sets, etc. Much of modern mathematics is not confined to this domain.
A lot of mathematicians would be in serious disagreement with me here. They would insist that their theorems are falisfiable, or even object that falsifiability is a nonsense concept in mathematics as everything is by definition true. I remain unconvinced of the validity of such world views, especially in the realm of science.
As someone who has read a lot of advanced mathematics, I can safely say that the standard of proof in modern mathematics is now very low. Most modern proofs essentially amount to proof by intimidation which most if not all readers must simply accept as an axiom. I recall recent stories [seedmagazine.com] about the "uncertainty" in many modern mathematical proofs. Apparently, the proofs were "unverifiable" by the academic referres assigned to validate them. To me, it sounded like the authors hadn't actually "proved" anything at all. But such is the state of modern mathematics.
I'd like to think that what I do is science. I really would. I endevour to make my proofs clear and above all repeatable, but I'm really just fighting the tide. Most advanced mathematics is a kind of pseudoscience. Undeservedly so, but that's the way it is.
Ramblings of a madman. (Score:3, Insightful)
Sigh... I'm irritated by people who think that their large vocabularies make them good communicators.
Re:I'm not that Smart! (Score:3, Insightful)
Re:I'm not that Smart! (Score:3, Insightful)
As an extremely simple example, (A union B) intersect C = (A intersect C) union (B intersect C) because [skipping trival steps] (A union B) intersect C is the set of all x such that x is in C (commutative) and (definition of intersection) x is in either A or B (definition of union), and (A intersect C) union (B intersect C) is the set of all x such that x is in C (distributive) and x is in either A or B (definition of union). Since these two formulae are identical, the sets are identical. This sometimes leads to very long notations of sets, but essentially makes solving a lot of complex proofs as simple as flipping things around to take the same form and applying that to the problem. Indirect by contradiction is done when you can subtract one set from the other and show that something remains, and inductive is done by applying the defining formula (say elements n and n + 1) to the definition of the problem set to show that their inclusion in the problem set is equivalent (which is essentially the same as a traditional inductive proof, induction is not the strong point of this method, but it CAN be done).