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Divine Proportions 192

David Halprin writes with a review of a new (and mighty odd sounding) mathematics book: "In my humble opinion, we have an unjustified polemic in the world of mathematics, yet again. My background is tertiary level mathematics and concomitant research in specialised areas, so when a friend e-mailed me the link to this book, I was so excited after reading the author's hype, that I ordered a pre-publication copy. My expectations have not been met, unfortunately, hence my analysis precipitated this review." Read on for Halprin's idiosyncractic take on Norman John Wildberger's Divine Proportions: Rational Trigonometry to Universal Geometry.
Divine Proportions - Rational Trigonometry to Universal Geometry
author Norman John Wildberger
pages 300
publisher Wild Egg Pty Ltd
rating 2
reviewer David Halprin
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.


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:

  1. The defining of these coordinates — either in words, with a geometrical description, or in clear mathematical symbology.
  2. 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.
  3. 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.


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?


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.


In its present format, a better title would be:-


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.





Triple Quad Formula for collinearity of three points

Triangular area degenerated to zero.


Pythagoras' Theorem for right triangles

Pythagoras' Theorem


Spread Law for any triangle

Sine Rule


Cross law for any triangle

Cosine Rule


Triple Spread Formula for any triangle (Quadrea)

16 x (Area)2

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Divine Proportions

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