Divine Proportions is not a textbook designed for a wide audience. At the most basic level Wildberger replaces the concepts of distance and ordinary angle measure with quantities he calls quadrance and spread :. However, since quadrance and spread are the basic quantities in the book, they are not defined from distance or from ordinary angle measure. And they certainly do not rely for their definition on transcendental functions such as sine or cosine.
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Divine Proportions is not a textbook designed for a wide audience. At the most basic level Wildberger replaces the concepts of distance and ordinary angle measure with quantities he calls quadrance and spread :. However, since quadrance and spread are the basic quantities in the book, they are not defined from distance or from ordinary angle measure.
And they certainly do not rely for their definition on transcendental functions such as sine or cosine. Instead, Wildberger assumes that his points are ordered pairs of numbers [ x , y ].
The primary advantages of quadrance and spread over distance and ordinary angle measure is that they are quadratic functions of the coefficients, and they do not involve square roots or transcendental functions. Hence, in particular, these quantities can be defined when the numbers under consideration are from fields more general than the fields of real or rational numbers. This generalization leads to what Wildberger labels as Universal Geometry.
Wildberger then proceeds to develop the basic and several advanced topics in Euclidean geometry and Universal Geometry with no significant reference to distance and absolutely no reference to ordinary angle measure.
Because of the quadratic nature of quadrance and spread this development is done with elementary though often messy arithmetic and algebra.
This does not lead to new results, but to alternate formulations of classical results. Given that the book is intended for a mathematically mature audience, the development of the material, the calculations, and the accompanying discussions are well organized, clear, and well-written.
I found no flaws in the mathematics although I did not read every argument in depth. The book presents an interesting and creative approach to the development of trigonometry and geometry. This could provide fertile ground for further investigation.
The intuitive geometric clarity of distance and angle measure does not carry over to quadrance and spread. Moreover, additivity for distance between collinear points and for angle measure between adjacent angles is natural, expected, and useful in the real world; this property does not hold for quadrance and spread.
There have to be extremely compelling reasons to replace the intuitively appealing, additive geometric quantities of distance and angle measure with less-intuitive, non-additive quantities. As one justification for this replacement Wildberger claims that computations with quadrance and spread are simpler than with distance and angle measure.
However, the increased simplicity comes not so much from reducing the number of computational steps needed to solve a problem indeed, the necessary algebraic steps can be extensive in rational trigonometry but from changing the nature of the computations: reliance on quadratic algebra and the elimination of transcendental functions. But would these changes make mastery of trigonometry and geometry easier for our students? The introduction of the sine and cosine is difficult for many, but not because these functions are transcendental but because they are unfamiliar and require the use of many sophisticated identities.
But the spread of two lines is also unfamiliar, and many of the rational trigonometry formulas are as complicated as their classical counterparts. Divine Proportions is unquestionably a valuable addition to the mathematics literature. It carefully develops a thought provoking, clever, and useful alternate approach to trigonometry and Euclidean geometry.
It would not be surprising if some of its methods ultimately seep into the standard development of these subjects. However, unless there is an unexpected shift in the accepted views of the foundations of mathematics, there is not a strong case for rational trigonometry to replace the classical theory. He received his Ph. Sigurdur Helgason in analysis on Lie groups.
A second volume is currently under development. He can be reached at barker bowdoin. Skip to main content. Search form Search. Login Join Give Shops. Halmos - Lester R. Ford Awards Merten M. Publication Date:. Number of Pages:. Log in to post comments.
Divine Proportions: Rational Trigonometry to Universal Geometry
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Divine Proportions : Rational Trigonometry to Universal Geometry
Rational trigonometry is a proposed reformulation of metrical planar and solid geometries which includes trigonometry by Canadian mathematician Norman J. Wildberger, currently a professor of mathematics at the University of New South Wales. Rational trigonometry avoids direct use of transcendental functions like sine and cosine by substituting their squared equivalents. Rational trigonometry follows an approach built on the methods of linear algebra to the topics of elementary high school level geometry. Distance is replaced with its squared value quadrance and ' angle ' is replaced with the squared value of the usual sine ratio spread associated to either angle between two lines.
Divine Proportions: Rational Trigonometry To Universal Geometry