# Divine Proportions: Rational Trigonometry to Universal Geometry

###### N. J. Wildberger
Publisher:
Wild Egg
Publication Date:
2005
Number of Pages:
300
Format:
Hardcover
Price:
0.00
ISBN:
0-9757492-0-X
[Reviewed by
William Barker
, on
07/6/2008
]

Divine Proportions is not a textbook designed for a wide audience. It’s more of a proof-of-concept, the development of the author’s unconventional approach to trigonometry and Euclidean geometry “intended for a mathematically mature audience.” However, N. J. Wildberger believes that his book “introduces a remarkable new approach to trigonometry and Euclidean geometry, with dramatic implications for mathematics teaching, industrial applications and the directions of mathematical research in geometry. […] Teachers and students will benefit from a simpler and cleaner theory… This book… provides the mathematical foundation for a dynamic and elegant new approach to teaching trigonometry and geometry.”

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]. Then the quadrance between two points A1 = [x1,y1] and A2 = [x2,y2] is defined to be

Q(A1,A2) = (x1x2)2 + (y1y2)2.

Similarly, given two lines l1 and l2  specified by equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 , then the spread between the two lines is defined to be

.

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. In particular, Wildberger labels the “five main laws of rational trigonometry” as the

4. cross law,

When reformulated using classical trigonometric notions, these laws correspond to the

1. additive formula for distance between three collinear points,
2. Pythagoras’ theorem (for distances),
3. Law of Sines,
4. Law of Cosines,
5. sum of angles of a triangle equals 180°.

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. I found Wildberger’s Universal Geometry, especially when applied to finite fields, particularly novel. This could provide fertile ground for further investigation.

However, I have reservations about the superiority of Wildberger’s methods over the classical theory. He believes his methods constitute a “simpler and more logical approach to trigonometry and geometry,” and should ultimately replace the classical approach in schools and colleges as the primary development of these subjects. I’m not convinced of the wisdom of attempting such a change.

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.

Individual evaluations of the importance of Wildberger’s new approach to trigonometry and geometry will probably depend on one’s viewpoints on the foundations of mathematics. Certainly Wildberger’s views on foundational issues have strongly motivated and influenced the directions taken in Divine Proportions. Wildberger believes there are many logical difficulties with “infinite set theory” — in his paper “Set Theory: Should You Believe?” Wildberger states “It is not clear that there are any sets that are not finite…” This motivates some rather non-standard definitions in Divine Proportions. For example, a line is not a set of points; it is a “3-proportion” with individual points either “on” or not “on” the line.

Wildberger’s dislike of “infinite set theory” leads to equally critical views concerning analysis and, in particular, the real number system. Consequently, as stated in the Introduction to Divine Proportions, his “new theory unites the three core areas of mathematics — geometry, number theory and algebra — and expels analysis and infinite processes from the foundations of the subject.” Anyone sharing Wildberger’s unconventional views will hail “rational trigonometry” for its lack of dependence on the field of real numbers and its avoidance of transcendental functions. However, if you do not share these critical views of analysis — I do not — then the claim of superiority of “rational trigonometry” over classical trigonometry loses much of its attraction.

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.

William Barker is the Isaac Henry Wing Professor of Mathematics at Bowdoin College. He received his Ph.D. at M.I.T. in 1973, writing a thesis under the guidance of Prof. Sigurdur Helgason in analysis on Lie groups. Barker was subsequently a John Wesley Young Research Instructor at Dartmouth College for two years, joining the Bowdoin faculty in 1975. His most recent work has been an undergraduate geometry textbook, Continuous Symmetry: From Euclid to Klein, co-authored with Roger Howe of Yale University. A second volume is currently under development. Barker has also been active with the MAA, taking part in the production of CRAFTY’s Curriculum Foundations Project and the CUPM’s Curriculum Guide 2004. He can be reached at barker@bowdoin.edu.

Intro (Divine Proportions) xv

I Preliminaries 1

1 Overview (Divine Proportions) 3

1.1 Introducing quadrance and spread . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Laws of rational trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Why classical trigonometry is hard . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Why rational trigonometry is easier . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Comparison example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Ancient Greek triumphs and difficulties . . . . . . . . . . . . . . . . . . . 17

1.7 Modern ambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Background 21

2.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Identities and determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Polynomial functions and zeroes . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Cartesian coordinate geometry 35

3.1 Points and lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Collinear points and concurrent lines . . . . . . . . . . . . . . . . . . . . . 38

3.3 Parallel and perpendicular lines . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Parallels and altitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Sides, vertices and triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.7 Affine combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.8 Perpendicular bisectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Reflections 51

4.1 Affine transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Lineations and reflection sequences . . . . . . . . . . . . . . . . . . . . . . 53

ix

x CONTENTS

II Rational trigonometry 57

5.1 Quadrances of triangles and quadrilaterals . . . . . . . . . . . . . . . . . . 59

5.2 Triple quad formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3 Pythagoras’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.4 Quadrance to a line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.5 Quadrea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.6 Archimedes’ formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.7 Quadruple quad formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.1 Spreads of triangles and quadrilaterals . . . . . . . . . . . . . . . . . . . . 73

6.2 Cross . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3 Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4 Ratio theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.5 Complementary spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.6 Spread law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.7 Cross law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.8 Spreads in coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.9 Vertex bisectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.1 Triple spread formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.2 Triple cross formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.3 Triple twist formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.4 Equal spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.5 Spread reflection theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.6 Examples using different fields . . . . . . . . . . . . . . . . . . . . . . . . 96

7.7 Quadruple spread formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8.1 Combining equal spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8.2 Spread polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.4 Explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.6 Composition of spread polynomials . . . . . . . . . . . . . . . . . . . . . . 110

8.7 Cross polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

9 Oriented triangles and turns 113

9.1 Oriented sides, vertices and triangles . . . . . . . . . . . . . . . . . . . . 113

9.2 Turns of oriented vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

9.3 Signed areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

CONTENTS xi

III Universal Geometry 119

10 Triangles 121

10.1 Isosceles triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

10.2 Equilateral triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

10.3 Right triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

10.4 Congruent and similar triangles . . . . . . . . . . . . . . . . . . . . . . . . 127

10.5 Solving triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

11 Laws of proportion 131

11.1 Triangle proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

11.2 Quadrilateral proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

11.3 Two struts theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

11.4 Stewart’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

11.5 Median quadrance and spread . . . . . . . . . . . . . . . . . . . . . . . . . 137

11.6 Menelaus’ and Ceva’s theorems . . . . . . . . . . . . . . . . . . . . . . . . 139

12 Centers of triangles 143

12.1 Perpendicular bisectors and circumcenters . . . . . . . . . . . . . . . . . . 143

12.2 Formulas for the circumcenter . . . . . . . . . . . . . . . . . . . . . . . . . 146

12.3 Altitudes and orthocenter . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

12.4 Formulas for the orthocenter . . . . . . . . . . . . . . . . . . . . . . . . . 149

12.5 Incenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

13 Isometries 155

13.1 Translations, rotations, reflections . . . . . . . . . . . . . . . . . . . . . . 155

13.2 Classifying isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

14 Regular stars and polygons 159

14.1 Regular stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

14.2 Order three stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

14.3 Order five stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

14.4 Order seven stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

14.5 Regular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

15 Conics 167

15.1 Centers of conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

15.2 Circles and ribbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

15.3 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

15.4 Quadrolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

15.5 Grammolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

16 Geometry of circles 177

16.1 Diameters and chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

16.2 Spreads in a circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

16.3 Parametrizing circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

xii CONTENTS

17.1 Cyclic quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

17.2 Circumquadrance formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

17.3 Cyclic quadrilateral quadrea . . . . . . . . . . . . . . . . . . . . . . . . . . 187

17.4 Ptolemy’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

17.5 Four point relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

18 Euler line and nine point circle 193

18.1 Euler line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

18.2 Nine point circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

19 Tangent lines and tangent conics 197

19.1 Translates and Taylor conics . . . . . . . . . . . . . . . . . . . . . . . . . . 197

19.2 Tangent lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

19.3 Higher order curves and tangents . . . . . . . . . . . . . . . . . . . . . . . 202

19.4 Foliumof Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

19.5 Lemniscate of Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

IV Applications 211

20.1 Spread ruler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

20.2 Line segments, rays and sectors . . . . . . . . . . . . . . . . . . . . . . . . 214

20.3 Acute and obtuse sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

20.4 Acute and obtuse triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 218

20.5 Triangle spread rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

21 Two dimensional problems 221

21.1 Harmonic relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

21.2 Overlapping triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

21.3 Eyeball theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

21.4 Quadrilateral problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

22 Three dimensional problems 227

22.1 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

22.2 Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

22.3 Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

22.4 Wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

22.5 Three dimensional Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . 232

22.6 Pagoda and seven-fold symmetry . . . . . . . . . . . . . . . . . . . . . . . 233

23 Physics applications 235

23.1 Projectilemotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

23.2 Algebraic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

23.3 Snell’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

23.4 Lorentzian addition of velocities . . . . . . . . . . . . . . . . . . . . . . . . 242

CONTENTS xiii

24 Surveying 243

24.1 Height of object with vertical face . . . . . . . . . . . . . . . . . . . . . . 243

24.2 Height of object with inaccessible base . . . . . . . . . . . . . . . . . . . . 244

24.3 Height of a raised object . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

24.4 Regiomontanus’ problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

24.5 Height fromthree spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

24.6 Vertical and horizontal spreads . . . . . . . . . . . . . . . . . . . . . . . . 248

24.7 Spreads over a right triangle . . . . . . . . . . . . . . . . . . . . . . . . . . 249

24.8 Spherical analogue of Pythagoras’ theorem . . . . . . . . . . . . . . . . . . 250

25 Resection and Hansen’s problem 251

25.1 Snellius-Pothenot problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

25.2 Hansen’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

26 Platonic solids 259

26.1 Tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

26.2 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

26.3 Octahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

26.4 Icosahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

26.5 Dodecahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

27 Rational polar coordinates 265

27.1 Polar spread and quadrance . . . . . . . . . . . . . . . . . . . . . . . . . . 265

27.2 Evaluating π2/16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

27.3 Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

27.4 Rational spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 271

27.5 Surfacemeasure on a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 276

27.6 Four dimensional rational spherical coordinates . . . . . . . . . . . . . . . 277

27.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

A Rational polar equations of curves 279

B Ellipson 287

Theorems with pages and Important Functions 289