''Divine Proportions: Rational Trigonometry to Universal Geometry'' is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

and

trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...

, called rational trigonometry. The book advocates replacing the usual basic quantities of trigonometry,

Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...

and

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

measure, by squared distance and the square of the sine of the angle, respectively. This is logically equivalent to the standard development (as the replacement quantities can be expressed in terms of the standard ones and vice versa). The author claims his approach holds some advantages, such as avoiding the need for

irrational numbers In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

. The book was "essentially self-published" by Wildberger through his publishing company Wild Egg. The formulas and theorems in the book are regarded as correct mathematics but the claims about practical or pedagogical superiority are primarily promoted by Wildberger himself and have received mixed reviews.

Overview

The main idea of ''Divine Proportions'' is to replace distances by the

squared Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...

, renamed in this book as ''quadrance'', and to replace angles by the squares of their sines, renamed in this book as ''spread'' and thought of as a measure of separation (rather than an amount of rotation) between two lines. ''Divine Proportions'' defines both of these concepts directly from the

Cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

s of points that determine a line segment or a pair of crossing lines, rather than indirectly from distances and angles. Defined in this way, they are rational functions of those coordinates, and can be calculated directly without the need for the square roots needed to calculate distances from coordinates or the

inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...

needed to calculate angles from coordinates. According to ''Divine Proportions'', this replacement has several key advantages: *For points given by rational number coordinates, the quadrance of pairs of points and spread of triples of points are again rational, avoiding the need for irrational numbers, or the concepts of

limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

used to define the real numbers. *By avoiding real numbers, it also avoids what Wildberger claims are foundational problems in the definition of angles and in the

computability Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is clo ...

of real numbers. *It allows analogous concepts to be extended directly to other number systems such as

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

s by using the same formulas for quadrance and spread that one would use for rational numbers. Additionally, this method avoids the ambiguity of the two supplementary angles formed by a pair of lines, as both angles have the same spread. This system is claimed to be more intuitive, and to extend more easily from two to three dimensions. However, in exchange for these benefits, one loses the additivity of distances and angles: for instance, if a line segment is divided in two, its length is the sum of the lengths of the two pieces, but combining the quadrances of the pieces is more complicated and requires square roots.

Organization and topics

''Divine Proportions'' is divided into four parts. Part I presents an overview of the use of quadrance and spread to replace distance and angle, and makes the argument for their advantages. Part II formalizes the claims made in part I, and proves them rigorously. Rather than defining lines as infinite sets of points, they are defined by their

homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...

, which may be used in formulas for testing the incidence of points and lines. Like the sine, the cosine and tangent are replaced with rational equivalents, called the "cross" and "twist", and ''Divine Proportions'' develops various analogues of

trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...

involving these quantities, including versions of the Pythagorean theorem,

law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and ar ...

and

law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...

. Part III develops the geometry of

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

s and

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...

s using the tools developed in the two previous parts. Well known results such as

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...

for calculating the area of a triangle from its side lengths, or the

inscribed angle theorem In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an i ...

in the form that the angles subtended by a chord of a circle from other points on the circle are equal, are reformulated in terms of quadrance and spread, and thereby generalized to arbitrary fields of numbers. Finally, Part IV considers practical applications in physics and surveying, and develops extensions to higher-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

and to

polar coordinate In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the ...

Audience

''Divine Proportions'' does not assume much in the way of mathematical background in its readers, but its many long formulas, frequent consideration of finite fields, and (after part I) emphasis on mathematical rigor are likely to be obstacles to a

popular mathematics Popular mathematics is the presentation of mathematics to an aimed general audience. The difference between recreational mathematics and popular mathematics is that recreational mathematics intends to be fun for the mathematical community, and ...

audience. Instead, it is mainly written for mathematics teachers and researchers. However, it may also be readable by mathematics students, and contains exercises making it possible to use as the basis for a mathematics course.

Critical reception

The feature of the book that was most positively received by reviewers was its work extending results in distance and angle geometry to finite fields. Reviewer Laura Wiswell found this work impressive, and was charmed by the result that the smallest finite field containing a regular pentagon is

\mathbb_

. Michael Henle calls the extension of triangle and conic section geometry to finite fields, in part III of the book, "an elegant theory of great generality", and William Barker also writes approvingly of this aspect of the book, calling it "particularly novel" and possibly opening up new research directions. Wiswell raises the question of how many of the detailed results presented without attribution in this work are actually novel. In this light, Michael Henle notes that the use of

"has often been found convenient elsewhere"; for instance it is used in

distance geometry Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based ''only'' on given values of the distances between pairs of points. More abstractly, it is the study of semimetric spaces and the isom ...

, least squares statistics, and

convex optimization Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization pr ...

. James Franklin points out that for spaces of three or more dimensions, modeled conventionally using

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

, the use of spread by ''Divine Proportions'' is not very different from standard methods involving

dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...

s in place of trigonometric functions. An advantage of Wildberger's methods noted by Henle is that, because they involve only simple algebra, the proofs are both easy to follow and easy for a computer to verify. However, he suggests that the book's claims of greater simplicity in its overall theory rest on a false comparison in which quadrance and spread are weighed not against the corresponding classical concepts of distances, angles, and sines, but the much wider set of tools from classical trigonometry. He also points out that, to a student with a scientific calculator, formulas that avoid square roots and trigonometric functions are a non-issue, and Barker adds that the new formulas often involve a greater number of individual calculation steps. Although multiple reviewers felt that a reduction in the amount of time needed to teach students trigonometry would be very welcome, Paul Campbell is skeptical that these methods would actually speed learning. Gerry Leversha keeps an open mind, writing that "It will be interesting to see some of the textbooks aimed at school pupils hat Wildbergerhas promised to produce, and ... controlled experiments involving student guinea pigs." , however, these textbooks and experiments have not been published. Wiswell is unconvinced by the claim that conventional geometry has foundational flaws that these methods avoid. While agreeing with Wiswell, Barker points out that there may be other mathematicians who share Wildberger's philosophical suspicions of the infinite, and that this work should be of great interest to them. A final issue raised by multiple reviewers is inertia: supposing for the sake of argument that these methods are better, are they enough better to make worthwhile the large individual effort of re-learning geometry and trigonometry in these terms, and the institutional effort of re-working the school curriculum to use them in place of classical geometry and trigonometry? Henle, Barker, and Leversha conclude that the book has not made its case for this, but Sandra Arlinghaus sees this work as an opportunity for fields such as her mathematical geography "that have relatively little invested in traditional institutional rigidity" to demonstrate the promise of such a replacement.

References

{{reflist, refs= {{citation , last = Arlinghaus , first = Sandra L. , author-link = Sandra Arlinghaus , date = June 2006 , hdl = 2027.42/60314 , issue = 1 , journal = Solstice: An Electronic Journal of Geography and Mathematics , title = Review of ''Divine Proportions'' , volume = 17 {{citation , last = Barker , first = William , date = July 2008 , publisher = Mathematical Association of America , title = Review of ''Divine Proportions'' , url = https://www.maa.org/press/maa-reviews/divine-proportions-rational-trigonometry-to-universal-geometry , work = MAA Reviews {{citation , last = Campbell , first = Paul J. , date = February 2007 , issue = 1 , journal =

Mathematics Magazine ''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...

, jstor = 27643001 , pages = 84–85 , title = Review of ''Divine Proportions'' , volume = 80, doi = 10.1080/0025570X.2007.11953460 , s2cid = 218543379 {{citation , last = Franklin , first = James , date = June 2006 , doi = 10.1007/bf02986892 , issue = 3 , journal =

The Mathematical Intelligencer ''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released qu ...

, pages = 73–74 , title = Review of ''Divine Proportions'' , url = https://philpapers.org/archive/FRADPR.pdf , volume = 28, s2cid = 121754449 {{citation , last = Henle , first = Michael , date = December 2007 , issue = 10 , journal = The American Mathematical Monthly , jstor = 27642383 , pages = 933–937 , title = Review of ''Divine Proportions'' , volume = 114 {{citation , last = Leversha , first = Gerry , date = March 2008 , doi = 10.1017/S0025557200182944 , issue = 523 , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, jstor = 27821758 , pages = 184–186 , title = Review of ''Divine Proportions'' , volume = 92, s2cid = 125430473 {{citation , last = Wiswell , first = Laura , date = June 2007 , id = {{ProQuest, 228292466 , doi = 10.1017/S0013091507215020 , issue = 2 , journal =

Proceedings of the Edinburgh Mathematical Society In academia and librarianship, conference proceedings is a collection of academic papers published in the context of an academic conference or workshop. Conference proceedings typically contain the contributions made by researchers at the conferen ...

, pages = 509–510 , title = Review of ''Divine Proportions'' , volume = 50, doi-access = free (n.b. surname Wisewell misspelled in source) Mathematics books 2005 non-fiction books Trigonometry Self-published books

Overview

Organization and topics

Audience

Critical reception

See also

References