''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 mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

and

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

, 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 the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...

and

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

measure, by squared distance and the square of the

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...

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 are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...

. 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, which Wildberger calls the ''quadrance'', and to replace angle measures by the squares of their sines, which Wildberger calls the ''spread'' between two lines. ''Divine Proportions'' defines both of these concepts directly from the

Cartesian coordinate In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

s of points that determine a line segment or a pair of crossing lines. Defined in this way, they are

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

s of those coordinates, and can be calculated directly without the need to take the

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

s or

inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted Domain of a functi ...

required when computing distances or angle measures. For Wildberger, a finitist, this replacement has the purported advantage of avoiding 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 2009 ...

and

actual infinity In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, involves infinite entities as given, actual and completed objects. The concept of actual infinity was introduced into mathematics near the en ...

used in defining the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, which Wildberger claims to be unfounded. It also allows analogous concepts to be extended directly from the rational numbers 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 (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

s using the same formulas for quadrance and spread. Additionally, this method avoids the ambiguity of the two

supplementary angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing ...

s 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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

law of sines In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\ ...

and

law of cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...

. Part III develops the geometry of

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

s and

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

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 Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), 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 ...

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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

and to polar coordinates.

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 rigour are likely to be obstacles to a

popular mathematics Popular mathematics is mathematical presentation aimed at a general audience. Sometimes this is in the form of books which require no mathematical background and in other cases it is in the form of expository articles written by professional mat ...

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 In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

\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 squared Euclidean distance "has often been found convenient elsewhere"; for instance it is used in distance geometry,

least squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...

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 problems ...

. James Franklin points out that for spaces of three or more dimensions, modelled 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 matrix (mathemat ...

, 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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...

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 sufficiently 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 , url = http://www-personal.umich.edu/~copyrght/image/solstice/sum06/wildberger.html , 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 The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

, 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 Science+Business Media that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes ...

, pages = 73–74 , title = Review of ''Divine Proportions'' , url = https://philpapers.org/archive/FRADPR.pdf , volume = 28, s2cid = 121754449 {{citation , last = Gefter , first = Amanda , year = 2013 , title = Mind-bending mathematics: Why infinity has to go , journal = New Scientist , volume = 219 , issue = 2930 , pages = 32–35 , doi = 10.1016/s0262-4079(13)62043-6 , bibcode = 2013NewSc.219...32G {{citation , last = Henle , first = Michael , date = December 2007 , issue = 10 , journal =

The American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposito ...

, 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 a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

, 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 , 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