Rational trigonometry


Rational trigonometry is a proposed reformulation of metrical planar and solid geometries by Canadian mathematician Norman J. Wildberger, currently a professor of mathematics at the University of New South Wales. His ideas are set out in his 2005 book Divine Proportions: Rational Trigonometry to Universal Geometry. According to New Scientist, part of his motivation for an alternative to traditional trigonometry was to avoid some problems that he claims occur when infinite series are used in mathematics. Rational trigonometry avoids direct use of transcendental functions like sine and cosine by substituting their squared equivalents. Wildberger draws inspiration from mathematicians predating Georg Cantor's infinite set-theory, like Gauss and Euclid, who he claims were far more wary of using infinite sets than modern mathematicians. To date, rational trigonometry is largely unmentioned in mainstream mathematical literature.

Approach

Rational trigonometry follows an approach built on the methods of linear algebra to the topics of elementary geometry. Distance is replaced with its squared value and 'angle' is replaced with the squared value of the usual sine ratio associated to either angle between two lines.. The three main laws in trigonometry – Pythagoras's theorem, the sine law and the cosine law – are given in rational form, and are augmented by two further laws – the [|triple quad formula] and the [|triple spread formula] –, giving the [|five main laws] of the subject.
Rational trigonometry is otherwise broadly based on Cartesian analytic geometry, with a point defined as an ordered pair of rational numbers
and a line
as a general linear equation with rational coefficients, and.
By avoiding calculations that rely on square root operations giving only approximate distances between points, or standard trigonometric functions, giving only truncated polynomial approximations of angles geometry becomes entirely algebraic. There is no assumption, in other words, of the existence of real number solutions to problems, with results instead given over the field of rational numbers, their algebraic field extensions, or finite fields. Following this, it is claimed, makes many classical results of Euclidean geometry applicable in rational form over any field not of characteristic two.
The book Divine Proportions shows the application of calculus using rational trigonometric functions, including three-dimensional volume calculations. It also deals with rational trigonometry's application to situations involving irrationals, such as the proof that Platonic Solids all have rational 'spreads' between their faces.

Notability and criticism

Rational trigonometry is mentioned in only a modest number of mathematical publications besides Wildberger's own articles and book. Divine Proportions was dismissed by reviewer Paul J. Campbell, in the Mathematics Magazine of the Mathematical Association of America : "the author claims that this new theory will take 'less than half the usual time to learn'; but I doubt it. and it would still have to be interfaced with the traditional concepts and notation." Reviewer William Barker, Isaac Henry Wing Professor of Mathematics at Bowdoin College, also writing for the MAA, was more approving: "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" New Scientist's Amanda Gefter described the approach of Wildberger as an example of finitism. James Franklin in the Mathematical Intelligencer argued that the book deserved careful consideration.
An analysis by Michael Gilsdorf of the example problems given by Wildberger in an early paper disputed the claim that RT required fewer steps to solve most problems, if free selection of classical methods is allowed to optimize the solution of problems. Concerning pedagogy, and whether using the quadratic quantities introduced by RT offers real benefits over traditional learning, the author observed that classical trigonometry was not initially based on use of Taylor series to approximate angles at all, but rather on measurements of chord and thus with a proper understanding students could reap continued advantages from use of linear measurement without the claimed logical inconsistencies when circular parametrization by angle is subsequently introduced.

Quadrance

Quadrance and distance both measure separation of points in Euclidean space. Following Pythagoras's theorem, the quadrance of two points and in a plane is therefore defined as the sum of squares of differences in the and coordinates:
The triangle inequality is expressed under rational trigonometry as.

Spread

Spread gives one measure to the separation of two lines as a single dimensionless number in the range for Euclidean geometry. It replaces the concept of angle discussed in the section below. Descriptions of spread may include:

Trigonometric

Suppose two lines, and, intersect at the point as shown at right. Choose a point on and let be the foot of the perpendicular from to. Then the spread is

Vector/slope (two-variable)

Like angle, spread depends only on the relative slopes of two lines and is invariant under translation. So given two lines whose equations are
we may rewrite them as two lines which meet at the origin with equations
In this position the point satisfies the first equation and satisfies the second and the three points, and forming the spread will give three quadrances:
The cross law – see below – in terms of spread is
which becomes:
This simplifies, in the numerator, to, giving:
Then, using the Brahmagupta–Fibonacci identity
the standard expression for spread in terms of slopes of two lines becomes
In this form a spread is the ratio of the square of a determinant of two vectors to the product of their quadrances

Cartesian (three-variable)

For a triangle, as opposed to a pair of lines or vectors, we can replaces points , and in the previous result with , and to obtain the spread at an appropriate vertex:
which, in symmetric numerator form, becomes:
and therefore for the other associated spreads, s1 and s2:

Spread compared to angle

Unlike angle, which can define a relationship between rays emanating from a point, by an arc measurement parametrization, and where a pair of lines can be considered four pairs of rays, forming four angles, 'spread' is more fundamental in rational trigonometry, describing two lines by a single measure of a rational function. Being equivalent to the square of a sine of the corresponding angle , the spread of both an angle and its supplementary angle are equal.
Spread is not proportional, however, to the separation between lines as angle would be; with spreads of 0,,,, and 1 corresponding to unevenly spaced angles 0°, 30°, 45°, 60° and 90°.
Instead, two equal, co-terminal spreads determine a third spread, whose value will be a solution of the triple spread formula for a triangle having spreads of, and :
giving the quadratic polynomial :
and solutions
This is equivalent to the trigonometric identity :
of the angles, and of a triangle, using
to denote a second spread polynomial in.
Finding the triple of a spread likewise makes use of the triple spread formula as a quadratic equation in the unknown third spread treating the known spreads and as constants. This turns out to be:
Further multiples of any basic spread of lines can either be generated by continuing use of the triple spread formula in this way, or by use of a recursion formula which applies it indirectly. Whereas any multiple of a spread that is rational will be polynomial in that spread, the converse does not apply. For example, by the half-angle formula, two lines meeting at a 15° angle have spread of:
and thus exists by algebraic extension of the rational numbers.

Turn and coturn

Twist

Spread polynomials

As seen for double and triple spreads, an th multiple of any spread, gives a polynomial in that spread, denoted, as one solution to the triple spread formula.
In the conventional language of circular functions, these th-degree spread polynomials, for, can be characterized by the identity:

Identities

Explicit formulas

From the definition it immediately follows that

Recursion formula

The triple spread formula
is an identity whose entries can themselves be spread polynomials of the form :, and,
So, taking a difference of the expressions
yields the recursive relation:

Relation to Chebyshev polynomials

The spread polynomials are related to the Chebyshev polynomials of the first kind,, by the identity
This implies
The second equality above follows from the identity
on Chebyshev polynomials.

Composition

The spread polynomials satisfy the composition identity

Coefficients in finite fields

When the coefficients are taken to be members of the finite field, then the sequence of spread polynomials is periodic with period. In other words, if, then, for all .

Orthogonality

When the coefficients are taken to be real, then for, we have
For, the integral is unless, in which case it is .

Generating functions

The ordinary generating function is
The exponential generating function is

Differential equation

satisfies the second-order linear nonhomogeneous differential equation

Spread periodicity theorem

For every integer and every prime, there is a natural number such that is divisible by precisely when divides. This number is a divisor of either or. The proof of this number theoretical property was first given in a paper by Shuxiang Goh and N. J. Wildberger. It involves considering the projective analogue to quadrance in the finite projective line.

Table of spread polynomials, with factorizations

The first several spread polynomials are as follows:

Laws of rational trigonometry

Wildberger states that there are five basic laws in rational trigonometry. He also states that these laws can be verified using high-school level mathematics. Some are equivalent to standard trigonometrical formulae with the variables expressed as quadrance and spread.
In the following five formulae, we have a triangle made of three points. The spreads of the angles at those points are, and, are the quadrances of the triangle sides opposite, respectively. As in classical trigonometry, if we know three of the six elements,, and these three are not the three, then we can compute the other three.

Triple quad formula

The three points are collinear if and only if:
where represent the quadrances between respectively. It can either be proved by analytic geometry or derived from Heron's formula, using the condition for collinearity that the triangle formed by the three points has zero area.
The line has the general form:
where the parameters can be expressed in terms of the coordinates of points and as:
so that, everywhere on the line:
But the line can also be specified by two simultaneous equations in a parameter, where at point and at point :
or, in terms of the original parameters:
If the point is collinear with points and, there exists some value of , call it, for which these two equations are simultaneously satisfied at the coordinates of the point, such that:
Now, the quadrances of the three line segments are given by the squared differences of their coordinates, which can be expressed in terms of :
where use was made of the fact that.
Substituting these quadrances into the equation to be proved:
Now, if and represent distinct points, such that, we may divide both sides by :

Pythagoras's theorem

The lines and are perpendicular if and only if:
where is the quadrance between and.
This is equivalent to the Pythagorean theorem.
There are many classical proofs of Pythagoras's theorem; this one is framed in the terms of rational trigonometry.
The spread of an angle is the square of its sine. Given the triangle with a spread of 1 between sides and,
where is the "quadrance", i.e. the square of the distance.
Construct a line dividing the spread of 1, with the point on line, and making a spread of 1 with and. The triangles, and are similar.
This leads to two equations in ratios, based on the spreads of the sides of the triangle:
Now in general, the two spreads resulting from dividing a spread into two parts, as line does for spread, do not add up to the original spread since spread is a non-linear function. So we first prove that dividing a spread of 1, results in two spreads that do add up to the original spread of 1.
For convenience, but with no loss of generality, we orient the lines intersecting with a spread of 1 to the coordinate axes, and label the dividing line with coordinates and. Then the two spreads are given by:
Hence
so that
Using the first two ratios from the first set of equations, this can be rewritten:
Multiplying both sides by :
Q.E.D.

Spread law

For any triangle with nonzero quadrances:
This is the law of sines, just squared.

Cross law

For any triangle,
This is analogous to the law of cosines. It is called the 'cross law' because, the square of the cosine of the angle, is called the 'cross'.

Triple spread formula

For any triangle,
This relation can be derived from the formula for the sine of a compound angle: in a triangle we have,
Equivalently, it describes the relationship between the spreads of three concurrent lines, as spread is unaffected when the sides of a triangle are moved parallel to themselves to meet in a common point.
Knowing two spreads allows the third to be calculated by solving the associated quadratic formula. Since two solutions are produced, further triangle spread rules must be used to select the appropriate one. While this appears more complex than obtaining a supplementary angle directly by subtraction, the irrational value of '' is avoided.

Trigonometry over arbitrary fields

As the laws of rational trigonometry give algebraic relations, they apply in generality to algebraic number fields beyond the rational numbers. Specifically, any finite field which does not have characteristic 2 reproduces a form of these laws, and thus a finite field geometry. The 'plane' formed by a finite field is the cartesian product of all ordered pairs of field elements, with opposite edges identified forming the surface topologically equivalent to a discretized torus. Individual elements correspond to standard 'points' and 'lines' to sets of no more than points related by incidence plus direction or slope given in lowest terms that 'wrap' the plane before repeating.

Example: (verify the spread law in )

The figure shows a triangle of three such lines in the finite field setting :
Each line has its own symbol and the intersections of lines is marked by two symbols present at points:-plane.|278x278px
Using Pythagoras's theorem with arithmetic modulo 13, we find these sides have quadrances of:
Rearranging the cross law as
gives separate expressions for each spread, in terms of the three quadrances:
In turn we note these ratios are all equal – as per the spread law :
Since first and last ratios match we just cross multiply, and take differences, to show equality with the middle ratio also:
Otherwise, the standard Euclidean plane is taken to consist of just rational points,, omitting any non-algebraic numbers as solutions. Properties like incidence of objects, representing the solutions or 'content' of geometric theorems, therefore follow a number theoretic approach that differs and is more restrictive than one allowing real numbers. For instance, not all lines passing through a circle's centre are considered to meet the circle at its circumference. To be incident such lines must be of the form
and necessarily meet the circle in a rational point.

Computation – complexity and efficiency

Rational trigonometry makes nearly all problems solvable with only addition, subtraction, multiplication or division, as trigonometric functions are purposefully avoided in favour of trigonometric ratios in quadratic form. At most, therefore, results required as distance can be approximated from an exact-valued rational equivalent of quadrance after these simpler operations have been carried out. To make use of this advantage however, each problem must either be given, or set up, in terms of prior quadrances and spreads, which entails additional work.
The laws of rational trigonometry, being algebraic, introduce subtleties into the solutions of problems, such as non-additivity of quadrances of collinear points or spreads of concurrent lines to give rational-valued outputs. By contrast, in the classical subject linearity is incorporated into distance and angular measurements to simplify these operations, albeit by 'transcendental' techniques employing real numbers entailing approximate valued output.