Divine Proportions: Rational Trigonometry to Universal Geometry

Rational trigonometry is a proposed reformulation of traditional metrical planar (and solid) geometry - which includes trigonometry - by Prof. Norman J. Wildberger in his book Divine Proportions: Rational Trigonometry to Universal Geometry (2005). The book is purposefully critical of traditional mathematics. It eschews transcendental operations entailed in the use of distance (taking square roots) and angle (evaluation of infinite polynomials) calculations central to current methods, in favour of purely algebraic ones. The change is accomplished by replacing ordinary distance with its square ('quadrance') and angular separation of lines by the sine ratio of quadrances in a right triangle ('spread') that, in turn, corresponds to the square of the usual sine ratio.

Following this approach of only using rational equivalences, much of Euclidean geometry is rebuilt without making assumptions of the underlying field. This is also the Universal Geometry aspect of Rational trigonometry: the claim that most results from classical geometry will be applicable, and possess analogs, over any field (not of characteristic two) not just the field of rational numbers.

The three main laws of trigonomtery (pythagoras' theorem, the sine law and the cosine law) are substituted with rational analogs and augmented by two further laws: one relating the quadrances of three collinear points and one relating the spreads of three concurrent lines (for a total of five main laws).

Wildberger holds a Ph.D. in mathematics from Yale University, and taught at Stanford University from 1984 to 1986 and at the University of Toronto from 1986 to 1989; he is currently an associate professor of mathematics at the University of New South Wales, Australia.

Quadrance and spread

Using quadrance instead of distance and spread instead of angle enables calculations to produce output results whose complexity matches that of the input data. In a typical trigonometry problem, for instance, rational values for quadrances and spreads will lead to calculated results (an unknown spread or quadrance) that will either be rational also or at most an expression containing the roots of only rational numbers. These computational gains (exact results, directly calculable) come at the expense of linearity. Doubling or halving a quadrance or spread does not double or halve as a length or a rotation. Similarly, the sum of two lengths or rotations will not be the sum of their individual quadrances or spreads. The mathematics of Rational trigonometry is, its applications aside, a special instance of a description of geometry in terms of linear algebra (using rational methods such as dot products and quadratic forms.) Students who are first learning trigonometry are often not taught about this use of linear algebra within geometry, and changing this state of affairs is (to paraphrase his comments) one aim of Wildberger's book .

Rational trigonometry is otherwise broadly based on Cartesian analytic geometry, with a point defined as an ordered pair (x, y) and a line as a general linear equation

$Ax+By+C=0.\,$

Trigonometry over arbitrary fields

A Triangle through the points (2, 8), (9, 9), and (10, 0) of the finite field-plane F₁₃ × F₁₃.

Rational trigonometry makes it possible to work in almost any mathematical field (fields of characteristic '2' are excluded for technical reasons) whether finite or infinite. The real numbers are not considered a true algebraic field and rational numbers fulfil their role in relation to a linear continuum. Where the output of a calculation would be the root of a rational number (an algebraic number) it can be added as a discreet element (extending the field) and does not require further evaluation: all results having 'exact' expressions.

Over finite fields, the 'plane' is actually a torus, corresponding to the elements of the cartesian product of ordered pairs, with opposite edges identified. An individual 'point' corresponds to one of these elements and a 'line' (which now 'wraps around' this region) corresponds to an initial point plus all exact multiples of the 'vector' (say '2 over and 1 up') giving the line its direction or slope.

example

The figure (right) shows such a 'triangle' of three lines in this finite field setting (F₁₃ x F₁₃). Each line has a separate symbol and the intersections of lines ('vertices') is marked by the appearance of two symbols together at the points: (2,8), (9,9) and (10,0).

Using Pythagoras' theorem (with arithmetic modulo 13) we find these sides have quadrances of:

(9 - 2)² + (9 - 8)² = 50 = 11 mod 13

(9 - 10)² + (9 - 0)² = 82 = 4 mod 13

(10 - 2)² + (0 - 8)² = 128 = 11 mod 13

Then, using manipulation of the Cross law - see below - to give an expression in s, the three (opposite) spreads of the triangle are found to be:

1 - (4 + 11 - 11)²/(4.4.11) = 1 - 3/7 = 8 mod 13

1 - (11 + 11 - 4)²/(4.11.11) = 1 - 12/3 = 10 mod 13

1 - (4 + 11 - 11)²/(4.4.11) = 1 - 3/7 = 8 mod 13

In turn we see that these ratios are all equal as per the Spread law - see below (at least in mod 13):

8/11 : 10/4 : 8/11

Since first and last ratios match (the triangle is 'isosceles') we just need to cross multiply and take differences to show equality with the middle ratio:

(11)(10) - (8)(4) = 78 (0 mod 13)

Quadrance

Quadrance and distance are concerned with the separation of points. Quadrance is the square of the distance. In the (x, y)-plane, the quadrance Q(A₁, A₂) for the points A₁ and A₂ is defined (following Pythagoras' theorem) as

$Q(A_{1},A_{2})=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}.\,$

Calculating a quadrance rather than a distance, there is no longer a need to find a square root for the sum of the squares of the differences in coordinates. When taking the square root of a quadrance (which is exact-valued) to give a result in terms of 'distance' therefore only one use is made of the a transcendental operation which may entail any degree of approximation. (Implicitly, conventional trigonometry makes use of such approximations constantly.)

Spread

Suppose ℓ ₁ and ℓ ₂ intersect at the point A. Let C be the foot of the perpendicular from B to ℓ ₂. Then the spread is s = Q/R.

The spread of two lines can be measured in four equivalent positions.

Spread gives one measure to the 'separation' of two lines (replacing 'angle'): a dimensionless number in the range '0' (parallel) to '1' (perpendicular), which can have several interpretations.

• Trigonometric: as the sine-ratio for the quadrances in a right triangle (and therefore equivalent to the square of the sine of the angle).

• Vector: as a rational function of the slopes (or directions) of two lines where they meet.

• Cartesian: as a rational function of the three co-ordinates used to describe these two vectors.

• Linear algebra: as a normalized rational function of the square of the determinant of two vectors (from three points) divided by the product of their quadrances.

Calculating spread

• Trigonometric

Suppose two lines, ℓ ₁ and ℓ ₂, intersect at the point A as shown at right. Choose a point B ≠ A on ℓ ₁ and let C be the foot of the perpendicular from B to ℓ ₂. Then the spread s is

$s(\ell _{1},\ell _{2})={\frac {Q(B,C)}{Q(A,B)}}={\frac {Q}{R}}.$

• Vector/slope (two-variable)

Like angle, spread depends only on the relative slopes of two lines (constant terms being eliminated) and spread with parallel lines is preserved. So given two lines whose equations are

a₁x + b₁y = constant and a₂x + b₂y = constant

we may rewrite them as two lines which meet at the origin (0,0) with equations

a₁x + b₁y = 0 and a₂x + b₂y = 0.

In this position the point (-b₁,a₁) satisfies the first equation and (-b₂,a₂) satisfies the second and the three points (0,0), (-b₁,a₁) and (-b₂,a₂) forming the spread will give three quadrances:

Q₁ = (b₁² + a₁²), Q₂ = (b₂² + a₂²) and Q₃ = (b₁ - b₂)² + (a₁ - a₂)²

The Cross law - see below - in terms of spread is:

1 - s = (Q₁ + Q₂ – Q₃)²/4Q₁Q₂

which becomes:

1 - s = (a₁²+ a₂² + b₁² + b₂² – (b₁ – b₂)² – (a₁ – a₂)²)²/4(a₁² + b₁²)(a₂² + b₁²).

This simplifies, in the numerator, to: (2 a₁a₂ + 2 b₁b₂)², giving:

1 - s = (a₁a₂ + b₁b₂)²/(a₁² + b₁²)(a₂² + b₂²)

Then, using the important identity due to Fibonacci (a₂b₁ - a₁b₂)² + (a₁a₂ + b₁b₂)² = (a₁² + b₁²)(a₂² + b₂²) the standard expression for spread in terms of slopes (or directions) of two lines becomes:

$s={\frac {(a_{1}b_{2}-a_{2}b_{1})^{2}}{(a_{1}^{2}+b_{1}^{2})(a_{2}^{2}+b_{2}^{2})}}.\,$

• Cartesian (three-variable)

$s={\frac {((y_{1}-y_{3})(x_{2}-x_{3})-(y_{2}-y_{3})(x_{1}-x_{3}))^{2}}{((y_{1}-y_{3})^{2}+(x_{1}-x_{3})^{2})((y_{2}-y_{3})^{2}+(x_{2}-x_{3})^{2})}}.\,$

This uses a general coordinate (x₃,y₃) in place of the origin, and (x₁,y₁) and (x₂,y₂) in place of (-b₁,a₁) and (-b₂,a₂), to specify the endpoints for a₁, b₁, a₂ and b₂.

Spread compared to angle

Unlike angle, 'spread' is a fundamental concept in rational trigonometry, describing a rational function of two lines, whereas 'angle' describes a relationship based on the circular measure between two rays emanating from a common point. Spread is equivalent to square of the sine of both an angle and its complement.

Degree	Radian	Spread
0	0	0
30	(1/6)π	1/4
45	(1/4)π	1/2
60	(1/3)π	3/4
90	(1/2)π	1
120	(2/3)π	3/4
135	(3/4)π	1/2
150	(5/6)π	1/4
180	π	0

Spread is not proportional to the separation between lines as angle would be, since sine squared is not a sawtooth curve; note that the values above of 1/4, 1/2, 3/4, and 1 correspond to degrees of 30, 45, 60, 90 (uneven steps). Further, spread need not be a rational number. For example, by the half-angle formula, two lines meeting at a 15° angle have spread of:

$\sin ^{2}(30^{\circ }/2)=(1-\cos 30^{\circ })/2=(1-{\sqrt {3}}/2)/2=(2-{\sqrt {3}})/4\approx 0.0667.$

This difference comes about from defining spread rationally. Angular measure achieves linearity by referencing circular motion but not true position (which spread does). Although not defined by rotation, spread does 'repeat' every 180 degrees (π radians) and in that sense is 'periodic'.

Laws of rational trigonometry

Wildberger states that there are five basic laws in rational trigonometry. He also states, correctly, 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 formulas, we have a triangle made of three points A₁, A₂, A₃, . The spreads of the angles at those points are s₁, s₂, s₃, , and Q₁, Q₂, Q₃, are the quadrances of the triangle sides opposite A₁, A₂, and A₃, respectively. As in classical trigonometry, if we know three of the six elements s₁, s₂, s₃, , Q₁, Q₂, Q₃, and these three are not the three s, then we can compute the other three.

Triple quad formula

The three points A₁, A₂, A₃,  are collinear if and only if:

$(Q_{1}+Q_{2}+Q_{3})^{2}=2(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}).\,$

It can either be proved by analytic geometry (the preferred means within rational trigonometry) or derived from Heron's formula, using the condition for collinearity that the triangle formed by the three points has zero area.

Proof (click at right to show/hide)

Illustration of nomenclature used in the proof.

The line $AB\,$ has the general form:

$ax+by+c=0\,$

where the (non-unique) parameters a, b and c, can be expressed in terms of the coordinates of points A and B as:

$a=A_{y}-B_{y}\,$

$b=B_{x}-A_{x}\,$

$c=A_{x}B_{y}-A_{y}B_{x}\,$

so that, everywhere on the line:

$(A_{y}-B_{y})x+(B_{x}-A_{x})y+(A_{x}B_{y}-A_{y}B_{x})=0.\,$

But the line can also be specified by two simultaneous equations in a parameter t, where t = 0 at point A and t = 1 at point B:

$x=(B_{x}-A_{x})t+A_{x}{\text{ and }}y=(B_{y}-A_{y})t+A_{y}\,$

or, in terms of the original parameters:

$x=bt+A_{x}\,$ and $y=-at+A_{y}.\,$

If the point C is collinear with points A and B, there exists some value of t (for distinct points, not equal to 0 or 1), call it λ, for which these two equations are simultaneously satisfied at the coordinates of the point C, such that:

$C_{x}=b\lambda \ +A_{x}$ and $C_{y}=-a\lambda \ +A_{y}.\,$

Now, the quadrances of the three line segments are given by the squared differences of their coordinates, which can be expressed in terms of λ:

${\begin{aligned}Q(AB)&\equiv (B_{x}-A_{x})^{2}+(B_{y}-A_{y})^{2}\\&=b^{2}+(-a)^{2}\\&=a^{2}+b^{2}\end{aligned}}$

${\begin{aligned}Q(BC)&\equiv (C_{x}-B_{x})^{2}+(C_{y}-B_{y})^{2}\\&=((b\lambda \ +A_{x})-B_{x})^{2}+((-a\lambda \ +A_{y})-B_{y})^{2}\\&=(b\lambda \ +(A_{x}-B_{x}))^{2}+(-a\lambda \ +(A_{y}-B_{y}))^{2}\\&=(b\lambda \ +(-b))^{2}+(-a\lambda \ +a)^{2}\\&=b^{2}(\lambda \ -1)^{2}+a^{2}(-\lambda \ +1)^{2}\\&=b^{2}(\lambda \ -1)^{2}+a^{2}(\lambda \ -1)^{2}\\&=(a^{2}+b^{2})(\lambda \ -1)^{2}\end{aligned}}$

${\begin{aligned}Q(AC)&\equiv (C_{x}-A_{x})^{2}+(C_{y}-A_{y})^{2}\\&=((b\lambda \ +A_{x})-A_{x})^{2}+((-a\lambda \ +A_{y})-A_{y})^{2}\\&=(b\lambda \ +A_{x}-A_{x})^{2}+(-a\lambda \ +A_{y}-A_{y})^{2}\\&=(b\lambda )^{2}+(-a\lambda )^{2}\\&=b^{2}\lambda ^{2}+(-a)^{2}\lambda ^{2}\\&=b^{2}\lambda ^{2}+a^{2}\lambda ^{2}\\&=(a^{2}+b^{2})\lambda ^{2}\end{aligned}}$

where use was made of the fact that $(-\lambda \ +1)^{2}=(\lambda \ -1)^{2}$.

Substituting these quadrances into the equation to be proved:

$(Q(AB)+Q(BC)+Q(AC))^{2}=2(Q(AB)^{2}+Q(BC)^{2}+Q(AC)^{2})\,$

$((a^{2}+b^{2})+(a^{2}+b^{2})(\lambda \ -1)^{2}+(a^{2}+b^{2})\lambda ^{2})^{2}=2((a^{2}+b^{2})^{2}+((a^{2}+b^{2})(\lambda \ -1)^{2})^{2}+((a^{2}+b^{2})\lambda ^{2})^{2})\,$

$(a^{2}+b^{2})^{2}(1+(\lambda \ -1)^{2}+\lambda ^{2})^{2}=2(a^{2}+b^{2})^{2}(1+((\lambda \ -1)^{2})^{2}+(\lambda ^{2})^{2})\,$

Now, if $A\,$ and $B\,$ represent distinct points, such that $(a^{2}+b^{2})\,$ is not zero, we may divide both sides by $Q(AB)^{2}=(a^{2}+b^{2})^{2}\,$:

$(1+\lambda ^{2}-2\lambda \ +1+\lambda ^{2})^{2}=2(1+(\lambda ^{2}-2\lambda \ +1)^{2}+\lambda ^{4})\,$

$(2\lambda ^{2}-2\lambda \ +2)^{2}=2(1+\lambda ^{4}-2\lambda ^{3}+\lambda ^{2}-2\lambda ^{3}+4\lambda ^{2}-2\lambda +\lambda ^{2}-2\lambda +1+\lambda ^{4})\,$

$4(\lambda ^{2}-\lambda \ +1)^{2}=2(2\lambda ^{4}-4\lambda ^{3}+6\lambda ^{2}-4\lambda +2)\,$

$4(\lambda ^{4}-\lambda ^{3}+\lambda ^{2}-\lambda ^{3}+\lambda ^{2}-\lambda +\lambda ^{2}-\lambda +1)=4(\lambda ^{4}-2\lambda ^{3}+3\lambda ^{2}-2\lambda +1)\,$

$\lambda ^{4}-2\lambda ^{3}+3\lambda ^{2}-2\lambda +1=\lambda ^{4}-2\lambda ^{3}+3\lambda ^{2}-2\lambda +1\,$

Pythagoras' theorem

The lines A₁A₃ (of quadrance Q₁) and A₂A₃ (of quadrance Q₂) are perpendicular (their spread is 1) if and only if:

$Q_{1}+Q_{2}=Q_{3}.\,$

where Q₃ is the quadrance between A₁ and A₂.

This is equivalent to the Pythagorean theorem (and its converse).

There are many classical proofs of Pythagoras' 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 ABC with a spread of 1 between sides AB and AC,

$Q(AB)+Q(AC)=Q(BC)\,$

where Q is the "quadrance", i.e. the square of the distance.

Proof

Illustration of nomenclature used in the proof.

Construct a line AD dividing the spread of 1, with the point D on line BC, and making a spread of 1 with DB and DC. The triangles ABC, DBA and DAC are similar (have the same spreads but not the same quadrances).

This leads to two equations in ratios, based on the spreads of the sides of the triangle:

$s_{C}={\frac {Q(AB)}{Q(BC)}}={\frac {Q(BD)}{Q(AB)}}={\frac {Q(AD)}{Q(AC)}}.$

$s_{B}={\frac {Q(AC)}{Q(BC)}}={\frac {Q(DC)}{Q(AC)}}={\frac {Q(AD)}{Q(AB)}}.$

Now in general, the two spreads resulting from dividing a spread into two parts, as line AD does for spread CAB, 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 $(x_{1},y_{1})$ and $(x_{2},y_{2})$. Then the two spreads are given by:

$s_{1}={\frac {(x_{2}-x_{2})^{2}+(y_{2}-y_{1})^{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}={\frac {(y_{2}-y_{1})^{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}},$

$s_{2}={\frac {(x_{2}-x_{1})^{2}+(y_{2}-y_{2})^{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}={\frac {(x_{2}-x_{1})^{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.$

Hence:

$s_{1}+s_{2}={\frac {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}=1.\,$

So that:

$s_{C}+s_{B}=1.\,$

Using the first two ratios from the first set of equations, this can be rewritten:

${\frac {Q(AB)}{Q(BC)}}+{\frac {Q(AC)}{Q(BC)}}=1.\,$

Multiplying both sides by $Q(BC)$:

$Q(AB)+Q(AC)=Q(BC).\,$

Q.E.D.

Spread law

For any triangle ${\overline {A_{1}A_{2}A_{3}}}$ with nonzero quadrances:

${\frac {s_{1}}{Q_{1}}}={\frac {s_{2}}{Q_{2}}}={\frac {s_{3}}{Q_{3}}}.\,$

This is the law of sines, just squared.

Cross law

For any triangle ${\overline {A_{1}A_{2}A_{3}}}$,

$(Q_{1}+Q_{2}-Q_{3})^{2}=4Q_{1}Q_{2}(1-s_{3}).\,$

This is analogous to the law of cosines. It is called 'cross law' because $(1-s_{3})$, the square of the cosine of the angle, is called the 'cross'.

Triple spread formula

For any triangle ${\overline {A_{1}A_{2}A_{3}}},$

$(s_{1}+s_{2}+s_{3})^{2}=2(s_{1}^{2}+s_{2}^{2}+s_{3}^{2})+4s_{1}s_{2}s_{3}.\,$

This corresponds approximately to the angle sum formulae for sine and cosine (in a triangle, whose angles sum to 180 degrees, sin a = sin(b+c) = (sin b)(cos c) + (sin c)(cos b)). Equivalently, it describes the relationship between the spreads of three concurrent lines, as spread (like angle) 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 spread to be calculated by solving the associated quadratic formula but, as two solutions are possible, further triangle spread rules must be used to select the appropriate one. (The relative complexity of this process contrasts with the much simpler method of obtaining a complementary angle.)

Ease of calculation

Rational trigonometry makes some problems solvable with only addition, subtraction, multiplication, and division, with fewer uses of other functions such as square roots, sine, and cosine compared to classical trigonometry. Such algorithms execute more efficiently on most computers, for problems such as solving triangles. Other computations, however, do involve more computations than their classical analogues; such computations include determining the quadrance of a line segment given the quadrance of two collinear line segments which compose it, or such as computing the spread of the sum of two angles with known spreads.

See also

• Spread polynomials
• Universal hyperbolic geometry

References

• Wildberger's rational trigonometry site, including downloadable papers and sections of his book
• A comparison of classical and rational trigonometry
• Alexander Bogomolny (2007). "A brief introduction to Rational Trigonometry". Cut-the-knot.
• N J Wildberger (2005), Divine Proportions: Rational Trigonometry to Universal Geometry, Wild Egg

External links

• N. J. Wildberger's homepage
• Print a protractor to measure spread
• N. J. Wildberger's YouTube channel—lectures on rational trigonometry
• James Franklin reviews "Divine Proportions: Rational Trigonometry to Universal Geometry"
• The ancient Greeks present: Rational Trigonometry
• One dimensional metrical geometry
• Euler Math Toolbox implementation of Rational Trigonometry