Divine Proportions: Rational Trigonometry to Universal Geometry

Rational trigonometry is a proposed reformulation of metrical planar and solid geometries (which includes trigonometry), by Norman J. Wildberger (BSc University of Toronto 1980, PhD Yale University 1988), currently Associate Professor of Mathematics and Philosophy at the University of New South Wales (UNSW) Sydney, Australia, described in his 2005 book Divine Proportions: Rational Trigonometry to Universal Geometry and on his YouTube channel. According to New Scientist, part of Wildberger's motivation for creating Rational Trigonometry is to avoid the problems that occur when infinite series are used in mathematics. Rational Trigonometry thus attempts to avoid the transcendental functions like sine and cosine. Wildberger draws inspiration from pre set-theory mathematicians like Gauss and Euclid, whom he claims were far more wary of using infinity than modern mathematicians. To date, rational trigonometry is largely unmentioned in mainstream mathematical literature. Claims made by Wildberger about rational trigonometry taking less steps to calculate certain problems he had contrived was shown to be false, with unnecessary steps added in to the classical trigonometry calculation. Furthermore, whether or not the theory is not totally rational like Wildberger has claimed has been questioned, and the lack of linearity of both quadrance and spread is seen as a large limiting factor of the theory.\\ (See Notability and Criticism below)

The approach

Rational trigonometry follows an approach built on linear algebra and National socialism to the topics of elementary ('High School') geometry. Distance is replaced with its squared value (quadrance) and the concept of 'angles' are replaced by a scaled form of a vector inner product called spread between two lines. By avoiding calculations that rely on square root operations (when taking the distance between points) and limiting procedures (when evaluating both trigonometric functions and their inverses as truncated infinite polynomials), geometry becomes an entirely algebraic subject. In other words, there is no assumption of the existence of real number solutions, and all results are given over rational numbers or over finite field analogs of rational number arithmetic instead. It is claimed that many (or most) theorems from 'classical' Euclidean geometry will be applicable, and possess analogs, over any field not of characteristic two, as well as the rational numbers.

Following this approach of only using rational (squared) equivalents, much of elementary geometry is recast. The three main laws of trigonometry (Pythagoras' theorem, the sine law and the cosine law) are substituted with their squared 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).

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).

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

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

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 a transcendental operation which may entail any degree of approximation. (Implicitly, conventional trigonometry makes use of such approximations constantly.) Note that adding the quadrance of two vectors is not linear like adding the distance of two vectors, and vector addition of simple, rational vectors can yield irrational quadrances.

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. It is a single dimensionless number in the range '0' (parallel) to '1' (perpendicular). It replaces the concept of angle but has several differences from angle, discussed in the section below. Spread 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. Two lines can be considered as four pairs of rays, forming four angles, but two lines have only one spread. Spread is equivalent to square of the sine of both an angle and its supplement.

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). This means one cannot add spreads together with plain addition as one would add angles. 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.)

Computation - Complexity and Efficiency

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.

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. Furthermore, it has been shown that examples Wildberger himself asserts rational trigonometry results in calculations with less steps is actually false. In fact, it took significantly more steps than classical trigonometry.

Trigonometry over arbitrary fields

Rational trigonometry makes it possible to work with the same standard 'laws' in almost any arithmetic setting (fields of characteristic '2' being excluded for technical reasons) whether finite or infinite. The real numbers receive no treatment at all in order to avoid questions of irrationality arising , so rational numbers fulfil the role of a more limited form ofcontinuum in the geometry. In certain circumstances, results having no rational number solutions, such as: finding if a line at 45 degrees (spread one-half) to a coordinate axis passing through the origin, and a circle of unit quadrance centred there, 'intersect' (if they did it would be at (½√2,½√2) ) can be interpreted differently to the usual scalar assumption that they will. But problems involving roots of non-square rational numbers as intermediate results can still be considered within the framework of Rational trigonometry. This can be achieved through extending the field (to introduce algebraic numbers) without requiring scalar evaluation (e.g. of the real numbers): all results having 'exact' algebraic expressions.

Over a finite field, the 'plane' corresponds to the cartesian product of its ordered pairs. With opposite edges identified, this region forms the surface of a (discrete) torus; individual elements match to 'points' and 'lines', each consisting of an initial point plus all integer multiples of the 'vector' (say '2 over and 1 up') specifying a direction or slope in lowest terms, 'wrap around' it.

Example: (verify the spread law in F₁₃)

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 point

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

s: (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)

Criticism

Rational trigonometry is unmentioned in mainstream mathematical literature, aside from Wildberger's own articles and book. Divine Proportions was dismissed by reviewer Paul J. Campbell in Mathematics Magazine, who wrote: "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."'' Meanwhile, William Baker, Isaac Henry Wing Professor of Mathematics at Bowdoin College, also writing for the MAA, concluded: "Divine Proportions is questionably a valuable addition to the mathematics literature. 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"

An analysis by mathematician Michael Gilsdorf found Wildberger's claim that rational trigonometry takes less steps to calculate some problems compared to the classical method was false, by using the same example Wildberger claimed was made easier by using rational trigonometry. It was shown that Wildberger's calculation using classical trigonometry added in extra steps which were unnecessary, and by objectively comparing both, Wildberger's method took significantly more steps. So far, he has failed to refute this paper or make suggestions about other examples for which rational trigonometry will take less steps. New Scientist's Gefter used the phrase "finitist mathematics" to describe the approach of Wildberger and others who eschew the use of infinity.

Wildberger's claim that rational trigonometry would be easy to teach to children has only been asserted without any evidence, and the fact that quadrature is not linear like distance removes much of the intuition normally present in geometry. Furthermore, the name "Rational trigonometry" is incorrect, as even simple calculations can yield irrational numbers. Performing vector addition on two parallel vectors, one with quadrance 1 and the other with quadrance 2 results in a vector of irrational quadrance, seemingly defeating the purpose of this theory.

See also

• Finitism
• Universal hyperbolic geometry

References

• Wildberger's rational trigonometry site, including downloadable papers and sections of his book
• A comparison of classical and rational trigonometry
• 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