{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

Portland, Ore. (2005)

– Engineers today model alternating current, magnetic flux and other quantities using trigonometric calculations that require the use of a computer, calculator or, at the least, lookup tables. Now mathematician Norman Wildberger proposes scrapping the angle representation in geometry in favor of a “rational” system.

By beginning with simpler definitions, all the familiar engineering calculations that currently require tables or a calculator can be done with simple arithmetic, argues Wildberger, whose “rational trigonometry” uses ratios of whole numbers in place of the sine, cosine and tangent functions.

“Many practical problems can be solved in an easier and more elegant fashion than classical trigonometry [affords],” said Wildberger, a professor at the University of New South Wales (Sydney, Australia) and author of Divine Proportions: Rational Trigonometry to Universal Geometry (see www.wildegg.com). “Tables or calculators are no longer necessary. It's a shame that it took until now, because accurate tables for many centuries were not widely available.”

Wildberger says his concepts of divine proportions, rational trigonometry and universal geometry will benefit EEs and anyone else whose work involves trigonometry. Starting with modern algebraic formulations of geometry in which points in a plane are represented by pairs of numbers, Wildberger's definitions of “quadrance” between points and “spread” between lines lead to simple polynomial expressions for basic geometric laws. He says the principles can be extended from Euclidean to non-Euclidean geometry, such as projective geometry or the hyperbolic geometry used in relativity theory.

Though Wildberger is a mathematician, he professes to “know how engineers think” because he counts several as immediate-family members. “My father, brother, sister and brother-in-law are all electrical engineers. Engineers don't want a lot of theory; they want to cut to the chase and get the job done. The usual formulation of trig is overly complicated, because it mixes the separate issues of measurements of triangles and circular motion. These are two different areas, and it is inappropriate to have to use the same system [for both].”

Wildberger's concept of divine proportions refers back to the work of Pythagoras circa 500 B.C. Pythagoras believed the universe was the result of a 'divine mind' that would express everything as a ratio of whole numbers. But the discovery of irrational numbers confounded that supposition. Dismayed at the discovery of square roots that were not rational numbers, Pythagoras and his followers believed they had failed to understand the true nature of the universe.

Wildberger now argues that Pythagoreans should have stuck with their original belief, “given up distance as a measurement and boldly concluded that the square of the length was more important than the lengths themselves.”

His approach drops the definition of distance, which requires the use of square roots, and instead uses the unit he calls quadrance, which compensates for irrational numbers by squaring all distances. Wildberger's second innovation is to define a degree of spread between two intersecting lines by using quadrances instead of angles. Combining quadrances and spreads, in place of distance and angles, yields rational trigonometry.

Had Pythagoras anticipated Wildberger's substitution, the ancient mathematician's famous theorem would have been written thus: The quadrance of the hypotenuse of a right triangle is equal to the sum of the quadrances of the two other sides. Pythagoras' familiar version-the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two other sides-requires the extraction of square roots to get the actual lengths of the sides. But by ditching distance for quadrance and angles for spreads, rational geometry allows just about anything related to trigonometry to be calculated with ratios of whole numbers, Wildberger argues.

“Quadrance is easier to work with than distance. Spread is more elementary than an angle,” he asserted.

To find the spread between two lines, just drop a line from a point on one line that is perpendicular to the other line. The spread is the ratio of the quadrance of the dropped side to the quadrance of the hypotenuse. Thus when lines are parallel, the spread is 0, because two sides of the triangle are 0; when lines are perpendicular, the spread is 1, because the dropped side and the hypotenuse of the triangle are equal.

Likewise, “45 degrees is a spread of 1/2, while 30 degrees and 60 degrees are spreads of 1/4 and 3/4, respectively. What could be simpler?” said Wildberger.

The upshot for “engineers, surveyors and scientists,” he said, is a set of tools that will “increase accuracy and reduce computation time for geometric problems. Now there are many more opportunities to effectively harness the power of modern mathematical software, such as Mathematica, Matlab, Maple, Magma and Mupad.”

The calculations in computer programs involving angles and distances will be simpler using rational trigonometry, since the formulas to perform the trigonometric functions will be arithmetic.

“Simpler is better, and this method is a lot simpler-not just by 10 or 20 percent but by a factor of two to five,” said Wildberger. “Some engineers today have forgotten their trigonometry. I hope this new system will provide them with a more natural and easy-to-use alternative, which will then stay in their minds longer.”

Classical or Euclidean geometry is taught today using axioms about distances and angles that can only be rigorously defined with calculus. But most people learn geometry and trigonometry before calculus, putting the cart before the horse.

“Divine proportions makes it clear why so many students are turned off by the mathematics of trigonometry,” Wildberger said. “The current curriculum attempts to arm you with machine guns to hunt rabbits. Rational trigonometry adds a metrical aspect with the notions of quadrance and spread but stays in the purely algebraic setting, thereby putting Euclidean geometry on a firmer foundation.”

Formulas simplified

Wildberger's formulation of geometry does not render sine, cosine and tangent obsolete, but it does make their use unnecessary except for problems that involve angular motion. “The essential roles of sine, cosine and tangent arise when considering angular motion around a circle,” he said, “but in rational trigonometry, points and lines are more fundamental than circles.”

He argues that rational geometry vastly simplifies many of the formulas EEs use. Even when calculus is used, if quadrance and spread are substituted for distance and angle, then the derivatives and integrals end up being much simpler, Wildberger says.

To demonstrate his point, Wildberger said Snell's law, which is used to determine everything from the direction of magnetic flux and current flow to the angle of reflection and refraction, can be applied with simple arithmetic. “The usual derivation of Snell's law uses the derivative of the square root function. But I give a derivation of Snell's law that shows that only derivatives of linear and quadratic functions are required, so it makes things more elementary.”

Wildberger said his book also offers “a fair number of integrals for flux, currents and other quantities that electrical engineers sometimes need to evaluate. The book also introduces rational alternatives to spherical and polar coordinates that I am sure will prove useful to all engineers in solving practical problems. I give a new form of spherical coordinates in two- and three-dimensional space with simpler, more intuitive terms. I also show how to use these for calculations of volumes and surface areas.”

Wildberger does not claim to have invented all these concepts, only to have codified them into a usable system. He gives historical credit to many others for the different aspects of rational trigonometry that have been used by various physicists to solve practical problems.

For instance, engineers already solve many trigonometric identities and integrals with various rational substitutions; linear algebra avoids angles by using dot products and cross products; and Einstein's theory of relativity makes use of quadratic intervals.