N J Wildberger Research Snapshot: Rational trigonometry and universal geometry

A brief introduction to the author's novel theory of rational trigonometry and how it allows us to extend Euclidean geometry to general fields and arbitrary quadratic forms.

[Achrononmaster]

For any teachers coming across this, I highly recommend studying Norman's book or his lecture series WildTrig. If you teach at a free school, or something close to it like a Steiner school, you will find it is easy to replace traditional trig with rational trigonometry. And I can give you motivation to not shy away, there are very few applications of trigonometry that are not covered completely by rational trig, so (as a practicing physicist) I can testify that I have never come across a problem where I explicitly needed cosines and since etc., except as an undergraduate where lecturers taught us in traditional trig.

Where transcendental functions are used in analysis and theoretical physics, sometimes you just have to get used to proofs using them, but they are also often easy to translate into rational trig. Not only that, but rational trigonometry also makes writing computer code a lot easier. And finally, I would like to make a plug for a unrelated but I think a cousin area of mathematics: Clifford or Geometric Algebra, which is a lot like a natural setting for geometry in the same spirit RatTrig is for trigonometry. and I believe Geometric (Clifford) algebra multivectors are really the way to do the most general rational trigonometry, universal geometry, and algebraic analysis especially computationally (avoids matrices).

[pauluk]

8 mins: it's impressive how easily spreads can be calculated (even in the head, for a practised student) given three quadrances in _vector_ form, since vectors naturally involve integral values, unlike standard trig examples using integral _distances_.

This shows an advantage of the Lin Alg approach over the 'High School' one using Pythagoras more implicitly and calculating spreads via quadrances. For instance, if you make up a triangle with three quadrances how do you _know_ that the slopes of the sides must always be rational? 

[fawzyhegab]

Woow! it seems very interesting ! I'm sure I'm going to give it a try in next months ! thankx alot .. 

[dreznik]

Norman I understand why an algebraist might be interested in the theorems derived from quadrance and spreads. But one notices that spreads (unlike angles) don't combine linearly but rather as polynomials. I would assume this would make a student's understanding of the topic even harder unless he is interested in algebraic properties of polynomials.

As for the irrationality of pi (sum of triangle's internal angles), using 360 degrees (or some other convenient integer) has been a clever "workaround". I would think adding 45+30 is simpler than applying on-linear sums of spreads.

What is wrong (for an applied scientist or engineer) to think of angles as ratios of arc lenght to the entire circumference? The discovery of the circle's connection with trigonometry (study of triangles) has (in my view) simplified (rather than made it harder) the topic for ordinary folks.

It's as if the circle's arc length acts as a "log" of spreads -- while spreads combine non-linearly, arc length is additive.

Most planar and 3d geometry programming I have done over the past 30 yrs have relied on representing angles as rational pairs (sin and cos) instead of the angle itself and storing the squares of distances instead of the distance itself. Occasionally I will need a square root to normalize a vector that comes out of a cross product. As for precision, with 64- and 128-bit floats are now available in most modern programming languages, many times supported by hardware.

[Achrononmaster]

I would not recommend "pen & paper" except as pedagogical methods. People make mistakes even in basic calculator math (pushing wrong buttons) about 1 in every 90 operations. It is much better to recommend using computer algebra packages where both symbolic and numeric computations can be written out in full and edited. I recommend Maxima for high school. And then SageMath which combines many free open source computer algebra packages under one roof, for doing most of the university work you will ever need to do (including Maxima as well as group theory (GAP) and number theory (FLINT) and statistics (R).).