[math-fun] "Rational Trigonometry" ??
Has anyone here dealt with Rational Trigonometry? I found it as a link through Chebyshev polynomials ?!? http://web.maths.unsw.edu.au/~norman/Rational1.htm http://en.wikipedia.org/wiki/Chebyshev_polynomial
On 4/16/11, Henry Baker <hbaker1@pipeline.com> wrote:
Has anyone here dealt with Rational Trigonometry?
I found it as a link through Chebyshev polynomials ?!?
I came across Wildberger's material a few years ago, but haven't pursued it. It is noteworthy that he treats only 2-space in his book, and mentions how much more difficult 3-space would be. His "quadrance" (length squared) and "spread" (sine squared --- would cosine not be a better choice?) hardly seem an earth-shaking improvement on traditional trigonometry ... Squares of length and cosine also turn up as magnitudes in the Euclidean geometric algebra which I've been buttonholing everybody about for the last ten years or so. See ‹TTT_EGA.txt› for n-space, and ‹teabag.pdf› for 3-space, at https://docs.google.com/leaf?id=0B6QR93hqu1AhZTcyM2EzNzItYWYwNi00NDU3LTk3NzQ... The difference is that the GA approach is to a considerable extent independent of the dimension (stand up the man who muttered "just as incomprehensible in two dimensions as ten!"). Admittedly, applying it to plane geometry does look such a pushover that I have never even bothered to try ... Fred Lunnon
My earlier reply perhaps gave the impression that Wildberger's approach is unmotivated. On the contrary, it's based on the very pertinant observation that the squares of lengths and sines (or cosines) occurring in elementary geometrical constructions are rational functions of the input coordinates, unlike traditionally employed lengths and angles. [He does himself no favours by the coy manner in which this crucial fact is concealed.] My criticism is rather that his work doesn't go nearly far enough towards rebuilding geometry as a (Clifford) algebra. The recently aired formula for distance between (partially parallel) flats in n-space is just one example of the remarkable power of this formalism. WFL On 4/17/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 4/16/11, Henry Baker <hbaker1@pipeline.com> wrote:
Has anyone here dealt with Rational Trigonometry?
I found it as a link through Chebyshev polynomials ?!?
I came across Wildberger's material a few years ago, but haven't pursued it. It is noteworthy that he treats only 2-space in his book, and mentions how much more difficult 3-space would be.
His "quadrance" (length squared) and "spread" (sine squared --- would cosine not be a better choice?) hardly seem an earth-shaking improvement on traditional trigonometry ...
Squares of length and cosine also turn up as magnitudes in the Euclidean geometric algebra which I've been buttonholing everybody about for the last ten years or so. See ‹TTT_EGA.txt› for n-space, and ‹teabag.pdf› for 3-space, at
https://docs.google.com/leaf?id=0B6QR93hqu1AhZTcyM2EzNzItYWYwNi00NDU3LTk3NzQ...
The difference is that the GA approach is to a considerable extent independent of the dimension (stand up the man who muttered "just as incomprehensible in two dimensions as ten!"). Admittedly, applying it to plane geometry does look such a pushover that I have never even bothered to try ...
Fred Lunnon
One proof I found compelling was a proof of the inability to trisect an angle given a compass & straightedge. http://web.maths.unsw.edu.au/~norman/papers/Trisection.pdf At 02:04 PM 4/17/2011, Fred lunnon wrote:
On 4/16/11, Henry Baker <hbaker1@pipeline.com> wrote:
Has anyone here dealt with Rational Trigonometry?
I found it as a link through Chebyshev polynomials ?!? http://web.maths.unsw.edu.au/~norman/Rational1.htm http://en.wikipedia.org/wiki/Chebyshev_polynomial
I came across Wildberger's material a few years ago, but haven't pursued it. It is noteworthy that he treats only 2-space in his book, and mentions how much more difficult 3-space would be. His "quadrance" (length squared) and "spread" (sine squared --- would cosine not be a better choice?) hardly seem an earth-shaking improvement on traditional trigonometry ... Squares of length and cosine also turn up as magnitudes in the Euclidean geometric algebra which I've been buttonholing everybody about for the last ten years or so. See â¹TTT_EGA.txt⺠for n-space, and â¹teabag.pdf⺠for 3-space, at
https://docs.google.com/leaf?id=0B6QR93hqu1AhZTcyM2EzNzItYWYwNi00NDU3LTk3NzQ...
The difference is that the GA approach is to a considerable extent independent of the dimension (stand up the man who muttered "just as incomprehensible in two dimensions as ten!"). Admittedly, applying it to plane geometry does look such a pushover that I have never even bothered to try ...
Fred Lunnon
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