Advice for prospective research mathematicians | Rational Trigonometry and spread polynomials 1

Here is a quick introduction / review of the essentials of Rational Trigonometry suitable for high school or first year uni students.

Our ultimate aim here is to explain the important spread polynomials / polynumbers, which are more pleasant variants of the Chebyshev polynomials of the first kind.

We cover the five main laws of rational trig: Pythagoras theorem, the Triple quad formula, the Cross law, the Spread law and the Triple spread formula. And we give self contained proofs of all of them.

The treatment is concise, relying on a prior theory of linear algebra and dot / scalar / inner products on a linear or vector space. This is really the proper starting point for an affine theory of metrical geometry, and it naturally incorporates the geometry of special relativity. So this theory should be of particular interest to physicists.

This is part of a series on Advice for prospective research mathematicians, mostly in the amateur direction, but this topic is quite important for professionals as well, as the implications of understanding Rational Trig are far and wide in contemporary geometry as well as mathematics education.

My book Divine Proportions: Rational Trigonometry to Universal Geometry (2005) which lays out the foundations of this exciting new development is available at https://www.wildegg.com/ in either e-format or as a physical hrad copy. At this website you can also find screenshot pdf's for many of my video series!

Find a lot more videos about this two dimensional number theoretic approach to important families of orthogonal polynomials in the Members section of the channel at the playlist:

Advice for Mathematics Exploration (for Members only):    • Advice for mathematics exploration (M...  


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Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...

My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.

Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!

If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated.

Here are the Wild Egg Maths Playlists (some available only to Members!)

   • Algebraic Calculus One  
   • Classical to Quantum (Members only)  
   • Algebraic Calculus Two  
   • Advice for mathematics exploration (M...  
   • Solving Polynomial Equations (Members...  
   • The Hexagrammum Mysticum (Members only)  
   • Algebraic Calculus and dCB curves (Me...  
   • Dynamics on Graphs (Members only)  
   • Плейлист  
   • Triangle Geometry  
   • Explorations with q-series (Members o...  
   • Six: An elementary course in Pure Mat...  
   • Maxel Inverses and Orthogonal Polynom...  
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Here are the Insights into Mathematics Playlists:

   • MathHistory: A course in the History ...  
   • WildTrig: Intro to Rational Trigonometry  
   • Math Foundations  
   • Wild Linear Algebra  
   • Famous Math Problems  
   • Box Arithmetic  
   • Elementary Mathematics (K-6) Explained  
   • Year9Maths  
   • Ancient Mathematics  
   • Wild West Banking  
   • Sociology and Pure Mathematics  
   • Sociology and Pure Physics  
   • Old Babylonian Mathematics  
   • Probability and Statistics: an introd...  
   • Boole's Logic and Circuit Analysis  
   • Universal Hyperbolic Geometry  
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