Towards a More Computational Mathematics - Rational Trigonometry and New Foundations for Geometry

Explore a revolutionary approach to mathematics in this 56-minute seminar that introduces rational trigonometry as a more computational alternative to traditional methods. Delve into the fundamental laws of rational trigonometry using elementary linear algebra, including the Cross law, Spread law, and Triple spread formula. Discover Paul Miller's spread protractor and examine examples from the Zome construction system. Investigate spread polynomials and their fascinating properties, along with quadruple quad and quadruple spread formulas. Learn about projective rational trigonometry and its applications in three-dimensional geometry, including the study of tetrahedra. Gain insights into a new mathematical framework that replaces real numbers with rational numbers, offering a more solid foundation for geometric calculations and understanding.

Intro to Rational Trigonometry
Two key examples
Outline of talk
Quadrance between points
Pythagoras and Triple quad formula
Spread between lines
Spread as a normalized squared determinant
Paul Miller's spread protractor
Laws of affine rational trigonometry
Thales' theorem
The ZOME construction system
ZOME and spreads
Two coloured primitive ZOME triangles
Three coloured primitive ZOME triangles
Proofs of main laws: Cross law
Proofs of spread law, and quadrea
Proof of Triple spread formula
Equal spreads and the logistic map
Spread polynomials
Formulas for spread polynomials
Examples of spread polynomials
Factorization of spread polynomials
Quadruple quad formula
Quadruple spread formula
Cyclic quadrilaterals
Application to a right tetrahedron
Projective Pythagoras theorem
Projective rational trigonometry
A projective triangle
Projective rational trigonometry
Projective quadrea
Solid geometry and tetrahedra
The regular tetrahedron