## Divine Proportions: Rational Trigonometry to Universal Geometry

This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. Rational trigonometry dramatically simplifies the subject by replacing transcendental trig functions with high school arithmetic and algebra to increase accuracy in practical problems, speed up computations, and allows metrical geometry to be systematically developed over a general field. This new theory brings together geometry, algebra and number theory and brings new ideas to algebraic and differential geometry, combinatorics and special functions.
The treatment is careful and precise, with over one hundred theorems, 170 diagrams, and two appendices. Applications include surveying and engineering problems, Platonic solids, spherical and cylindrical coordinate systems, and selected physics problems, such as projectile motion and Snell's law. Examples over finite fields are also included. Some calculus appears in the latter part of the book. ISBN: 0-9757492-0-X Wild Egg books, Sydney 2005 Order it from Wild Egg books here. It is also available at amazon.com. |

The book's content is largely elementary, but is presented concisely. It requires some mathematical maturity and skill at algebraic manipulation, along with an interest in geometry and its applications. It will be valuable to

- Undergraduate mathematics or physics majors
- Professional mathematicians, especially those with an interest in geometry, number theory, combinatorics and special functions
- Scientists with an interest in mathematics, i.e. physicists, chemists
- Engineers and some computer scientists interested in graphics
- Mathematically talented high school students and their teachers
- Amateur mathematicians

- Introduction and overview chapter motivate and explain the key features of the new theory, along with some of the weaknesses of traditional trigonometry
- Preliminary chapter provides review of necessary background material
- More than one hundred theorems (most of them new) over four parts
- More than one hundred and seventy illustrative diagrams
- Index with separate table of theorems
- Two appendices.

- Two dimensional geometry problems
- Surveying problems (Elementary)
- Surveying problems (Advanced)- Resection and Hansen's problems
- Boxes, pyramids, wedges and pagodas
- Physics---Projectile motion, Snell's law, Algebraic dynamics, Lorentzian addition of velocities (as in Einstein's special theory of relativity)
- Platonic solids
- Folium of Descartes and Bernoulli's lemniscate
- Five fold symmetry and regular polygons
- Calculus problems involving rotational and spherical symmetry (volumes, surface areas, centroids, moments of inertia of spheres, caps, hyperboloids, toroidal rings etc.)
- New formulas for classical curves using rational polar coordinates.

- Two dimensional geometry problems
- Surveying problems (Elementary)
- Surveying problems (Advanced)- Resection and Hansen's problems
- Boxes, pyramids, wedges and pagodas
- Physics---Projectile motion, Snell's law, Algebraic dynamics, Lorentzian addition of velocities (as in Einstein's special theory of relativity)
- Platonic solids
- Folium of Descartes and Bernoulli's lemniscate
- Five fold symmetry and regular polygons
- Calculus problems involving rotational and spherical symmetry (volumes, surface areas, centroids, moments of inertia of spheres, caps, hyperboloids, toroidal rings etc.)
- New formulas for classical curves using rational polar coordinates.

- Introduction and overview chapter motivate and explain the key features of the new theory, along with some of the weaknesses of traditional trigonometry
- Preliminary chapter provides review of necessary background material
- More than one hundred theorems (most of them new) over four parts
- More than one hundred and seventy illustrative diagrams
- Index with separate table of theorems
- Two appendices.