Wildberger N.J. Divine Proportions: Rational Trigonometry to Universal Geometry

Sydney: Wild Egg Books, 2005. - 320 p.This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. It shows how to replace transcendental trig functions with high school arithmetic and algebra to dramatically simplify the subject, increase accuracy in practical problems, and allow metrical geometry to be systematically developed over a general field. This new theory brings together geometry, algebra and number theory and sets out new directions for algebraic geometry, combinatorics, special functions and computer graphics. The treatment is careful and precise, with over one hundred theorems and 170 diagrams, and is meant for a mathematically mature audience. Gifted high school students will find most of the material accessible, although a few chapters require calculus. 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.PreliminariesOverview
Introducing quadrance and spread
Laws of rational trigonometry
Why classical trigonometry is hard
Why rational trigonometry is easier
Comparison example
Ancient Greek triumphs and difficulties
Modern ambiguities
Background
Fields
Proportions
Identities and determinants
Linear equations
Polynomial functions and zeroes
Quadratic equations
Cartesian coordinate geometry
Points and lines
Collinear points and concurrent lines
Parallel and perpendicular lines
Parallels and altitudes
Sides, vertices and triangles
Quadrilaterals
Affine combinations
Perpendicular bisectors
Reflections
Affine transformations
Lineations and reflection sequencesRational trigonometryQuadrance
Quadrances of triangles and quadrilaterals
Triple quad formula
Pythagoras’ theorem
Quadrance to a line
Quadrea
Archimedes’formula
Quadruple quad formula
Spread
Spreads of triangles and quadrilaterals
Cross
Twist
Ratio theorems
Complementary spreads
Spread law
Cross law
Spreads in coordinates
Vertex bisectors
Triple spread formula
Triple spread formula
Triple cross formula
Triple twist formula
Equal spreads
Spread reflection theorem
Examples using different fields
Quadruple spread formula
Spread polynomials
Combining equal spreads
Spread polynomials
Special cases
Explicit formulas
Orthogonality
Composition of spread polynomials
Cross polynomials
Oriented triangles and turns
Oriented sides, vertices and triangles
Turns of oriented vertices
Signed areasUniversal GeometryTriangles
Isosceles triangles
Equilateral triangles
Right triangles
Congruent and similar triangles
Solving triangles
Laws of proportion
Triangle proportions
Quadrilateral proportions
Two struts theorem
Stewart’s theorem
Median quadrance and spread
Menelaus’and Ceva’stheorems
Centers of triangles
Perpendicular bisectors and circumcenter
Formulas for the circumcenter
Altitudes and orthocenter
Formulas for the orthocenter
Incenters
Isometries
Translations, rotations, reflections
Classifying isometries
Regular stars and polygons
Regular stars
Order three stars
Order five stars
Order seven stars
Regular polygons
Conics
Centers of conics
Circles and ribbons
Parabolas
Quadrolas
Grammolas
Intersections with lines
Geometry of circles
Diameters and chords
Spreads in a circle
Parametrizing circles
Quadrilaterals
Cyclic quadrilaterals
Circumquadrance formula
Cyclic quadrilateral quadrea
Ptolemy’s theorem
Four point relation
Euler line and nine point circle
Euler line
Nine point circle
Tangent lines and tangent conics
Translates and Taylor conics
Tangent lines
Higher order curves and tangents
Folium of Descartes
Lemniscate of Bernoulli
IV Applications
Triangle spread rules
Spread ruler
Line segments,rays and sectors
Acute and obtuse sectors
Acute and obtuse triangles
Triangle spread rules
Two dimensional problems
Harmonic relation
Overlapping triangles
Eyeball theorem
Quadrilateral problem
Three dimensional problems
Planes
Boxes
Pyramids
Wedges
Three dimensional Pythagoras’ theorem
Pagoda and seven-fold symmetry
Physics applications
Projectile motion
Algebraic dynamics
Snell’s law
Lorentzian addition of velocities
Surveying
Height of object with vertical face
Height of object with inaccessible base
Height of a raised object
Regiomontanus’problem
Height from three spreads
Vertical and horizontal spreads
Spreads over a right triangle
Spherical analogue of Pythagoras’ theorem
Resection and Hansen’s problem
Snellius-Pothenot problem
Hansen’s problem
Platonic solids
Tetrahedron
Cube
Octahedron
Icosahedron
Dodecahedron
Rational spherical coordinates
Polar spread and quadrance
Evaluating pi^/
Beta function
Rational spherical coordinates
Surface measure on a sphere
Four dimensional rational spherical coordinatesRational polar equations of curves
Ellipson
Theorems with pages and Important Functions