divine proportions: rational trigonometry wildberger [pdf]
Rational Trigonometry to Universal Geometry by Norman Wildberger
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About this book :-
Rational Trigonometry to Universal Geometry written by
This text introduces a remarkable new approach to trigonometry and Euclidean geometry, with dramatic implications for mathematics teaching, industrial applications and the direction of mathematical research in geometry.
The key insight is that geometry is a quadratic subject. So rational trigonometry replaces the quasi-linear notions of distance and angle with the related, but more elementary, quadratic concepts of quadrance and spread, thus allowing the development of Euclidean geometry over any field. This text covers the key definitions and results of this new theory in a systematic way, along with many applications including Platonic solids, projectile motion, Snell’s law, the problems of Snellius-Pothenot and Hansen, and three-dimensional volumes and surface areas.
Book Detail :-
Title: Rational Trigonometry to Universal Geometry
Author(s): Norman John Wildberger
Publisher: Wild Egg Books
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About Author :-
The author Norman John Wildberger is Canadian Australian mathematician. He is an Associate Professor in mathematics at University of New South Wales (UNSW) in Sydney Australia. He has also taught at Stanford University and the University of Toronto.
He earned his Ph.D. from Yale University in 1984. His main research interests are Lie theory, representation theory and hypergroups but he has also done work in number theory, combinatorics and mathematical physics.
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Book Contents :-
Rational Trigonometry to Universal Geometry written by
cover the following topics.
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
Fields, Proportions, 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
Affine transformations, Lineations and reflection sequences
Quadrances of triangles and quadrilaterals, Triple quad formula, Pythagoras’ theorem, Quadrance to a line, Quadrea, Archimedes’ formula, Quadruple quad formula
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
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 areas
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’s theorems
Centers of triangles
Perpendicular bisectors and circumcenter, Formulas for the circumcenter, Altitudes and orthocenter, Formulas for the orthocenter, Incenters
Translations, rotations, reflections, Classifying isometries
Regular stars and polygons
Regular stars, Order three stars, Order five stars, Order seven stars, Regular polygons
Centers of conics, Circles and ribbons, Parabolas, Quadrolas, GrammolasIntersections with lines
Geometry of circles
Diameters and chords, Spreads in a circle, Parametrizing circles
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
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
Projectile motion, Algebraic dynamics, Snell’s law, Lorentzian addition of velocities
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
Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron
Rational spherical coordinates
Polar spread and quadrance, Evaluating π2/16, Beta function, Rational spherical coordinates, Surface measure on a sphere, Four dimensional rational spherical coordinates, Conclusion
Rational polar equations of curves
Theorems with pages and Important Functions
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