source: njwildberger 2017年9月23日
WildTrig: Intro to Rational Trigonometry:
An introduction to Rational Trigonometry and Universal Geometry: simpler and more powerful for calculations, easier to learn, more general, and a richer theory of Euclidean geometry leading to many new discoveries. Also the basis for Universal Hyperbolic Geometry. This series is the first one on my channel, and in some sense the motivation for a lot of the direction of the future of mathematics, in my opinion. With rational trigonometry, we learn that we can do lots of things algebraically that formerly required transcendental functions and irrationalities. Not only liberating, this is also empowering, as a lot of calculations run now faster and more smoothly. And all of metrical geometry gets recast in a much more elegant and general framework.
8:38 0: An Invitation to Geometry: the WildTrig series
9:48 1: Why Trig is Hard
9:59 2: Quadrance via Pythagoras and Archimedes
9:21 3: Spread, angles and astronomy
9:49 4: Five main laws of rational trigonometry
9:28 5: Applications of rational trigonometry
8:37 6: Heron's formula viewed rationally
6:33 7: Solving triangles with rational trigonometry
9:56 8: Centers of triangles with rational trigonometry
8:39 9: The laws of proportion for a triangle
8:37 10: Geometry of circles with rational trigonometry
9:52 11: Applications of rational trig to surveying (I)
9:20 12: Cartesian coordinates and geometry
9:05 13: Why spreads are better than angles
9:46 14: Rational parameters for circles
10:17 15: Complex numbers and rotations
7:19 16: Rational Trigonometry Quiz 1
10:01 17: Rational trigonometry: Solutions to Quiz 1
9:34 18: Medians, altitudes and vertex bisectors
10:01 19: Trigonometry with finite fields (I)
10:02 20: Trigonometry with Finite Fields (II)
10:00 21: Trigonometry with Finite Fields (III)
9:36 22: Highlights from triangle geometry (I)
8:12 23: Highlights from triangle geometry (II)
8:09 24: Spread polynomials
9:58 25: Pentagons and five-fold symmetry
8:35 26: Applications of rational trig to surveying (II)
9:05 27: Stewart's theorem
9:07 28: What size ladder fits around a corner?
8:10 29: Trisecting angles and Hadley's theorem
9:57 30: Polar coordinates and rational trigonometry
7:22 31: Introduction to Projective Geometry
9:51 32: Projective geometry and perspective
7:57 33: Projective geometry and homogeneous coordinates
8:19 34: Lines and planes in projective geometry
10:10 35: Affine geometry and barycentric coordinates
10:05 36: Affine geometry and vectors
10:11 37: The cross ratio
8:47 38: More about the cross ratio
9:22 39: Harmonic ranges and pencils
9:17 40: The fundamental theorem of projective geometry
8:46 41: Conics via projective geometry
9:01 42: An algebraic framework for rational trigonometry (I)
10:02 43: An algebraic framework for rational trigonometry (II)
6:34 44: How to learn mathematics
9:50 45: Einstein's special relativity: an introduction
10:03 46: Red geometry (I)
10:00 47: Red geometry (II)
8:35 48: Red geometry (III)
9:45 49: Circles in red geometry
9:47 50: Green geometry (I)
8:50 51: Green geometry (II)
8:36 52: Pythagorean triples
9:41 53: An introduction to chromogeometry
8:19 54: Chromogeometry and Euler lines
7:37 55: Chromogeometry and the Omega triangle
7:54 56: Chromogeometry and nine-point circles
9:58 57: Proofs in chromogeometry
9:49 58: Triangle spread rules
9:58 59: Triangle spread rules in action
9:17 60: Acute and obtuse triangles
8:03 61: Proofs of the Triangle spread rules
5:46 62: Rational trigonometry Quiz #2
9:35 63: Hints for solutions to Quiz #2
8:11 64: The 6-7-8 triangle
9:36 65: Barycentric coordinates and the 6-7-8 triangle
9:39 66: Squares in a pentagon
9:41 67: Trisecting a right triangle
9:56 68: Euler's Four Point Relation
9:56 69: What is geometry really about?
10:02 70: Determinants in geometry (I)
10:05 71: Determinants in geometry (II)
19:23 72 NEW: Determinants in Geometry (III)
19:26 73: Spreads, determinants and chromogeometry I
20:26 74: Spreads, determinants and chromogeometry II
19:54 75: Spreads, determinants and chromogeometry III
28:52 76: Coloured spreads and generalizations I
26:56 77: Coloured spreads and generalizations II
23:27 78: Geometry with a general dot product
30:12 79: The general rational laws of trigonometry
28:07 81: Rheticus and 17th century trig tables
23:19 82: Maths Education and Rational Trigonometry II
30:54 83: Maths Education and Rational Trigonometry III
22:00 84: Rational trigonometry and mathematics education IV
24:25 85: The true role of the circular functions
28:30 86: Understanding uniform motion: are radians really necessary?
