9/2 Introduction, syllabus, course policies; Euclid's elements; straight edge and compass constructions; constructing an equilateral triangle; which reguar n-gons are constructible?; bisecting a line segment and an angle. References: Textbook, 1.1 - 1.3, https://mathcs.clarku.edu/~djoyce/java/elements/elements.html
9/4 Precise formulas for which reguar n-gons are constructible; video of the construction of the 17-gon; addition and subtraction of line segment lengths; constructions of perpendicuars and parallels. References: Textbook, 1.1 - 1.3, https://en.wikipedia.org/wiki/Constructible_polygon, https://www.youtube.com/watch?v=zpFoVjYbQo8, https://mathcs.clarku.edu/~djoyce/java/elements/elements.html
9/9 Division of a line segment into n equal parts. Thales' theorem. Multiplication and division of lengths; the unit element. The rational numbers. References: Textbook, 1.3 - 1.4
9/11 Similar triangles. The diagonal of the unit square has length \sqrt{2}; irrationality of \sqrt{2}. Irrational numbers. Euclid's parallel postulate and Playfair's axiom. References: Textbook, 1.5, 2.1. https://en.wikipedia.org/wiki/Irrational_number#Open_questions
9/16 The sum of angles in a triangle. Congruent triangles. SAS, SSS, ASA. The Isoceles triangle theorem. Opposite sides of a parallelogram are equal. Reference: Textbook, 2.1 - 2.2.
9/18 Geometric algebra: the square of a sum. The area of a parallelogram. The area of a triangle. The Pythagorean theorem. Reference: Textbook, 2.3 - 2.5
9/23: Proof of the Thales theorem using areas, invariance of angles in a circle, alternative proof of the Pythagorean theorem using similarity of triangles. Reference: Textbook, 2.6 - 2.8
9/25: Constructibility of square roots, constructibility of the regular pentagon, the Cartesian coordinate system, equations of straight lines and circles. Reference: Textbook 2.8, 3.1 - 3.3
9/30: Equation of the equidistant line between two points, intersections of lines and circles, a length is constructible if and only if it is given by rational operations and square roots, angle between two lines, rotations and isometries. Reference: Textbook 3.3 - 3.6
10/2 Proof of the Three Reflections Theorem. An isometry is determined by the images of three points. Introduction to vectors. Reference: Textbook, 3.7, 4.1.
10/7 The geometry of vectors: addition and scalar multiplication. The line through a vector. Linear independence. Line segments in vector notation. The relative direction between vectors. Vector proof of Thales' Theorem. Midpoints and the center of mass. The medians of a triangle pass through the centroid. Reference: Textbook, 4.1 - 4.3.
10/9 The inner product: lengths, angles. The altitudes of a triangle intersect. Inner product and cosine. Cauchy-Schwartz and the triangle inequality. The inner proudct in higher dimensions. Reference: Textbook, 4.4 - 4.6.
10/16 Rotations as matrices. Multiplication of matrices and trigonometric identities. Practicalities regarding the midterm. (Complex numbers and the plane, though not tested.) Reference: Textbook, 4.7
10/21 MIDTERM 1
10/23 Perspective. Axioms for projective geometry. A realization of the axioms through the projective plane. Homogeneous coordinates. Other fields, other models of projective geometry. Reference: Textbook, 5.1, 5.3, 5.4
10/28 More about other models of projective geometry. The projective plane over the field of two elements. Projection; the case of lines. Reference: Textbook, 5.5, 5.9
10/30 Projection onto parallel lines. Projection onto perpendicular lines. Project from infinity. Projection onto arbitrary configurations of lines. Linear fractional transformations. Reference: Textbook, 5.5, 5.6
11/4 The cross-ratio. Invariance under fractional linear transformations. A visual example. Analogies with isometries of the plane. A point is determined by its cross ratio with 3 other points. There is a unique fractional linear transformation sending three fixed points to three other fixed points. Fractional linear transformations are characterized by preserving the cross ratio. Reference: Textbook, 5.7 and 5.8
11/6 Spherical geomtetry. Lines, triangles, and angles. Spherical isometries. Reference: Textbook, 7.4. For extra review with computing angles between curves, this youtube video: https://www.youtube.com/watch?v=ZbliSSiDVWc
11/11 Some interesting formulas/results (but no proofs) from spherical geometry; SAS, SSS, ASA. AAA! The sum of the angles in a spherical triangle. The spherical Pythagorean theorem. Introduction to hyperbolic geometry. Any two points are connected by a "line". Textbook 8.1 and https://www.math.purdue.edu/~arapura/460/spherical.pdf
11/13 The failure of the parallel postulate. Extending linear fractional transformations to the complex plane. Isometries take lines to lines, beginning of the proof. Rewriting the equations for our lines. Reference: Textbook, 8.1, 8.2, 8.4
11/18 Isometries take lines to lines, completion of the proof. Mobius transformations. Reference: Textbook, 8.4
11/20 More detalis about why Mobius transformations preserve angle. Hyperbolic distance. Reference: Textbook, 8.5, 8.6