Geometry III/IV
2020 - 2021
- V. V. Prasolov, V. M. Tikhomirov, Geometry, American Maths. Soc., 2001.
(the e-book is available on DUO)
- V. V. Prasolov, Non-Euclidean Geometry ---- (the paper copies of this book was distributed in Michaelmas, thanks to the kind permission of the author and the publisher).
Who is who
Preliminary course content (subject to change):
Euclidean geometry, spherical geometry, affine and projective geometries, Möbius transformations, hyperbolic geometry, further topics (geometric surfaces, discrete groups).
Schedule:
- Week 1: Introduction. Axioms: Euclid and Hilbert.
- Week 2: Euclidean geometry: isometry group, its generators, conjugacy classes.
- Week 3: More on Euclidean isometry group: fixed points of isometries, conjugacy classes of isometries, orthogonal transformations as isometries preserving the origin. Discrete groups acting on Euclidean plane. Problems class on geometric constructions and on using reflections for solving problems.
- Week 4: Euclidean geometry in 3 dimensions. Spherical geometry: distance, triangle inequality, geodesics.
- Week 5: Spherical geometry: polar correspondence,
congruence of triangles, sine and cosine rules. Problems class on discrete actions.
- Week 6: Spherical geometry: area of a triangle; isometries on the sphere.
- Week 7: Affine geometry. Projective line. Problems class on spherical geometry.
- Week 8: Projective line and projective plane.
- Week 9:
Classical theorems: Pappus and Desargues. Projective plane: its topology, polarity on
projective plane. Problems class on projective geometry.
- Week 10:
Hyperbolic geometry: Klein disc model (distance, isometries, perpendicular lines).
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- Week 11: Möbius transformations, Inversion.
- Week 12: Inversion.
Möbius transformations and cross-ratios. Inversion in space and stereographic projection.
- Week 13: Stereographic projection. Conformal models of hyperbolic geometry (PoincarĂ© disc). Problems class on Inversions and Möbius geometry.
- Week 14: Isometries of PoincarĂ© disc. Circles in PoincarĂ© disc. Upper half-plane model.
- Week 15: Elementary hyperbolic geometry: sine and cosine rules. Problems class on Poincare disc model.
- Week 16: Area of a triangle. Projective models of hyperbolic geometry: Klein model.
- Week 17: Hyperboloid model.
Types of isometries of hyperbolic plane. Problems class on computations in hyperbolic geometry.
- Week 18: More on isometries. Horocycles and equidistant curves. Discrete reflection groups.
- Week 19: Taming infinities with horocycles. Family of geometries: sphere-plane-hyperbolic plane. More on discrete groups acting on the hyperbolic plane. Problems class: computations in the Klein model.
- Week 20: Hyperbolic surfaces. Review via 3D hyperbolic geometry.
If you have any questions you are very welcome to ask (during the lectures, after a lecture,
during office hours, in any other convinient time or via e-mail)!!!
Assignments:
- There will be 4 sets of marked assignments per term (to submit through Gradescope on Fridays by 5pm, weeks 3,5,7,9 and 14,16,18,20). This will constitute 10% of module credits. --
- There will be also weekly unmarked sets of exercises. Please solve them timely!
- In addition, there will be additional reading material for students that are enrolled in MATH4141 (Geometry IV) and for MSc students, see
here for the instructions.
- Weeks 1-2: ---- Exercises ---- Hints ---- ------ ----- (Isometries of Euclidean plane)
- Weeks 3-4: ---- Exercises ---- Hints ---- ------ ------(Isometries; actions of groups; a bit of spherical geometry)
- Weeks 5-6: ---- Exercises ---- Hints ---- ------ ------(Spherical geometry)
- Weeks 7-8: ---- Exercises ---- Hints ---- ------ -----
(Affine and Projective geometry)
- Weeks 9-10: -. Exercises ---- Hints ---- ------ --------------------- (Projective geometry, Klein model of hyperbolic geometry)
~~~~~~~~MERRY~CHRISTMAS~~~~~AND~~~~~HAPPY~NEW~YEAR~!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- Weeks 11-12: Exercises ---- Hints ---- ------- ------ (Möbius geometry, Inversion)
- Weeks 13-14: Exercises ---- Hints ---- ------- -------(Poincare disc, upper half-plane)
- Weeks 15-16: Exercises ---- Hints ---- ------- -------(Computations in the upper half-plane. Klein disc and hyperboloid models)
- Weeks 17-18: Exercises ---- Hints ---- ------- -------(Classification of isometries. Horocycles and equidistant curves)
Lecture Notes:
Handouts:
- Euclidean geometry: axioms (page 2) and basic theorems (pages 3-4), as well as the contact details on page 1.
Some tables to review the course:
Additional handout:
(not an essential part of the course, it is non-examinable!)
- Here you can find an instruction on how to construct vertices of dodecahedron/icosahedron with spherical ruler and compass.
If you find any mistakes/misprints in lecture notes, solutions or handouts, please let me know - Thanks!