Reading and Some on-line Resources
- A. D. Gardiner, C. J. Bradley, Plane Euclidean Geometry, UKMT, Leeds 2012.
- M. J. Greenberg,
Euclidean and Non-Euclidean Geometries,
San Francisco: W. H. Freeman, 2008.
- V. V. Prasolov, V. M. Tikhomirov, Geometry, American Maths. Soc., 2001.
- A. B. Sossinsky, Geometries,Providence, RI : American Mathematical Soc. 2012.
- H. S. M. Coxeter, Introduction to Geometry, Wiley, published 1963.
- J. W. Anderson, Hyperbolic Geometry, Springer Undergraduate Mathematics Series, 1999.
- E. Rees, Notes on Geometry, Universitext, Springer, 2004.
- P. M. Neumann, G. A. Stoy, E. C. Thompson, Groups and Geometry, Oxford University Press, 1994.
- A. F. Beardon, Algebra and Geometry, Cambridge University Press, 2005.
- E. B. Vinberg (Ed.), Geometry II .
(Chapter 3 of Part I: Geometry of Spaces of Constant Curvature ), Encyclopaedia of Mathematical Sciences, Vol. 29, Springer-Verlag.
- S. Katok, Fuchsian Groups, University of Chicago Press, 1992.
Books and Lecture Notes available on-line:
- Caroline Series, Hyperbolic geometry. Lecture notes (2008).
- Anton Petrunin, Euclidean plane and its relatives. A minimalist introduction, Lecture notes (2020).
- Charles Walkden, Hyperbolic geometry, Lecture notes (2019).
- J. W. Cannon, W. J. Floyd, R. Kenyon, W. R. Parry, Hyperbolic geometry. MSRI Publications, Volume 31 (1997).
- Danny Calegary, Chapter 2: Hyperbolic Geometry}of a forthcoming book on 3-manifolds (2015).
- T. K. Carne, Geometry and groups. Lecture notes.
- Michail Gromov, Sign and Geometric meaning of curvature, IHES (1990).
- Nigel Hitchin, Projective Geometry, Lecture notes. Chapters 1, 2, 3, 4.
- Michael Kapovich, Hyperbolic Manifolds and Discrete Groups. Lectures on Thurston's hyperbolization. (1994).
- Wolfgang Meyer, Toponogov's Theorem and Applications, Lecture Notes, (1989).
- William Thurston, The geometry and topology of three-manifolds. Lecture notes (1979).
- John R. Parker, Hyperbolic spaces. Hyperbolic spaces via quaternionic, Clifford and and p-adic Möbius transformations.
- Norbert Peyerimhoff, Geometry, Lecture notes (Euclidean, affine and projective geometries).
- Mark Lackenby, Hyperbolic Manifolds. Lecture notes, 2000. Continues
- Alexander Barvinok, Combinatorics of polytopes, Lecture notes, 2010. ----
- Bruno Martelli, An Introduction to Geometric Topology, version 3 (2023).
- Oleg Viro, "Defining relations for reflections. I. Good source for understanding the role of reflections in the isometry group.
- Oleg Viro, Biflippers and head to tail composition rules. A new graphical calculus for operating with isometries.
- Richard Evan Schwartz, Serge Tabachnikov, Elementary Surprises in Projective Geometry.
- Kostiantyn Drach, Richard Evan Schwartz, A Hyperbolic View of the Seven Circles Theorem.
- Athanase Papadopoulos, Weixu Su, On hyperbolic analogues of some classical theorems in spherical geometry.
- Tom Davis, Inversion in a circle. An article showing how to use inversions (but not giving the proofs of basic properties of inversion).
- Jeff Weeks, Non-Euclidean billiards in VR, Bridges 2020 Conference Proceedings.
- William Thurston, On Proof and Progress in Mathematics.
- Roger C. Alperin, Barry Hayes, Robert J. Lang, Folding the hyperbolic crane. See p.14 for the picture!
More on-line Reading:
Webpages, websites, portals:
- Euclid's "Elements" complete text with all proofs, with illustration in Geometry Java applet, website by David E. Joyce.
- Geometry on "cut-the-knot" portal ---- Theorems, problems and
puzzles in Euclidean geometry (often with hands-on pictures!), by Alexander Bogomolny.
- Geometry Website with supporting materials to a course in Euclidean, inversive and projective geometries, by Vladimir V. Kisil.
- Mathematical Imagery Galleries on different aspects of geometry (by Jos Leys).
- Hyperbolic tesselations, webpage by David E. Joyce (includes some printable tilings).
- Introduction to hyperbolic geometry, by the Institute for Figuring. Webpage including hyperbolic soccer ball and crochet models.
- Illustrating Mathematics: webpage including illustrations of Thurston’s eight geometries, fractals, Penrose tilings, pantograph, etc... , by Rémi Coulon.
- Software visualising geometry: Tesselations, games, including "Rubik's cube" on surfaces, by Roice Nelson.
- Hyperbolic Tesellations by Don Hatch.
- Tiling page and Hyperbolic geometry page on The Geometry Junkyard by David Eppstein.
- Geometry and the Imagination ---- Handouts from the workshop led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston in Minneapolis, 1991. Lets of activities with scissors, cabbage, kale, flashlights, sewing, and polyhedra.
- Why slicing cone gives an ellipse ---- Video on Grant Sanderson's YouTube channel 3Blue1Brown.
- Möbius tranformations revealed, 2-min video on Möbius transformations and stereographic projections, by Douglas Arnold and Jonathan Rogness.
- Loxodromic transformation, in the page by Paul Nylander.
- Animation demonstrating Inversion in circles, by Malin Christersson.
- 1-minute video illustrating stereographic projection, by Henry Segerman.
- Playing Sports in Hyperbolic Space, Dick Canary in a
Numberphile video (by Brady Haran).
- Illuminating hyperbolic geometry, Short video (4:25 min) by Henry Segerman and Saul Schleimer on projecting the hemisphere model to Klein disc, Poincare disc and upper half-plane.
- Spherical Droste video, Short video (1:56 min) by Henry Segerman. (Notice that you can move the frame while viewing this Droste effect in spherical space!)
- Video, comparing spherical, Euclidean and hyperbolic geometry.
- Hyperbolic tiling
, 1 minute animation by
- Mathematical Etudes 2-min colorful films on geometry (and many more activities and visualisations with explanations in Russian).
- Applet for creating hyperbolic drawings in Poincar\'e disc.
- Applet for creating hyperbolic tesselations from your pictures, by Malin Christersson.
- Inversion Tool, hands-on demonstration of inversion on cut-the-knot portal.
- Spherical geometry Applet by David Little.
- Hyperbolic geometry Poincare disc, Applet by David Little.
- moebinv Symbolic, numeric and graphic manipulations in Non-Eclidean geometry, C++ libraries by Vladimir V. Kisil (and others).
- Not Knot, a short film on Thurston’s Geometrization Conjecture, by the Geometry Centre in U. of Minnesota.
- Jos Leys, Étienne Ghys, Aurélien Alvarez, Dimensions (9 short films exploring geometry in various spaces).