Geometry

Reading and Some on-line Resources

Library books:

• 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 here.
• Alexander Barvinok, Combinatorics of polytopes, Lecture notes, 2010. ----
• Bruno Martelli, An Introduction to Geometric Topology, version 3 (2023).

Papers:

• 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:

• Axioms and Postulates: Euclid, Hilbert and many more... ---- Overview of axiomatics: Euclid, Hilbert. Kant's ideas of space in time, Ensteins' relativity (Warning: Euclid and Hilbert is maths, but Kant and Einstein is philosophy, read at your own risk!)
• "History of Non-Euclidean Geometry" ---- In 25 slides.
• Circle inversion, Nice illustrated introduction (with proofs) by Malin Christersson.
• How to use a square and two nails to draw a circle ----
• R. Nelson, H. Segerman, Visualising Hyperbolic Honeycombs.

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.
• Geometrikon gallery of Topics in Geometry, website by Paris Pamfilos.
• 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.

Videos:

• 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 Vladimir Bulatov.
• Mathematical Etudes 2-min colorful films on geometry (and many more activities and visualisations with explanations in Russian).

Software:

• Applet for creating hyperbolic drawings in Poincare disc.
• Applet for creating hyperbolic tesselations from your pictures, by Malin Christersson.
• Inversion Tool, hands-on demonstration of inversion on cut-the-knot portal.
• Cayley Graph Generator Online group visualiser by Jean-Baptiste Bellynck.
• moebinv Symbolic, numeric and graphic manipulations in Non-Eclidean geometry, C++ libraries by Vladimir V. Kisil (and others).

Artwork:

• M. C. Escher, official website.
• Polyhedra and Art. Webpage by George W. Hart.
• Polyhedral sculptures by George W. Hart.
• Stereographic projection and models for hyperbolic geometry. By Henry Segerman. (3-D toys: move the source of light to get different models)
• Tilings by (and after) Escher. Webpage on Mathematical Imagery by Jos Leys.
• Hyperbolic Geometry Artworks, by Paul Nylander.
• How to create repeating hyperbolic patterns, by Douglas Dunham (based on Escher's patterns). See also here
• D. Taimina, ``Crocheting Adventures with Hyperbolic Planes''. Published by A K Peters (2009).

Games:

• Zeno Rogue webpage.
• Hyperbolica, Non-Euclidean adventure games.
• H2Snake, a game of snake on a hyperbolic surface of genus 2.

Films:

• 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).