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,
. Lecture notes (2008).*Hyperbolic geometry* - Anton Petrunin,
, Lecture notes (2020).*Euclidean plane and its relatives. A minimalist introduction* - Charles Walkden,
, Lecture notes (2019).*Hyperbolic geometry* - J. W. Cannon, W. J. Floyd, R. Kenyon, W. R. Parry,
. MSRI Publications, Volume 31 (1997).*Hyperbolic geometry* - Danny Calegary,
__Chapter 2: Hyperbolic Geometry}__of a forthcoming book on 3-manifolds (2015). - T. K. Carne,
. Lecture notes.*Geometry and groups* - Michail Gromov,
, IHES (1990).*Sign and Geometric meaning of curvature* - Nigel Hitchin,
*Projective Geometry*, Lecture notes. Chapters__1__,__2__,__3__,__4__. - Michael Kapovich,
. (1994).*Hyperbolic Manifolds and Discrete Groups. Lectures on Thurston's hyperbolization* - Wolfgang Meyer,
, Lecture Notes, (1989).*Toponogov's Theorem and Applications* - William Thurston,
. Lecture notes (1979).*The geometry and topology of three-manifolds* - John R. Parker,
*Hyperbolic spaces* - Norbert Peyerimhoff,
, Lecture notes (Euclidean, affine and projective geometries).*Geometry* - Mark Lackenby,
. Lecture notes, 2000. Continues*Hyperbolic Manifolds*__here__. - Alexander Barvinok,
, Lecture notes, 2010. ----*Combinatorics of polytopes* - Bruno Martelli,
, version 3 (2023).*An Introduction to Geometric Topology*

__ Papers:__

- Oleg Viro, "
. Good source for understanding the role of reflections in the isometry group.*Defining relations for reflections. I* - Oleg Viro,
. A new graphical calculus for operating with isometries.*Biflippers and head to tail composition rules* - 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,
. An article showing how to use inversions (but not giving the proofs of basic properties of inversion).*Inversion in a circle* - Jeff Weeks,
, Bridges 2020 Conference Proceedings.*Non-Euclidean billiards in VR* - William Thurston,
.*On Proof and Progress in Mathematics* - Roger C. Alperin, Barry Hayes, Robert J. Lang,
. See p.14 for the picture!*Folding the hyperbolic crane*

__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__onby David Eppstein.**The Geometry Junkyard**__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).