Jun 15
  1. Euclidean and Non-Euclidean Geometry



  1. The Greeks made Space the subject-matter of a science of supreme simplicity and certainty. Out of it grew, in the mind of classical antiquity, the idea of pure science. Geometry became one of the most powerful expressions of that sovereignty of the intellect that inspired the thought of those times. At a later epoch, when the intellectual despotism of the Church, which had been maintained through the Middle Ages, had crumbled, and a wave of skepticism threatened to sweep away all that had seemed most fixed, those who believed in Truth clung to Geometry as to a rock, and it was the highest ideal of every scientist to carry on his science 'more geometrico.' Hermann Weyl

  2. Vectors and Plane Geometry Karl Heinz Dovermann Professor of Mathematics University of
    Hawaii January 27, 2011
  3. Synthetic Plane Projective Geometry notes by R.Street McQuarrie University
  4. PLANE GEOMETRY:AN ELEMENTARY TEXTBOOK BY SHALOSH B. EKHAD, XIV (CIRCA 2050) DOWNLOADED FROM THE FUTURE BY DORON ZEILBERGER Rutgers University  (PG?) This is a very bizarre book written by the creative mind of Doron Zeilberger at Rutgers. Essentially it's an attempt to implement the whole of classical Euclidean geometry as a Maple program, where the program essentially derives all the main results from modules (programming modules, not the algebraic kind) containing the embedded definitions. It has an interesting framing narrative: It's supposed to be a textbook from the year 2050, where all mathematics is done by sentient computers who effectively prove theorems entirely algorithmically. I've heard this idea for years-that human proofs in mathematics will someday be replaced by computer implemented proofs that will allow virtually any logically solvable problem, no  matter how large and complex, to be solved in finite time. The debate on the validity of this prediction is a book in and of itself.  For now,I'll just say although I believe computers can and will have a very important role in mathematics in the future, I'm quite skeptical that scrap paper spade work by mathematicians will become a thing of the past. To be perfectly honest, I don't know enough about Maple to be able  to actually "run" the book for myself to see if the results are instructive.(I'm much more familar with Wolfram's Mathematica. )  So I'll reserve judgement for now. But the idea of a dynamic book on geometry that can be "run" on a PC is quite interesting and I'm glad there are mathematicians like Zeilberger who are trying to actually be creative with something as old-fashioned as the classical geometry course. If you know Maple, then by all means, check it out.
  5. Geometry N.I.Shepherd-Barron Easter 1996 Oxford University  (PG/PG-13) A brief set of notes on spherical and hyperbolic geometry in the plane and space. They're incomplete, but what's here is lucid, concise and very visual. Complete proofs. No exercises, however. so they're of limited use. However, coupled with a good teacher, they can be very useful for a post-linear algebra geometry course. Recommended.
  6. Geometry 2010/11 Simon Salamon Dipartimento di Matematica Politecnico di Torino Corso Duca degli    (PG)This is a strange set of notes for a combined course in linear algebra, classical plane vector geometry and some basic differential geometry. To be honest, I found them rather puzzling and disorganized. There's some nice stuff here about the connection between vector spaces and thier geometric counterparts, like quadratic forms, but there's nothing here that can't be found in the excellent linear algebra notes above or in other post-linear algebra geometry notes. Pass on these.
  7. Plane Geometry by George Wentworth and David Eugene Smith (G) A classical book from 1890(!) now available online  It's basically a standard high school textbook on Euclidean geometry in the plane and three-dimensional space. It's very clear and detailed with lots of pictures, but it's excruciatingly old fashioned-there's really nothing in it you can't find in an old copy of Euclid's Elements or one of a thousand other books written in the last 200 years. You may like that approach-if you do, by all means, help yourself. But frankly, I prefer Giventhal's translation of Kiselev's Geometry if I want an old fashioned book on geometry or one of the other notes here for a modern presentation.
  8. Geometry - Summer 2012 Neal Nelson Evergreen University (PG)These are supplemental, fragmentary lecture notes to accompany a modern geometry course for teachers. They are in bullet-point format and cover the 3 of the classical axiomatic geometries (Euclidean, Spherical and Hyperbolic) with many historical notes and insights, particularly into axiomatics and the role that logic played in the modern formulation of these axiomatic systems, as well as both the analytic and synthetic approaches to these geometries. Be warned that they are not intended as a main course text, they specifically supplement  Gerald Venema's Foundations of Geometry and make many specific references to that textbook for proofs and specific axioms. You certainly could use another standard text in conjunction, but it would be more problematic to use them. Still, lots of nice insights and sidebars by Nelson. It would certainly make a solid study aid for such a course. Recommended.
  9. Geometry  Ken Monks University of Scranton Revised: Fall 2006 (PG)A beautiful,very unusual and remarkably careful set of notes for a university geometry course that doubles as an introduction to mathematical proof. There are an enormous number of careful definitions and axioms for Euclidean and non-Euclidean geometric systems,but theorems are given without proofs. Monks encourages his students to work in teams to prove most of these theorems for homework.  Emphasizes the use of logic in developing axiomatic systems in geometry without the notes turning into a course on logic- which gives much needed practice in a very relevant area of mathematics.Instructors using the notes can decide how much to prove in class and how much to leave to the students depending on the relative strength of those students, giving them enormous flexibility. It can also act as a source of problems or review for students with a background in proof. Either way, the notes are beautifully constructed and students who work through them will learn a great deal even if the instructor proves most results for them. Highly recommended.
  10. Vector Geometry William Greenberg Virginia Tech       (PG)This is an online textbook designed for a "vector geometry" course presuming a course in multivariable calculus that covers vectors in the plane and 3-space, although it doesn't require any background in linear algebra per se.  This is a very accessible and gentle course that presumes very little and develops basic vector algebra very visually.The purpose of this course seems to be to give an introduction to concrete vectors in low dimensions as preparation for either mathematical methods courses for physics and engineering majors or vector analysis and abstract linear algebra for mathematics majors.The contents: Parametric equations in the plane and space, polar, cylindrical and spherical coordinates,complex numbers and their geometry in the plane,vector spaces in low dimensions, cross and dot products, lines and planes in Euclidean space and applications to physics such as projective motion. Personally, I think all this material should-and usually is-covered in a serious vector analysis course and it's kind of silly to present it all without the accompanying calculus.But it's all presented well with good pictures and diagrams. It'll be a very useful and versatile set of notes
  11. for both mathematics and physical science students. Recommended.
  12. Fundamentals of Geometry Oleg A. Belyaev Moscow State University 2007 Homepage    (PG)This deliberately unfinished book is one of the most complete and visually beautiful geometry textbooks I've ever seen. Seriously. Although there are short sections towards the end on various non-Euclidean geometries such as hyperbolic and projective, the vast bulk of the text centers on a complete development of absolute incidence geometry i.e. the general geometry developed via the Hilbert axioms. The development is beautiful and extremely careful and buttressed with many pictures. The main classical geometries-Euclidean, hyperbolic and projective-are then developed as variants of the abstract geometry where very specific axioms are altered.  The result is immensely informative and clear. Although there are no explicit exercises, there are many minor results which are left unproven for the reader to prove from earlier results. Overall, it covers much the same ground as Monks above, but more visually and intuitively. In fact, the union of the 2 notes can serve as a very challenging and thorough course in the foundations of geometry for serious undergraduates. By itself, though, Belyaey is a wonderful source and I highly recommend it for anyone interested in classical geometry from a modern perspective.
  13. Geometry Foundations  Inna Sysoeva University of Pittsburgh SPRING 2013 Course Materials (PG) A bullet point collection of notes pieced together from across the web. Uneven in detail and nothing that can't be gathered better in other sources. Pass.
  14. Foundations of Geometry Bruce Shapiro California State University, Northridge Revised for Spring 2013  (PG)These are the most recent version of the notes that Shapiro has written to teach college geometry for many years at Northridge. They are specifically geared to train aspiring high school teachers in accordance with The National Council of Teachers of Mathematics official recommendation guidelines for the university training of mathematics teachers.Shapiro gives in the first 4 chapters a fascinating summary and commentary on the Council's recommendations and how it shaped his writing of the notes in the Introduction. If teaching matters to you even if you're not interested in geometry per se, I heartily recommend reading his comments.The text proper begins with Chapter 5 with a discussion of basic logic and naive set theory. Chapter 6 gives an axiomatic development of the real numbers similar to the one that usually begins a standard real analysis course.The text then proceeds for the next 7 sections to in depth axiomatic and historical discussions of  the various proposed axiom systems for classical geometry: Euclid's Elements,the Hilbert axioms, the Birkoff-McLane metric axioms, and- in a rather strange inclusion-the The School Mathematics Study Group (SMSG) at Yale University's axioms developed in the 1950's that served as the basis of the famous book by E.E. Moise and Floyd Downs,  the 1983 University of Chicago School Mathematics Project axioms and finally, the hybrid system proposed by Gerald Venema in his aforementioned book. All these axiom systems are used freely and interchanged between in the course of the rest of the book depending on which system, in the author's opinion, gives the clearest and most insightful constructions and proofs.Shapiro then proceeds in the rest of the notes to develop all the classical abstract geometries and thier properties this way with many pictures, historical notes and detailed proofs-incidence geometry,neutral geometry, Euclidean ruler and protractor constructions, hyperbolic geometry, metric geometry and much more. In my opinion, this may be the single best free online text on the foundations of geometry that currently exists. I believe there are sources that match it in quality-but none surpass it. The highest possible recommendation and hopefully it will remain available for free for years to come.
  15. Geometry Claudiu C. Remsing RHODES UNIVERSITY 2006 (PG-13) A very nice, advanced and complete set of lecture notes that develop both classical Euclidean geometry and basic differential geometry in Euclidean space from the standpoint of linear algebra and emphasizing the role of geometric transformations i.e. isometries and similarities. Requires a comfortable background in linear algebra, some real analysis and group theory, so really not appropriate as an introductory course. Curves, vector fields, the Serret-Frenet equations, curvature, submanifolds in Euclidean spaces, the inverse and implicit mapping theorums and a very good introduction to matrix groups, particularly Lie groups.There are also applications to relativity, such as the Lorentz transformation group. This is a sophisticated unifying presentation that shows how the various perspectives on Euclidean space are related and integrated into each other by linear algebra and group theory. Best used by advanced students as a supplement to both classical geometry and differential geometry courses or as a very good second course. Recommended for students with the appropriate background.
  16. Hyperbolic Geometry Charles Walkden Univerity Of Manchester September, 2012(PG-13)
  17. Comprehensive set of notes on hyperbolic geometry that begins with a nice review of Euclidean geometry that establishes the differences between the 2 geometries as relates to the different metrics on the spaces, something a lot of sources on this subject sorely lack. The notes then move on to the traditional topics in the subject: basic Mobius transformations, geodesics in hyperbolic space, the Poincare model, the hyperbolic Gauss-Bonnet Theorem,fixed points, classification of Mobius transformations, a complete discussion of Fuchian groups and much more. Very careful and not too wordy while still containing many  examples and exercises. For an introduction to this important subject to follow up a course on Euclidean geometry or to supplement such a course, you'd be hard pressed to find a better source online. Highly recommended.
  18. Coordinate Geometry Joel W Robbin  University of Wisconsin Tuesday 2005  (PG) An excellent set of notes for an undergraduate course that emphasizes, like Remsing's above, the connection between the classical geometries and linear algebra, but at a much more introductory level and with a complete focus on the classical Euclidean and non-Euclidean geometries. Affine and projective geometries are covered towards the end. Sadly, there are unwritten sections on inversive and Klienian group theoretic geometry,but these are easily found in other sources.The emphasis is on classical Euclidean geometry through linear transformations, although synthetic methods are freely used when the author believes it clarifying.  Very clear and complete, covers several topics not usually seen in the usual undergraduate foundations of geometry course, such as the theorems of Morley, Napoleon and the Fermat Point. Not as complete as some of the other notes here, but still an excellent source for a college geometry course and highly recommended.
  19. Notes on Euclidean Geometry Paul Yiu Florida Atlantic University 2006  (G)A very complete and lucid set of notes for an introductory course in Euclidean geometry that presumes a high school course in geometry. These notes emphasize synthetic and constructable methods and hardly use linear algebra, they are clearly intended to expand greatly a student's command of basic classical geometry building on what's learned in a first course.  Many classical results, such as the Euler line and the nine point circle, the Shoemaker's knife, tangent circles and quadrilateral constructions and much more. It also contains a chapter on complex numbers and how many classical results can be greatly simplified by expressing points as complex numbers and using the arithmetic and  calculus of the complex plane.For those who want to master classical geometry after a first course, this and the follow up notes on the geometry of circles and triangles will provide a wealth of information. Not appropriate for a general course, but a great follow up to deepen students' appreciation of the methods of ancient Greek geometry. Highly recommended.
  20. Geometry of the Triangle Paul Yiu Summer 2001  Florida Atlantic University Version 13.0411 April 2013 (PG) Exactly what the title says it is. A very detailed, visual but very classical development of the Euclidean and non-Euclidean geometry of triangles and circles using both linear transformations and ruler and protractor methods. This gives a host of important results that usually aren't covered in depth in a traditional geometry course for lack of time.Also contains a detailed presentation of the geometry of conic sections in the plane. A perfect follow up course to Yiu's Euclidean geometry notes above, although they aren't necessary as prerequisite. Together, they form as complete a presentation of advanced classical Euclidean geometry without group theory as you're likely to find. Highly recommended for students who have had a first course and a deep interest in classical methods.
  21. Geometry I  notes author Simon Salamon Lecturer: Konstanze Rietsch King's College University of London 2013  (PG) Sophisticated and concise "British" style lecture notes on Euclidean and non-Euclidean geometry from the metric point of view of the Birkoff/ MacLane axioms.Similar to Robbins' notes above,but more complete and concise. Very visual with careful proofs, linear algebra is used throughout to develop the geometry of transformations. Similar triangles, similarity theorems,  the Pythagorean theorem and the parallel postulate, Cartesian coordinate geometry through linear transformations, groups of isometries, spherical and hyperbolic spaces and more. This version of the notes comes complete with the problem sets, giving a complete course. For students who want to learn geometry from a modern point of view quickly and not to get sidetracked with minutia, these notes will be just the ticket.  Highly recommended.
  22. Planar Circle Geometries an Introduction to Moebius–, Laguerre– and Minkowski–planes Erich Hartmann Darmstadt University of Technology (PG-13) A comprehensive and rigorous set of notes on inversive geometry and it's relations to Mobius,Laguerre and Minkowski plane geometry. The notes begin with affine and projective geometry in the plane, then moves on to ovals and conics in the plane. The rest of the notes focus on the essential results of the 3 plane geometries of the title. The notes are very algebraic, focusing on the underlying fields and their linear transformations while still remaining quite visual-it reminded me in some ways of Robin Harsthorne's classic The Foundations of Projective Geometry.  It's a very challenging set of notes, best suited for strong undergraduates or first year graduate students with good command of basic abstract algebra and plane geometry. Still, this is very important material in geometry that serves as good preparation for graduate courses in algebraic geometry. Recommended for the right audience.
  23. CLASSICAL GEOMETRY  DANNY CALEGARI University of Chicago (then at Caltech) 2003 (PG-13) An interesting and advanced set of notes on the classical geometries that emphasize the Klienian approach to geometry via matrix groups of isometries and similarities. Very well written and organized, but uncompromisingly abstract and modern-there are no pictures, but many explicit matrix form group theoretic computations and careful proofs of the main results of Euclidean, hyperbolic, projective and spherical geometries. Additionally, there is material on Coxeter reflection groups and the bare bones of point set topology and smooth manifolds! Think of it as a more advanced, more concise version of Remsing's notes.  A strong set of notes for strong students, but definitely not a beginner's course! Still, a good source for advanced students looking for a modern presentation to deepen their understanding of geometry . Highly recommended.
  24. Classical Geometries Robert Connelly Cornell University Math 452 Spring 2010   (PG) Concise but quite lucid, deep and exercise-driven presentation of the classical geometries with emphasis on both the axiomatic approach andlinear algebra. Rather short on detail and examples, but what examples it does have are very well chosen. It also has quite a few illustrations and historical notes, which help illuminate the presentation greatly.There is an enormous number of exercises, all quite good and not too difficult. If the student is willing to learn actively and draw many ofhis or her own pictures, this can be the basis for a very strong courseindeed. Recommended.
  25. Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD RICHARD KENYON, AND WALTER R. PARRY  Brown University(PG-13) A wonderfully written presentation of the essentials of hyperbolic geometry for mathematics students and professionals who have never learned it and need to understand it for research in topology, geometry or modern physics. It presumes the reader has a good command of both Euclidean geometry, calculus and the basics of topology.  The real strength of these notes is their strong historical context; the authors clearly have enormous command of not only the geometry but the history of the development of the subject and the key examples that motivated it. This gives a deep understanding of concepts through their key examples which is invaluable to a neophyte. A self contained development of the essentials of differentiable manifolds and Riemannian geometry allows the authors to jump directly from the basic axioms of hyperbolic space to modern developments such as hyperbolic geodesics and inversive geometry.  Many pictures and careful proofs further increase the value of this notes. In the past,students learning hyperbolic geometry had to choose between superficial treatments in introductory geometry books and exhaustive treatises that required a second year graduate student background. With these notes,  students now have a very good "middle ground" option that provides all essential facts in an extremely readable and informative form. Highly recommended!!!
  26. Projective Geometry Simon Salamon Oxford University 2001  (PG-13) Detailed and careful set of notes on the essentials of projective geometry assuming linear and abstract algebra and complex numbers. Integrates the algebraic and visual aspects of the projective plane and projective spaces very well. Very good exercises sets. Good for students who want a rapid but detailed presentation. Another very good set of notes from across the pond. Highly recommended.
  27. Euclid's Elements with coloured diagrams by Oliver Byrne, 1847.   (G)Yes, that's correct, the printed version of this book was published in 1847, before the Civil War. It's a striking demonstration of the durability of this work of the Ancient World that it's older editions are now receiving new life on the internet. The colored diagrams and symbols are great addition. Anyone who was educated before the 20th century was expected to have a least read Euclid,if not mastered him. Sadly,it's no longer the case. This version is one of several currently online that can help to rectify this great tragedy of history. Recommended.
  28. Geometry in Low Dimensions Mark Steinberger Math 432-532 University of Alberta 2011 
  29. A course in low-dimensional geometry Mark Steinberger University of Alberta 2014 (PG-13) The first link is to the homepage of the course, which gives exams and commentary on the online text Steinberger has written and continues to draft. As for the notes themselves, linked at the second link, they're the real jewel here. It's a fascinating and beautifully written online text giving a modern development of Euclidean, spherical and hyperbolic geometry using linear algebra, analysis and topology in an "analytic" approach rather than the axiomatic approach. For example, hyperbolic geometry is developed using Poincare's original view of hyperbolic geometry as the plane with a Riemannian rather than the standard Euclidean metric.Steinberger also develops just enough of the theory of smooth manifolds in Euclidean space to discuss spherical and hyperbolic geometry in a totally modern way- this will give additional depth for students interested in differential topology and geometry. This is an extremely powerful approach to classical geometry that isn't nearly as popular as it used to be.Yet, he doesn't neglect the important topics of classical geometry-the Klienian group theoretic approach is also discussed in depth. There are lots of good exercises and diagrams. That being said, because of the high  bar on prerequisites, it's quite a bit more intensive than the average course of this type usually is. While I think good,hard working mathematics majors would generally have no trouble with it, I wonder if the average high school teacher or education student in such a course would have the background needed, let alone be able to keep up. Steinberger also has an annoying tendency to dip into more advanced areas of mathematics without warning, which would further exacerbate this problem. For example, he proves the theory of eigenvalues over a general commutative ring when discussing a change of basis. These quibbles aside- for students with a very strong linear algebra and calculus/ real analysis background, this will be a wonderfully deep and modern course on an important subject. It will also serve as excellent geometric preparation for a graduate course in differential geometry.Highly recommended.
  30. Topics in Geometry Ben H.Smith McGill University  May 2014  (PG-13)Another undergraduate "foundations" course that tries to recast Euclidean geometry in the modern mathematical language of linear algebra and group theory. These focus specifically on Euclidean geometry and the Klienian perspective of groups of transformations. Symmetry is the main theme here and it is illuminated with many examples and beautiful sidebars by the author. They are extremely well written and entertaining, containing many historical notes and intuitive discussions. Where else are you going to find a geometry lecture where the author quote both The Sound of Music and the original Willie Wonka And The Chocolate Factory when discussing Euclidean reductionism? The proofs of the basic results of Euclidean geometry are particularly clear, gentle and inviting, as they are done simultaneously with actively drawing the geometric objects stepwise. Sadly, the original problem sets have been lost at the website. Still, with so much to offer in a relatively short package, the notes are definitely worth downloading. Highly recommended.UPDATE: The most recent version of the course by Smith-2014-has restored the exercises and included full solutions! This means these materials are even more valuable to the serious self-study student then earlier versions. This could change at any second- so download them now! 
  31. Geometry 1 Boris Springborn TECHNISCHE UNIVERSIT AT BERLIN Winter Semester 07/08 (PG-13) This is another modern course on geometry. The notes are nearly entirely about non-Euclidean geometries. This one is pitched at about the same level as Steinberger's above, but it's far less comprehensive and with considerably less detail and with more emphasis on linear algebra and group theory. So it's rather challenging. It's also organized in a  rather unusual fashion. On the other hand, it's very visual with a lot of diagrams to accompany careful proofs and good examples. The exercises are meaty and interesting without being too difficult. A good course for serious mathematics majors to follow up a strong course on Euclidean geometry. Recommended.
  32. Geometry Meighan I. Dillon Southern Polytechnic State  University  November 16, 2009(PG)A much more standard and thorough course in the classical geometries. Uses traditional axiomatic methods and gives careful proofs and clear diagrams of most important results in Euclidean, neutral, hyperbolic and projective geometry. An unusual and welcome content in the notes is a gentle and informed discussion of the Hilbert axioms. A ton of good exercises, well written and straightforward without being too wordy or lengthy. In other words, a fine source for self-study in geometry for mathematics students and even honors students in high school. Highly recommended.
  33. Ideas in Geometry Alison Ahlgren and Bart Snapp University of Illinois Urbana-Champaign 2010 (PG) This is a Open Source undergraduate book on classical geometry in progress, which is available currently under CreativeCommons License. It richly deserves to be a popular textbook for high school honors and undergraduate "foundations of geometry" courses -the title is very appropriate, because this is an introduction to classical geometries motivated by and constructed primarily by ideas  The content is that of a standard undergraduate level course in Euclidean geometry-Euclidean geometry via axioms,non-Euclidean geometry by altering Euclidean axioms, geometric transformations, ruler and compass constructions and more. But the presentation most definitely is not.  Not only is this beautifully written presentation particularly example and problem driven, there are many wonderful historical and visual side discussions and insights.Some examples: The book opens with a detailed overview of Euclid's Elements and the remarkably significant role the text played in the evolution of Western scientific thought and includes many examples of that role not commonly known outside of geometers, such as Eratosthenes' use of the basic results to determine the circumference of the Earth. There are not only many explicit exercises, but many minor facts that are thrown out without proof that the student is encouraged to attempt to prove themselves. There is a very nice and clear discussion of so-called "taxicab geometry"-which is essentially using combinatorial and rectangular coordinate methods to analyze geometric objects that are partitioned into grids. (It occurs to me this is actually a very simple example of a chart imposed on an abstract Euclidean space!) There's a fascinating section on "picture proofs" of classical results on Euclidean geometry i.e. how to take the essential axioms of Euclidean geometry and combine them with detailed drawings of planar geometric objects to convert intuitive diagrams into rigorous proofs. This is askill many mathematics students lack and a careful reading of this chapter and doing it's exercises will assist them greatly in improving that skill. Other wonderful tidbits here are a comparison of analytic, algebraic (i.e. coordinate and formula based) and synthetic geometry, classical constructions like doubling the cube and a final chapter on plane convex sets. This is a jewel and here's hoping it remains online indefinitely for all students and teachers of geometry. Most highly recommended!
  34. Shaping  Modern Mathematics: From One to Many Geometries Raymond Flood Gresham Professor of Geometry (PG)  Very nice,concise lectures on the history of classical geometry from the ancient Greeks to the development of non-Euclidean geometry and it's role in general relativity. The actual lectures delivered by Professor Flood can be found on YouTube-showing  just about anything that's been filmed in human history can be found on the Web these days if you're willing to spend time searching. Worth a look .
  35. .GEOMETRY AND GROUPS Notes T. K. Carne University of Cambridge Michaelmas 2012  (PG-13)
  36.  GEOMETRY Notes T. K.Carne University of Cambridge  Easter 2002 (PG)Professor Carne has taught from and written several versions of the notes for the second year and third year geometry notes at The University of Cambridge over the years. These courses are essentially similar in spirit, but very different in level, emphasis and scope from each other. The first is the 2002 version, which is a modern course in Euclidean and non-Euclidean geometry from a modern point of view with a working knowledge of linear algebra and complex numbers asprerequisites. The prerequisite for this course is in fact the Algebra & Geometry Part I Mathematical Tripos IA  course, of which a version can be found in the Linear/Multilinear Algebra And Matrix Analysis section above here. The contents of the 2002 course:  Isometries as linear transformations,spherical geometry, inversions, the projective plane, the hyperbolic plane, Mobius transformations as conformal maps, tesselations,and finite symmetry groups  The second set of notes is for a third year course that's pitched at a considerably higher level and has substantially greater prerequisites and coverage-namely the abstract algebra or group theory course. The previous course isn't required as a prerequisite, but the course would be far more demanding without it. The second course is also a modern course in the classical geometries, but it vastly expands the group-theoretic perspective and relates the classical geometries to modern algebra and topology.  The contents: The isometry group of Euclidean space and it's component transformations, the Platonic solids, lattices, the crystallographic groups, extended chapters on Mobius transformations and the hyperbolic plane, Fuchian and modular groups,hyperbolic 3 space and its transformations,  3 dimensional Euclidean and hyperbolic isometries, involutions, Klein groups, orbits,Hausdorff dimension and an introduction to fractal geometry and they finish with a discussion of the Schotty groups. Carne shows enormous command of the subject in both lectures and writes very well indeed, giving many examples, good pictures and detailed proofs of great clarity. The exercises for both courses are equally good and can be found at Carne's webpage.  Both of these notes will be excellent sources for astudent looking to learn about the modern perspective on geometry and needs a source that's relatively brief but detailed. Both are very highly recommended.
  37. Modern Geometry D Joyce Clark University Fall 2005  (PG) Very brief but lucid set of notes on classical geometries focusing on classical proofs and isometry groups. Not bad, but really not substantial enough to use as course materials. You can check it out, but really nothing here you can't find in more comprehensive sources listed here. 
  38. Projective Geometry Nigel Hitchens Oxford University 2003  (PG-13) These are Hitchens' lecture notes for the projective geometry course at Oxford and you can see why they've been shamelessly copied by professors all over the United Kingdom and Europe. They are very rigorous and at the same time, very intuitive, concrete and example driven.  Hitchen spends a very large chunk of the notes motivating the need for projective geometry with both real world examples such as perspective drawing and computer graphics and actual geometric constructions such as projection of lines through the origin of the sphere.  The connections between the projective plane, Euclidean space and general vector spaces is also emphasized-the idea of projective transformations being generalizations of linear transformations is a very effective one for a student with a linear algebra background. The connections with linear algebra are the focus throughout the notes and many sophisticated concepts of advanced linear algebra are given visual analogues in projective space. For example, the exterior algebra is introduced in connection with decomposable vectors in the projective plane and the Klein quadratic.The notes end with a discussion of axiomatic geometries and how the parallel postulate can be derived from the axioms of the projective plane. One of the best written, sophisticated and readable introductions I've ever seen to any subject. The highest recommendation possible.
  39. Geometry of Surfaces Nigel Hitchens Oxford University 2013   (PG-13)  Another beautiful set of notes by Hitchens focusing on the deep geometry and topology of abstract surfaces. The prerequisites are again a strong background in linear algebra and some group theory. The topic selection is fairly predictable,such as the combinatorial classification of compact surfaces, the Euler characteristic, orientability, Riemann surfaces and much more. There's considerable overlap with other works on the classical topology/geometry of surfaces, such as John Stillwell's The Geometry of Surfaces and  Lectures on Surfaces by Anatole Katok and Vaughn Climenhaga (and of course, the earlier online draft reviewed earlier and can be found below). But these notes have Hitchens' amazing gift as a geometer and teacher to explain concisely and clearly the intuitive properties of very challenging concepts through pictures and without sacrificing rigor. This allows him to give glimpses into more advanced topics without
  40. details that greatly enhance the insights of the presentation. A good example is his discussion of how the Riemannian metric arises out of the need to rigorize the concept of curve length in a variety of spaces. Another outstanding set of notes by Hitchen and a must read for any student of geometry or lecturer.Again,the highest recommendation possible.
  41. Notes on Euclidean Geometry Kiran Kedlaya based on notes for the Math  Olympiad Program (MOP) Version 1.0, last revised August 3, 1999  (PG) An advanced course in Euclidean and non-Euclidean geometry for honors high school or undergraduate students with a solid background in basic Euclidean geometry. Again, a problem and example driven course, designed for preparing gifted high school students for the Math Olympiad. Surprisingly concise, most of the results beyond the bare bones are in the exercises. The exercises vary in difficulty. from doable with effort to impossibly difficult. There are also virtually no pictures-all diagrams are part of the solution of exercises.The metric approach is regulated to an appendix. If you like very terse presentations where all but the most elemental results are given as exercises, you'll like this presentation. But for those looking for a more substantial presentation, you're better off using Yiu, Stienberger, Carne or one of the others listed here. And if you want a problem driven approach, Gilbert's above is the way to go. There's a later,expanded version of this same course available, with much more detail but still with an emphasis on problem solving-you can see it and my review of it here.
  42. EUCLIDEAN GEOMETRY ANCA MUSTATA University Core Cork (Ireland) Spring 2013   (G)When a set of notes begins with,. Warning: please read this text with a pencil at hand, as you will need to draw your own pictures to illustrate some statements. -well, I get nervous. That's usually a polite warning that this is a Moore method course that'll make you do all the work whether or not you get it. I'm happy to report that's not the case at all. The author does make the students check some results, but it's usually fairly straightforward diagram drawing and spade work.And if you've got access to Sketchpad, it'll be easier then that. What's great here is that author presents a very clear and beautifully written introduction to both basic and advanced Euclidean geometry that really assumes no real background. In this age where Euclid is a lost art, this is an enormous help to the self studying student. It'll also serve as a terrific primer for some of the more sophisticated treatments covered above. There are many wonderful diagrams and sidebars. This is the perfect introduction to Euclidean geometry for a self studying student that wants to learn it quickly and clearly. Highly recommended.
  43. Some Fundamental Ideas in Analytic & Euclidean Geometry Lecture Notes David Maslanka Illinois Institute of Technology Math 119 Fall 2011   (G) A brief but very careful, visual and rich primer made up of some of the most important definitions, results and basic ideas in both Euclidean and Cartesian plane geometry, such as conic sections and the classical results on triangles. It doesn't cover much, but what it does cover, it covers extremely lucidly with many pictures and it requires virtually no prerequisites. A really nice set of notes that can act as either a supplement or review for a more comprehensive course at the undergraduate level or for a strong high school level class. Very much worth a look.
  44. Euclid's 'Elements' Redux by Daniel Callahan (G) This fascinating and immensely useful open source textbook's intent is best described by the author:  0.1. Statement of Purpose To rewrite portions of Euclid's Elements so that it may be used as a basic textbook on mathematical logic and geometry (in terms of the American educational system, for use in grades 7-12 and undergraduate college courses on proof writing). While it is desirable that these rewritten proofs follow Euclid's originals, this is not as essential as the results themselves. The text is open source-which means not only is the text itself freely available for download with no restrictions, but readers and instructors are free to make whatever additions they want. My sincere hope is that open source books will be the wave of the future, but the finances of such books is for now quite limited. Consequently, the number of such texts is also quite limited. Still, this is an excellent one. The first chapter cobbles together from Wikipedia and other sources the essentials on open source textbooks-which is terrific resource for educators or people interested in free source academics. In the main text, Callahan then takes John Casey's 1885 version of Euclid,which is now in the public domain, and redraws all the diagrams,reorganizes the topics somewhat and creates a complete solutions manual at the end. The result is a beautiful version of part of Euclid that will assist any high  school student or college geometry student weak in the basics master this critical material once and for all.  A wonderfully lucid and accessible source for students of all levels in geometry and Callahan's to be commended highly for creating it. Very highly recommended.
  45. Modern Geometry Michael Filaseta University of South Carolina Math 532 Material    (PG)Again, very terse, bullet point notes-nothing that can't be gotten far better from some of the other sources here.
  46. Mirrors Reflections: The Geometry of Finite Reflection Groups Incomplete Draft Version 01 Alexandre V. Borovik  and Anna S. Borovik 25 February 2000  (PG-13) A nice set of notes on the finite reflection groups with a good first chapter reviewing Euclidean and non-Euclidean geometry from a modern point of view. This material is a bit more advanced then most of the other links in this section, but it didn't really fit in anywhere else. This is a very important application of the group theoretic approach to geometry and this remains one of the most accessible presentations available. Recommended.
  47. Geometry and the Imagination John Conway, Peter Doyle, Jane Gilman, and Bill Thurston Version 0.941,Darthnouth University Winter 2010 (PG-13)
  48. Geometry and the Imagination Peter Doyle University of Wisconsin Urbana-Champlain lecture notes 1994 (PG-13)This is a fascinating pair of resources on the Web both stemming from the same source- an intensive geometry seminar course given by the authors to gifted students and geometrically weak mathematicians at Princeton University and the Geometry Workshop in Minnepolis. The purposes of the seminar and its resulting notes and problem sets were to provide a broad introduction to modern geometry with minimal prerequisites and to provide a foundation for the beginning research on the subject.  The first link gives an early draft of annotated research projects that comprised most of the work in the course, that also contains a statement of the purpose and organization of the course,which I think is important for people to understand before beginning to work through them. The second greatly expands the projects and removes all preliminary expository material. The notes and exercises are quite visual and cover a large range of classical and modern geometry/ topology: Knot theory, polyhedra, surfaces and their classifications, symmetry patterns, orbifolds, the Euler characteristic Show that the connected sum of two projective planes is a Klein bottle. Consider the great dodecahedron with self-intersections removed. Is it orientable? What is its topological type?(a) a cabbage leaf.(c) a piece of banana peel. If you take two adjacent regions, bounded by a -shape, is the total curvature in the whole equal to the sum of the total curvature in the 2 subregions?  This is a treasure trove of problems for any geometry and topology student which will stretch their visual intuition. My one objection is that it's not clear to me what students can actually use them, what mathematical prerequisites would actually prepare students for these brutal problems. Clearly, students would need at minimum a strong high school course in Euclidean geometry and/ or some experience with proofs. But for strong students, it's a gift. Highly recommended.
  49. Geometry Balazs Szendroi Oxford University 2014   (PG)Very brief and concise, but very well written set of notes for a classical geometry course based on elementary linear algebra. These are supplementary notes and exercises, so they really aren't intended to serve as a full course text. That being said, there's some very nice stuff here, such as the geometric interpretation of the scalar triple product as a parallelopiped in 3 dimensional spaces and isometries as linear transformations, that are very well presented. The notes are balanced with 2 pages of nice exercises. These will be very useful notes in conjunction with a full course text. Highly recommended.
  50. Geometry Unbound Kiran  S. Kedlaya  MIT  version of 17 Jan 2006: (PG) This is a greatly expanded and revised version of Kedlaya's Notes on Euclidean Geometry already reviewed above.Basic and advanced Euclidean geometry, problem solving techniques in plane geometry, isometries as linear transformations, spherical geometry, the projective plane, inversive geometry,the hyperbolic plane, geometric inequalities and much more. It was developed by Kedlaya for the same clientele as the earlier notes, namely Mathematics Olympiad high school honor students. So despite having much more detail then the previous version, they are still problem and exercise centered and contain many intentional gaps for students to fill as well as many basic facts of geometry that are stated without proof. Also, as with the earlier text, most diagrams are left for the reader to construct. The resulting text reminds me of  Gilbert's notes, although nowhere near as comprehensive and with more emphasis on problem solving and Euclidean geometry. For a college geometry course where the students have a background in Euclidean geometry and proof, these notes would prove a very helpful text, particularly for mainly problem solving courses which undergraduates should begin to take toward the end of their training. Highly recommended for prepared students.
  51. Transformation Geometry Claudiu C. Remsing Rhodes University    (PG-13) This is a vastly expanded version of the first chapter of Remsing's geometry notes discussed above. It's one of the most complete and historically grounded presentations of transformation geometry that currently exists and it's available free for download, which is amazing. Unfortunately for the causal reader, the prerequisites are about the same as the other course-namely a strong background in linear algebra, basic Euclidean geometry, calculus and some group theory. But good students who have a good background in linear algebra and are willing to pick up the rest as they go, they won't find a better source then Remsing's notes on the subject. Contents: basic definitions of the Euclidean plane and geometric transformations, translations and halfturns, reflections and rotations, isometries, products of reflections, fixed points and involutions, the classification of plane isometries and explicit formulas for them, group theory basics, cyclic and dihedral groups, symmetry groups, similarities and their classifications and affine transformations. The amazing thing about these notes is their completeness-you simply won't find a source that contains just about everything you ever wanted to know or prove about transformation geometry in the plane and 3 space and their groups.Explicit formulas, group tables and full proofs are given for  just about everything introduced and it's done with enormous clarity and rigor. My one quibble is that unlike the more general geometry notes by Remsing, there are almost no pictures. That being said,students at this level will find it really helpful to draw their own pictures, especially with Geogebra or Sketchpad. An outstanding source every geometry student and teacher needs to know about. Very highly recommended.
  52. Introduction to Geometry Richard Blecksmith Northern Illinois University Spring 2013  (PG)  A terrific,extensive and original set of notes for a college level geometry course that doubles as an introduction to post-calculus rigorous mathematics. There are 2 things about Blecksmith's notes that make them so interesting and original. First is the dual purpose of the "book". It's not just a course in classical geometry, although the bulk of the contents certainly is about that far-ranging subject. But it doesn't begin with Euclid or any of the usual beginning places for such a course.It begins with 70 pages on logic, real analysis and topology that certainly won't replace undergraduate courses on these critical subjects, but will act as introductions to them that will make for an easier transition from plug and chug courses to those courses. These chapters also provide a deep scaffolding from which to carefully present classical geometry in a modern manner. An axiomatic definition of the real, rational and irrational numbers is given in a conversational style in the first chapter.The second chapter gives the basics of formal logic in the same manner and the third chapter gives a short presentation of the essentials of metric and topological spaces. When one finally gets to geometry, all these concepts are used freely to develop geometry using the Birkoff/ MacLane metric postulates. Most of the notes concern a rigorous development of Euclidean geometry based on the distance function, a development quite similar to that given by Edwin E. Moise in his classic textbook on geometry. Only in the final chapter is non-Euclidean geometry broached-the spherical, projective and hyperbolic  geometries are developed quickly and tersely, but quite clearly. There are many pictures and very careful proofs, along with many exercises, wonderful historical facts and sidebars. Which brings us to the second and more important aspect of these notes- Blecksmith wants to make geometry fun and entertaining and he succeeds very well The notes are a joy to read, as the author inserts many witty and wonderful comments, reflections and historical notes, as well as creative touches. From the many dialogues between an imaginary math professor named Ptofessor Flappenjaw and his students, who have such amusing handles as Ernest Noddsoff and Hedda Earful to the author's wonderful digression on John Harrison and the story of the discovery of the calculation of longitude to the discussion of checking into the Hotel of Infinite Rooms where there are no vacancies and  there's a shockingly simple solution to the problem. As I said, a joy to read and students will learn a gigantic amount of mathematics by carefully working through them. They are clearly a book draft in progress and I hope that even when it's completed, the author continues to make it available for free online. It's a wonderful work and one day after it's published,  it may very well join Greenberg and Moise as one of the all time great geometry books. Most highly recommended!
  53. Geometry African Virtual University  (PG)Another solid entry in the online workbooks for the AFU. After a brief digression into the history of Euclidean and non-Euclidean geometry, these notes provide an old fashioned but very clear development of classical geometry to a student with a basic Euclidean geometry background with plenty of examples and exercises. The contents: Axiomatic development of Euclidean geometry, Euclidean plane isometry,triangles,circles, similarity of objects, two dimensional Euclidean vector spaces, the straight line, coordinate mappings in the plane, conic sections, three dimensional Euclidean space and vector geometry, planes and lines in Euclidean 3 space, transformations of coordinates in 3D-space, quadric surfaces, non-Euclidian geometry,affine and projective transformations and more. A lot of the required study materials, as usual, are patchworked from online sources like Wikipedia. But the exercises and examples are quite nice and will strengthen the student's background in geometry. Personally, I think it'll probably be best used as an exercise supplement for one of the substantial sources above. Recommended.
  54. Geometric Transformations and Wallpaper Groups Lance Drager Texas Tech University Course Materials  (PG)A substantial set of notes on symmetry and the transformation groups in Euclidean space. Originally presented as a Power Point lectures, now available in PDF format, these notes give a careful and comprehensive treatment of the classical isometries and similarities in both the plane and 3-space as linear transformations using substantial matrix algebra. Since the course remains in the plane and 3 space, abstract vector spaces and linear transformations are not used and the essential group theory is developed as needed for the course. Matrix algebra and determinants, rotations and reflections, isometries and their groups, point and symmetry groups and more. These notes are very visual and elementary- they will be quite useful as a study aid in learning transformation geometry. Recommended.
  55. Groups and Geometry The Second Part of Algebra and Geometry T.W.Korner University of Cambridge April 20, 2007 (PG-13)This is an earlier version of the course taught by T.K. Carne which is listed and reviewed above. I find it interesting the 2 different versions of what is basically a geometry and algebra course is taught in each case by analysts. (Korner, of course, is one of the world's preeminent analysts and he's well known at Cambridge for teaching a host of different classes on a regular basis.  At European universities, there seems to be less pressure to specialize to extremely narrow niches in both research and teaching as in this country's academia. Whether or not this results in superior training is a debate for another time and place-but I digress.)  In any event, Korner's course is far more algebraic and less visual then Carne's and focuses quite a bit more on the group theory and linear algebra aspects of the materials. In fact, the course is much more like a group theory course then a geometry course and Korner likes to "preview" other courses in the Cambridge tripos he draws the material from, like the linear algebra course. It's an ok set of notes, very careful with Korner's usual entertaining writing style. But frankly, Carne's version has much deeper coverage of this material and if you want to learn geometry from the Kleinian perspective, you're much better off using those.
  56. Projective Geometry Lecture Notes Thomas Baird University of Newfoundland March 26     (PG-13)Another "British" style set of notes on projective geometry. Concise,nicely written and crystal clear with many pictures, again emphasizing the connection between the projective plane and abstract vector space theory. Similar to content to Hitchen's notes , but focuses more on the algebraic aspects,gives more detailed proofs and focuses much less on the big picture. I love Hitchens' notes, but these would make self-study in projective geometry a bit easier as a supplement to those notes. By themselves, they're another fine set of notes on the subject from across the pond. Highly recommended.
  57. Geometry  Unpublished Notes - Darij Grinberg Karlsruhe Institute of Technology (PG)Disconnected and disorganized collection of proofs, theorems and diagrams. The author himself says to not expect too much. I thank the author for his candor in the matter. You might find some of this useful, but I'd pass.
  58. GEOMETRY Lecture notes by Razvan Gelca Texas Tech University  (PG) A brief set of notes to supplement a college geometry course on absolute or incidence geometry. Absolute geometry is basically Euclidean geometry without the parallel postulate. Too brief to serve as complete course notes, but does have some nice examples and exercises and may be useful as a study tool in a college geometry course.
  59. A Quick Introduction to Non-Euclidean Geometry A Tiling of the Poincare Plane From Geometry: Plane and Fancy , David Singer, page 61. Robert Gardner Presented at Science Hill High School March 22, 2006 (G)A brief essay lecture given by the late Robert Gardner to high school students on the basic ideas of non-Euclidean geometry. Nice bathroom reading with some good insights on the basic ideas.
  60. Euclidean Geometry Tai-Peng Tsai University of British Columbia MATH 308 Fall2011 (PG) Ridiculously terse presentation of classical geometry.  Some nice examples, but really nothing you can't find in any of the recommended sources here. I'd pass.
  61. Modern Geometry: a Dynamic Approach John E. Gilbert University of Texas  (PG) These are Gilbert's lecture notes for a "foundations of geometry" course for upper-level mathematics majors at the University of Texas.The "dynamic" aspect of the course is that all concepts are presented using The Geometer's Sketchpad software package as a tool to visualize concepts. The use of computer software in classical geometry has become very standard except with the most stubborn of the old school holdout professors and teachers (who probably won't be active much longer)-and the Sketchpad has largely become the most popular to use. I've never used it myself, I'm sorry to say. But software only improves a course if the underlying presentation is sound for conveying the mathematics itself. Gilbert gives a sophisticated presentation of the standard axiomatic Euclidean and non-Euclidean geometry course-it presumes the student has a very good working knowledge of basic Euclidean geometry from high school.   Of course, this would be ideal to assume, as it was automatic up until about 25 years ago. But as I've said earlier, the degeneration of mathematical training in the United States really makes this a tough assumption to make.  Be that as it may, Gilbert uses Sketchpad to present advanced Euclidean geometry and non-Euclidean geometry in an unusual manner: He presents geometry an example and exercise driven manner where he primarily uses careful diagrams to prove results-and show how such diagrams, if made central without precise axioms and definitions, can lead to incorrect results despite logically correct reasoning. The reason, of course,is because even the most carefully drawn computer generated figure introduces an element of subjectivity because no 2 geometers construct a diagram in exactly the same manner. The classic example, which Gilbert gives in addition to many other excellent counterexamples-is the false "proof" that all triangles are isosceles. He proceeds in this fashion, constructing and testing axiom systems for all the classical geometries: First advanced Euclidean geometry with the parallel postulate, then Euclidean geometry without the parallel postulate (neutral geometry), and finally, abstract geometries where the parallel postulate is replaced with a weaker version that allows more then one unique line to be parallel to a point ( hyperbolic, projective and spherical geometries). He also develops the Birkoff-MacLane  metric axioms along the way. He then develops geometric transformations and in the final chapter, inversive geometry. It is very well written and challenging and the use of Sketchpad throughout gives an enormous number of beautiful and instructive diagrams.One drawback of course is that Sketchpad isn't a free program. Fortunately, there's a nearly equal free substitute I think would work just as well- Geogebra. But make no mistake-this is presentation based on active learning and nearly half the course consists of exercises that range from fair to very challenging indeed. It's a very good course, but you better make sure you or your students are up for the challenge. Highly recommended for such students.
  62. Projective  Geometry Nigel Hitchens Oxford University 2003  (PG-13) These are Hitchens' lecture notes for the projective geometry course at Oxford and you can see why they've been shamelessly copied by professors all over the United Kingdom and Europe. They are very rigorous and at the same time, very intuitive, concrete and example driven. Hitchen spends a very large chunk of the notes motivating the need for projective geometry with both real world examples such as perspective drawing and computer graphics and actual geometric constructions such as projection of lines through the origin of the sphere.  The connections between the projective plane, Euclidean space and general vector spaces is also emphasized-the idea of projective transformations being generalizations of linear transformations is a very effective one for a student with a linear algebra background. The connections with linear algebra are the focus throughout the notes and many sophisticated concepts of advanced linear algebra are given visual analogues in projective space. For example, the exterior algebra is introduced in connection with decomposable vectors in the projective plane and the Klein quadratic.The notes end with a discussion of axiomatic geometries and how the parallel postulate can be derived from the axioms of the projective plane. One of the best written, sophisticated and readable introductions I've ever seen to any subject. The highest recommendation possible. GEOMETRY Notes T. K. Carne University of Cambridge  Easter 2002 (PG) Ideas in Geometry Alison Ahlgren and Bart Snapp University of Illinois Urbana-Champaign 2010 (PG) This is a Open Source undergraduate book on classical geometry in progress, which is available currently under Creative Commons License. It richly deserves to be a popular textbook for high school honors and undergraduate "foundations of geometry" courses-the title is very appropriate, because this is an introduction to classical geometries motivated by and constructed primarily by ideas  The content is that of a standard undergraduate level course in Euclidean geometry-Euclidean geometry via axioms, non-Euclidean geometry by altering Euclidean axioms, geometric transformations, ruler and compass constructions and more. But the presentation most definitely is not.  Not only is this beautifully written presentation particularly example and problem driven, there are many wonderful historical and visual side discussions and insights.Some examples: The book opens with a detailed overview of Euclid's Elements and the remarkably significant role the text played in the evolution of Western scientific thought and includes many examples of that role not commonly known outside of geometers, such as Eratosthenes' use of the basic results to determine the circumference of the Earth. There are not only many explicit exercises, but many minor facts that are thrown out without proof that the student is encouraged to attempt to prove themselves. There is a very nice and clear discussion of so called "taxicab geometry"-which is essentially using combinatorial and rectangular coordinate methods to analyze geometric objects that are partitioned into grids. (It occurs to me this is actually a very simple example of a chart imposed on an abstract Euclidean space!) There's a fascinating section on "picture proofs" of classical results on Euclidean geometry i.e. how to take the essential axioms of Euclidean geometry and combine them with detailed drawings of planar geometric objects to convert intuitive diagrams into rigorous proofs. This is a skill many mathematics students lack and a careful reading of this
  63. chapter and doing it's exercises will assist them greatly in improving that skill. Other wonderful tidbits here are a comparison of analytic, algebraic (i.e. coordinate and formula based) and synthetic geometry, classical constructions like doubling the cube and a final chapter on plane convex sets. This is a jewel and here's hoping it remains online indefinitely for all students and teachers of geometry. Most highly recommended!
  64. Modern Geometry S. A. Fulling Texas A & M  2009    (PG) Truly bare bones supplemental notes for a course based on Greenberg's classic.  They really aren't intended as a course in and of themselves-but really, if the professor wanted the students to work primarily out of the book and the exercises, why he'd post notes at all  is a mystery to me. You may find them useful,but personally I didn't see the point. I'd rather just work out of an old copy of Greenberg with my limited student time.
  65. The Geometry of the Sphere John C. Polking Rice University (PG) These are the "notes" for a course on spherical geometry given to high -school teachers at the High School Teachers Program at the IAS/Park City Mathematics Institute at the Institute for Advanced Study at the Rice Institute during July of 1996. Polking creates a nonlinear set of notes purely in HTML and LaTEX where one can click back and forth between embedded pages at will. This creative format allows selective reading of the material online. As for the notes themselves, they use a metric style to describe the elements of spherical geometry and always relate these to their counterparts in the plane:  Lines and spheres, planes, spheres, circles, and great circles,incidence relations on a sphere, distance and isometries, lunes, angles on the sphere, area on the sphere, the area of a lune,spherical triangles, Girard's Theorem, the Euler Characteristic and more. The notes are well written,visual and pitched at a fairly low level-anyone with a basic understanding of plane geometry and calculus should have no trouble with them. It's a nice set of notes, gentle and clear. If you don't want to learn spherical geometry in great depth and just want the basics, this should work well for you. Recommended.
  66. Surfaces And Almost Everything You Wanted To Know About Them Anatole Katok Vaughn Climenhaga Penn State University (PG-13) This is preliminary lecture note version of Katok and Climenhaga's recent textbook of the same name, issued through the AMS's excellent Student Mathematical Library series available here. It's considerably briefer then the finished version and presumably has many more errors. That being said, it's a great boon that the authors have continued to make available online this late draft. The notes covers many of the topological and especially geometric aspects of surfaces embedded in Euclidean space-classification of compact surfaces and the Euler characteristic, triangulation, smooth structures, the round sphere, flat torus, Mobius strip, Klein bottle, elliptic plane, representation by equations in  three-dimensional space, parametric representation, representation as factor spaces, the metric geometry of surfaces, isometry groups of surfaces and much more. There are many pictures, careful proofs and challenging exercises. This would make a great supplement to an undergraduate course in topology as it develops many of the ideas of graduate algebraic topology and abstract manifolds in the low dimensional setting. That being said, it's not for the timid- the background needed is a good command of linear algebra, some group theory and a rigorous understanding of calculus. At most colleges, this means either honors students or seniors.For such students, it would be a treasure and a great preparation for graduate school. Highly recommended.
  67. Introduction to Geometry David Mond University of Warwick 2003(PG)
  68. From Geometry To Groups David Mond University of Warwick 2003 (PG-13) I'm considering both these notes by Mond together since they naturally form 2 halves a single year long course. I strongly recommend they be used in that manner by students, although it's not necessary. These are 2 solid courses in geometry and group theory in the "Cambridge" style-all business, concise and clear, with all essential details but no extraneous side chit-chat. They also have Mond's unique style, which is quite visual and in geometry, this is quite important. Indeed, Mond  declares early in the first notes that he intends for diagrams to be an integral part of the proof process. He believes that elementary geometry is different from other branches of mathematics in that careful proofs can be given entirely in terms of diagrams. While this is somewhat debatable, there's no doubt that an elementary geometry course without diagrams is a little like remaking King Kong without the ape. Something essential is lost. He then proceeds to give a modern geometry course entirely through transformations with a legion of diagrams in the plane. He covers Euclidean plane geometry ( congruence, triangles, circles,polygons), isometries and similarities, spherical geometry, hyperbolic geometry, polyhedra and Euler's formula. Basic set theory is assumed. Axiomatic geometry is downplayed, but not eliminated and there are many discussions of the history and various axiom systems in the history of geometry. Lots of good exercises, most not too difficult.The result is a very careful, modern yet very readable geometry course, oneof the best I've seen of relatively short length for undergraduates. The second course is essentially a group theory course where basic results are motivated completely through the connections to classical geometry. It is very similar in spirit to the geometry course and as I said. forms a very natural sequel.  The isometry and symmetry groups of various geometric objects in the plane and 3-space are the main focus of the notes, building on the mastery of geometric transformations in the "Introduction". Explicitly calculated group tables as examples abound. The notes are deep and have many surprising additions of interest in both mathematics and physics. For example, Mond gives an excellent discussion of "breaking symmetry" , which is of such importance in quantum field theory but is rarely discussed at an introductory level. Essentially breaking symmetry is the process of decomposing a given group of symmetries into specific subgroups of symmetries by either removing or adding specific mappings. Mond gives a very simple and clear example of this somewhat perplexing concept: the group of symmetries of the square can be seen as a subgroup of the symmetries of a rectangle. It turns out this is also a very powerful method of constructing new examples from previous ones. Any student studying group theory, either in basic algebra or a group theory course, could benefit highly from supplementing their readings with these notes. In conclusion, Mond's  notes provide a text for a remarkable course in geometry and groups and are both highly recommended!
  69. Links on  Geometry (and Mathematics in general)     A number of micellaneous papers, notes and draft texts on geometry ranging from basic level to graduate. Worth a look as there's a lot of good stuff here.
  70. Geometry I J.N.Bray Queen Mary College of London University 2012–13 (PG) A relatively brief set of notes on vector geometry in 3 space. Matrix algebra and nonrigorous calculus is assumed. All the basics-parallelogram law, coordinate computations, dot and cross products- are here and done nicely. Visual and intitutive, with good examples and exercises. That being said, there's nothing here you can't find done as well or better in the more substantial sources here. Good but nothing earth shaking.
  71. Geometry for Computer (Graphic)s William Goldman University of Maryland Fall 2013 (PG-13) An interesting development of all the geomeric material needed for either computer science students specializing in graphics algorithm development or undergraduate/ graduate mathematics students doing research in that burgeoning field. The course presumes a good working knowledge of linear algebra. The content of the notes includes projective geometry; geometric transformations (rotations, reflections, translations,projections), homogeneous coordinates and data types for lines and basic topology. Rather surprisingly, it doesn't explicitly develop commutative algebra or algebraic geometry using Grobner bases, although it certainly develops all the background and more one needs for such a course. It's quite a bit more careful and rigorous then one would expect from such a course. By necessity, since material is developed as  needed and not in a systematic way, it isn't as organized as mathematics courses usually are. In any event, there's a wealth of material here that both kinds of students will find very helpful for further study. There are lots of nice exercises and computer projects, none too difficult. In addition, Goldman writes beautifully and amusingly, so it's a lot of fun to read. If you don't believe me, just look at some of the chapter titles. Mathematics students are sadly weak in computer science these days and courses like this will help quite a bit. Highly recommended.
  72. College Geometry Christopher Cooper MCQuarrie University 2013   (PG) One of the best of Cooper's excellent lecture notes. This is a broad yet selective set of lecture notes on college level geometry. Unlike a lot of the lecture notes/ online books here, Cooper's notes don't try and cover the entirety of the subject of classical geometry. Instead, he chooses to focus on three topics: projective geometry, transformational geometry (symmetry) and ruler and compass geometric constructions. This allows him to focus on each topic and develop it in great depth in a reasonably short "book", one that can realistically be covered in a semester. The development is very visual without sacrificing rigor-the prerequisites are a course in linear algebra and some working knowledge of basic Euclidean geometry. One of the things Cooper insists on, like Mond and Hitchens'  notes earlier, is to creatively develop conceptual intuition,first before beginning a careful development so the students understand why things are defined this way. For example, he draws the analogy of the real projective plane as a necessary extension of the real affine plane the same way the complex numbers were created as an extension of the real number so that every real number could have 2 roots (allowing for the roots of negative numbers). In the same way, the real projective plane extends the affine plane so that all lines have at least one point of intersection (parallel lines in Euclidean space don't intersect). There are deep geometric insights like this throughout Cooper's notes, working hand and hand with linear algebra for rigorous proofs. Many other parts of the presentation are original as well. For example, in the sections on geometric transformations, all mappings are defined on inner product spaces rather then general vector spaces.This gives orthonormal transformations and groups immediately rather then needing a full development of inner products. There are literally one or 2 diagrams on every page.  There's also hundreds of exercises that range from very easy to quite challenging-and all exercises are presented with complete solutions. Focused, selective in content and beautifully written, I can't think of a better free source for self-learning this material. One of the very best free geometry sources currently available. Highest possible recommendation.
  73. Lectures on Groups and Their Connections to Geometry Anatole Katok Vaughn Climenhaga Penn State University   (R) Another course on the connections between geometry and groups. But unlike say, Mond or Carne's courses, these notes are much more general and sophisticated and go well beyond the Klienian program and it's relations to classical geometry.  It's really a first year graduate course and it's hard to imagine any but the very best undergraduates being able to handle the material beyond the first 3 sections. A considerably larger amount of basic group theory is initially developed then in most sources of this kind, including monoids, homomorphisms and isomorphisms, commutative diagrams, cyclic and permutation groups and nilpotent groups. The second chapter covers the standard material such courses usually cover, but again, in greater depth and at a much more advanced level: symmetry in Euclidean space,groups of isometries in the plane and 3 space, groups of symmetries,symmetries of bodies in R2 and R3,isometries of the plane,even and odd isometries ,isometries are products of reflections,
  74. the group structure of Isom(R2),finite symmetry groups, classifying isometries of R3,isometries of the sphere,the structure of SO(3),the structure of O(3) and odd isometries, odd isometries with no fixed points, finite subgroups of  Isom(R2) and Isom(R3), regular polyhedra, convex polytopes and their transformations. And that's just in chapter 2!! Chapter 3 continues into advanced linear algebra on complex inner product spaces and the resulting description of both the projective plane and projective space, including a brief introduction to Lie groups and algebras and their role in both algebra and manifold theory. Chapters 4-6 continue into far more sophisticated terrain, such as an introduction to the fundamental group that covers the barest bones while the authors take care ensure it doesn't turn into an algebraic topology course(but would serve as a very nice introduction for students as the final section of a point-set topology course) , an introduction to the geometric properties of finite groups,including a modern presentation of Cayley graphs and Deck transformations (which could clearly supplement an algebraic topology course, and  geometric perspective is actually deeper then Hatcher or Bredon's presentation and in some respects clearer) and concludes with a very accessible presentation of the "large scale" or "growth" geometric properties of finitely generated groups. This is an area of much active recent research and although  it's a bit difficult, it's very good to have a geometric presentation available.  Overall, a very deep, well written and visual course on the title subject and containing vitally important material-but definitely not for standard undergraduates. It would make a wonderful supplement to a graduate course in group theory or algebra. Recommended for the right audience.
  75. Euclid's Elements Mark Reader Boston University Spring 2009   (G) This is a very inventive and interesting take on the basic undergraduate geometry course.The course assigned selections from Heath's translation of Euclid's Elements, and the students had to present these sections with complete proofs, commentary and pictures in modern language in LaTeX with figures drawn using GeoAlgebra. This is a very powerful way of creating a Moore-method type course based on material that isn't too difficult. The resulting lecture notes are concise, but beautifully clear and visual treatment of many topics in The Elements. I'm generally suspicious of do-it-yourself mathematics courses, as I've said. But Reader is to be really commended for using the Moore method course to create a means to master it quickly, effectively and without trying to cover 5 terms of material on one semester the way some of the
  76. other examples of this type course listed here do. Highly recommended.