TU Berlin Fakultät II
Institut für Mathematik

Geometrie I



Geometry Group



Vergangene Semester


Images, Videos, and Games

Virtual Math Labs



Vorlesungen Prof. Dr. Ulrich Pinkall Di 12-14 MA 141
Mi 10-12 MA 144
Uebung Charles Gunn Fr 12-14 MA 313

  • [27.01.06] Check out this survey article on Octonions.
  • [03.01.06] I have posted a solution to the exercise from Assignment 8 regarding the axis of a linear complex.
  • [26.12.05] Charles Gunn will not have office hours on January 2. Instead, he will offer office hours on January 4, 2:00-3:00 pm, in MA 319 as usual.
  • [21.12.05] There is now a short document dealing with isometries of the hyperbolic plane with some worked out examples.
  • [15.12.05] There is a short document describing the matrix form for a null system and its inverse (dual).
  • [02.11.05] Regarding the final test (Klausur): students should be prepared to pass a verbal (muendlich) test at the end of the course. The teaching staff will decide on an individual basis if a student is required to pass such a test.
  • [28.10.05] Small typo in Hausaufgabe 1.ii): There is a missing ''with'' in the first sentence. The correct phrase is " ...such that there exists a line segment joining P and Q with no point in common with ..."
  • [28.10.05] There is a guide to notation used in this course for download in case you weren't present at the practice session.
  • [21.10.05] There is a correction to Exercise 4-i) from the first assignment sheet.
  • [20.10.05] The Uebung will take place on Friday, 12-14, in MA-313. The first assignment is also due at that time (not on Thursday as originally printed on the assignment sheet).

Projective Geometry is one of the foundation disciplines of modern mathematics. In this course we will aim to acquire a basic understanding of its content, from a variety of perspectives. After this introduction, we will proceed to study how euclidean and non-euclidean geometry can be derived as specializations of projective geometry -- using the theory of quadradic forms from linear algebra. Finally, we'll turn to other related topics such as Moebius geometry, which studies geometric properties left unchanged by inversions in spheres.
Wichtige Hinweis: Sie brauchen eine Lineal, Geo-dreieck, und eine harte Bleistift fuer die Uebung.

Assignment 1(corrected)
Assignment 2
Assignment 3
Assignment 4
Assignment 5
Assignment 6
Assignment 7
Assignment 8
Assignment 9
Assignment 10
Assignment 11
Assignment 12
Assignment 14

Theorem of Desargues
Projective Generation of a Point-wise Conic
Pascal's Theorem
The Punctured Torus Explorer allows the user to explore some discrete subgroups of PSL(2,C) corresponding to once-punctured tori.

Punctured Torus Explorer

  • There are some related books on reserve in the mathematics library under the label GeometrieI (the books under Pinkall are for another course!).
    • Coxeter, The Real Projective Plane. A classic text, which develops real projective geometry in the plane from first principles.
    • Coxeter, Non-euclidean Geometry. A classic text, in which non-euclidean geometry is developed from first principles out of projective geometry.
    • Dorwart, The Geometry of Incidence. A simplified introduction to the basic concepts of projective geometry.
    • Samuel, Projective Geometry. A modern, abstract approach.
    • Pedoe, An Introduction to Projective Geometry. More geometric approach; first chapter has good examples.
    • Beutelspacher, Projektive Geometrie. A modern approach, with lots of examples and some new applications to structural analysis and cryptology.
  • A pdf file containing the soon-to-be-published book Projektive and Cayley-Kleinsche Geometrien by A. L. Onishchik and R. Sulanke. This is available for use by students in this course only.
  • A related script by Nigel Hitchin, University of Texas.
  • A survey article on octonions by John Baez, University of California, Riverside.
  • Zirkel, dynamic geometry software by R. Grohlmann.

  • Successful completion of the course will be determined by completion of 50% of the homework problems, participation in the practice sessions, and (possibly) passing a final verbal test (muendliche Klausur). The test will be given on an individual basis as determined by the professor and assistant.
  • New material needed for the exercises may be introduced in the practice sessions. Please speak with the assistant if you have problems with attending.
  • In principle, working in groups is encouraged. However, the write-up of the homework problems is to be completed by each student individually.
  • The assistant is American and may use English when writing; however, he will try to speak German as much as you can bear.
  • Problem sheets will be handed out at the Uebungen; they are also available at the same time on this web page above.
  • Due dates for homework assignments are sharp: no extensions will be given without a medical excuse.
  • Depending on the time available for correction, it may be that not all problems are corrected each week, but rather a random sub-sample of the homework problems are chosen to be corrected. The solutions for ALL problems will in any case be posted on the web-site below. Only those problems which are corrected will be included in calculating the 50% limit required for receiving a certificate.
Student projects:
Sprechstunden Pinkall ?? MA 301
Gunn Mo 16-17 MA 319

Charles Gunn . 03.05.2006.