# Geometry textbook with an abstract algebra emphasis

I'm teaching a variety of undergraduate and graduate geometry classes (mostly for in-service teachers) which range from elementary axiomatic geometry to more advanced transformational geometry. I'm looking to bring various levels of abstract algebra into these classes that incorporate basic ring and field theory to bridge the gap between these two traditional approaches to geometry but without teaching a full algebraic geometry course. In particular, I'm looking for suggestions of textbooks that give a good treatment of:

1. Abstract affine spaces including finite and rational affine spaces,
2. The relationship between affine spaces defined using abstract algebra and affine spaces defined by axioms (particularly affine planes and 3-dimensional affine spaces),
3. Projective spaces using homogeneous coordinates from affine spaces.

Are there any books out there that cover this material in a way that doesn't get too heavy into advanced algebraic geometry?

• I don't have a full answer at my fingertips, but I would suggest having a look at the masters: both Klein and Gelfand have written books which might be helpful. Neither is exactly what you ask for, but they might be a good place to starts. Both are cheap and eminently readable. Feb 18 '19 at 20:29
• I haven't read them myself, but John Stillwell's books The Four Pillars of Geometry and/or Numbers and Geometry might be of interest.
– J W
Feb 19 '19 at 19:40
• I wonder if you could say a bit more about the exact background you are expecting of these students. If these are in-service teachers in the US (I assume this based on your wording) it is reasonably unlikely they will remember any of the abstract algebra they had - and if they took a different route to licensure than a standard undergrad math+ed, they never have taken such a course. Anyway, knowing that will make a big difference as to what is really appropriate for your audience. Feb 20 '19 at 3:08
• @kcrisman Assume my students are proficient in abstract algebra, at least the basics of groups, rings, and fields. I think my description of the content I'm looking for is very specific and could fit somewhere in an undergrad major course or a masters level course (for teachers or otherwise). Presumably you have an idea and are worried it's too advanced. Please share anyway and I can decide the level with which to apply it. Feb 20 '19 at 5:21
• On further thought, some of the material in the opening chapters of Gallier's Geometric Methods and Applications could also be useful, as could Joswig & Theobald's Polyhedral and Algebraic Methods in Computational Geometry. I do have these books, but have not used them as the main text for a course, just for additional reading.
– J W
Feb 20 '19 at 8:50

I think you are looking for a "French style" approach. So maybe Geometry by M. Audin is good for you. Note that a new edition in French is available. It covers pretty well points 2 and 3 but not 1. Also interesting is the book of P. Gabriel, Matrizen,Geometrie, Lineare Algebra, unfortunatedly only available in German or a French translation. I hope this helps.

Permit me to endorse @halfbloodprince's recommendation.

Stillwell's four pillars are: Euclidian straight-edge/compass constructions, linear algebra & coordinates, projective geometry, transformational groups & non-Euclidean geometry. The most important aspect is that he views geometry from several different viewpoints, which to my mind is much more effective than focussing solely on one viewpoint. This allows him to show how algebra emerges from projective geometry, and how projective geometry leads to the hyperbolic plane.

Audin's book is great but may require a little effort I suggest FOUR PILLERS OF GEOMETRY - JOHN STILLWELL apart from transformation group it will give broad aspect of non-euclidean geometry.

• Welcome to ME.SE. Could you expand a bit on your answer? It would benefit from more discussion about the books in question, so that people reading the answer can form an informed opinion about them. Jun 2 '19 at 11:03