Euclid with Birkhoff - MathOverflow

Euclid with Birkhoff

2010-07-05T13:27:27Z

I'm looking for an short and elementary book which does Euclidean geomety with Birkhoff's axioms.

It would be best if it would also include some topics in projective (and/or) hyperbolic geometry.

About the course. The students suppose know some basic calculus, but they did not see real proofs.
Most of the students in my course want to become math teachers. The course description says: "Euclidean and Hyperbolic geometries and their development from postulate systems".

I choose Birkhoff's axioms sinse they use real numbers as a building block. This makes possible to do intro without cheating and without borring details. I know some good books for school students, but I am looking for something bit more advanced.

P.S. I want to thank everyone for comments and answers.

As I stated in the comments, I did not find an appropriate book and wrote the lecture notes my-self. They are available on arXiv, here is the link: Euclidean and Hyperbolic Planes.

Answer by Joseph Malkevitch for Euclid with Birkhoff

2010-07-05T14:21:54Z

I am not sure this qualifies under short and elementary but I recommend:

Geometry: A Metric Approach with Models

Richard Millman and George Parker; Springer-Verlag, NY, 1981

(there may be a more recent edition, I did not check).

This book does treat hypberbolic geometry.

Answer by Andrey Rekalo for Euclid with Birkhoff

2010-07-05T14:24:41Z

How about Basic Geometry and Basic Geometry - Manual for Teachers by George D. Birkhoff and Ralph Beatley?

In the spring of 1923, Professor Birkhoff was invited to deliver in Boston a series of Lowell Lectures on Relativity. In order to present this subject with as few technicalities as possible he decided to devise the simplest possible system of Euclidean geometry he could think of, and... he hit upon the framework of the system that, with all the details filled in, is now BASIC GEOMETRY.

Answer by Gerry Myerson for Euclid with Birkhoff

2010-07-06T01:59:52Z

Chapter 14 of Prenowitz and Jordan, Basic Concepts of Geometry, begins, "In this chapter a tentative treatment of congruence is given based on a proposal of G. D. Birkhoff (1884-1944) that the real number system should be assumed in the treatment of Euclidean geometry at an elementary level. Birkhoff's development was modified and simplified by the School Mathematics Study Group. Our treatment is an adaptation of theirs and assumes a modification of their Ruler Postulate employed by MacLane."

References are given to Birkhoff and Beatley, to the School Mathematics Study Group textbook Geometry (Yale U. Press, 1961), and to S. MacLane, Metric postulates for plane geometry, American Mathematical Monthly 66 (1959) 543-555.

There is no discussion of projective or hyperbolic geometry in Chapter 14 (but there is an extensive discussion of hyperbolic geometry in earlier chapters).

You might also be interested in Moise, Elementary Geometry From An Advanced Standpoint. In Chapter 8 you find out that the way he has been presenting plane geometry "is not the classical one. It was proposed in the early 1930's by G. D. Birkhoff, and has only recently become popular." In later chapters he does hyperbolic geometry, but I don't know whether he follows the Birkhoff path. I see no mention of projective geometry in this book.

Answer by Mark Meckes for Euclid with Birkhoff

2010-07-06T15:43:47Z

Geometry: Plane and Fancy doesn't exactly fit what you describe (it moves on to the fancy stuff a lot sooner than 2/3 of the way through), but might nevertheless be worth a look.

Answer by roy smith for Euclid with Birkhoff

2011-05-11T01:15:38Z

Have you taught this course before? After teaching it several times from Millman/Parker and other materials using Birkhoff's axioms, I suggest you consider using Euclid himself plus Hartshorne's guide, Geometry: Euclid and beyond, which uses a form of Hilbert's axioms.

The problem for me is that real numbers are much more sophisticated than Euclidean geometry, and the Birkhoff approach is thus a bit backwards except for experts like us who know what real numbers are.

When we covered as much of Millman/Parker as we could manage, the most enjoyable part for the class was the section on neutral geometry, which I learned recently was lifted bodily from Euclid Book I.

If you like assuming that every line in the plane is really the real numbers R, what about going the rest of the way and assuming the plane itself is R^2? Then you can use matrices to define rigid motions and do a lot that connects up to their calculus courses.

Moise is more succinct than the 500 pages suggests as I recall, and is an excellent text from a mathematician's standpoint, but very forbidding probably from a student's. I noticed Moise went from 1.4 to 1.9 pounds from 1st edition to third so maybe the first is also 25% shorter.

The old SMSG books in the 1960's were based on Birkhoff's approach, but are not short. They are also available free on the web.

I just looked at the old SMSG book and found the following circular sort of discussion of real numbers: "if you fill in all those other non rational points on the line, you have the real numbers."

Clint McCrory spent several years developing his own course using Birkhoff's approach at UGA, and made it very successful. Here is a link to his course page. The students loved his class at least in its evolved form after a couple years. they especially appreciated the GSP segment at the start. Apparently many students had little geometric intuition and used that to acquire some. Clint apparently never found an appropriate book to use though.

http://www.math.uga.edu/~clint/2008/geomF08/home.htm

After teaching this course myself from Greenberg, Millman/Parker, Clemens, supplemented by Moise, and the original works of Saccheri, my own Birkhoff axioms, I finally found Euclid and Hartshorne to be my favorite, by a large margin.

But the beauty of the topic is that there is no perfect choice. You will likely enjoy the search for your favorite too. There is a reason however that Euclid has the longevity it has.

Answer by John Palmieri for Euclid with Birkhoff

2011-05-11T03:58:28Z

If you're willing to use an unpublished manuscript, from the little I've looked at it, this book by Matthew Harvey looks pretty good. However, he uses Hilbert's axioms rather than Birkhoff's. Jack Lee at the University of Washington is writing another book, using a variant on the SMSG postulates, designed for a geometry course for math majors who are considering teaching high school. His book spends several chapters on hyperbolic geometry, but doesn't have any projective geometry. The book is not publicly available, but you could email him and ask him about it.