From: Allan Adler Subject: certain axioms for geometry Date: 16 Sep 2000 09:30:12 -0500 Newsgroups: sci.math.research Summary: [missing] At the end of Krause's book Taxicab Geometry, he gives some axioms for Euclidean geometry. The axioms involve a metric and a function which measures angles and he explicitly considers them to be part of a modern formulation of geometry. The axioms seem to be somewhat in the spirit of Hilbert's axioms but are also somewhat different. One correspondent suggested that they might be due to Birkhoff around 1940. I looked up some work of Birkhoff from that time and he does have a metric approach to geometry in which he characterizes elliptic, euclidean and hyperbolic geometries by simple metric properties, but I didn't see a place for an angle measurement function in his system. In some treatments of geometry, for example that of Szmielew and Borsuk, there is some discussion of angle measurement but it isn't really axiomatized and they seem to determine a canonical one uniquely from metric properties. I think that in "length spaces" there is also a definition of angle in terms of metric properties. But I think there is really a more general set of axioms that people work with that gives a more independent status to angle measurement. The reason I think so is that in Yaglom's book "A simple non-euclidean geometry and its physical basis", he discusses 9 plane geometries, where 9=3 x 3, by choosing a kind of metric in 3 ways and by choosing a kind of angle measurement in 3 ways. Similarly, he remarks that one can get 27 = 3 x 3 x 3 solid geometries by also having 3 ways of defining the measure of dihedral angles. So my question is this: where is this more general axiom system, based on a "metric space" together with an independent notion of angle measurement, discussed in a general setting? Ignorantly, Allan Adler ara@zurich.ai.mit.edu **************************************************************************** * * * Disclaimer: I am a guest and *not* a member of the MIT Artificial * * Intelligence Lab. My actions and comments do not reflect * * in any way on MIT. Morever, I am nowhere near the Boston * * metropolitan area. * * * **************************************************************************** ============================================================================== From: lrudolph@panix.com (Lee Rudolph) Subject: Re: certain axioms for geometry Date: 16 Sep 2000 15:30:04 -0500 Newsgroups: sci.math.research Allan Adler writes: >At the end of Krause's book Taxicab Geometry, he gives some axioms for >Euclidean geometry. The axioms involve a metric and a function which measures >angles and he explicitly considers them to be part of a modern formulation >of geometry. The axioms seem to be somewhat in the spirit of Hilbert's axioms >but are also somewhat different. One correspondent suggested that they might >be due to Birkhoff around 1940. I looked up some work of Birkhoff from that >time and he does have a metric approach to geometry in which he characterizes >elliptic, euclidean and hyperbolic geometries by simple metric properties, >but I didn't see a place for an angle measurement function in his system. You may have been looking too late. The stuff you want is (it seems to me) in "A set of postulates for plane geometry, based on scale and protractor", Annals of Mathemeatics, April, 1932, vol. 33, pp. 329-345. Lee Rudolph