From: Allan Adler Subject: Whose axioms are these? Date: 08 Sep 2000 17:27:47 -0400 Newsgroups: sci.math Summary: [missing] In the book Taxicab Geometry, by Eugene F. Krause, the axioms for Euclidean geometry are stated as follows, there being 13 of them. For Krause, a geometry is a quadruple [P,L,d,m] consisting of a set P (of "points"), a set L (of "lines"), and two functions d,m described in more detail below. (1) Given any two points, there is exactly one line containing them. (2) Every line contains at least two points; P contains at least 3 noncollinear points. (3)-(5) (P,d) is a metric space with distance function d. (6) Given any line l the set of points of l is isometric (with respect to d) to the space of real numbers. Betweenness is defined as follows: if P,A,B are distinct points, then P is between A and B if A,B,P are collinear and d(A,P)+d(B,P)=d(A,B). From this, one can define "segment" and "convex". (7) If l is any line then the complement of l in P is the union of two convex sets H1, H2 such that any segment joining a point of H1 to a point of H2 meets l. The convex sets H1, H2 in (7) are called half-planes. Likewise, one can define "ray" and "angle". (8) m assigns to each angle a real number between 0 and 180. (9) Given a ray AB on the edge of a half-plane H and given a real number r between 0 and 180, there is exactly one ray AP such that P belongs to H and m(angle PAB)=r. (10) If D is in the interior of angle ABC then m(angle ABD) + m(angle DBC) = m(angle ABC) (11) If B is between A and C and D does not belong to the segment AC then m(angle ABD) + m(angle DBC) = 180. (12) (SAS) Given a 1-1 correspondence between the vertex sets of two triangles. If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence. (13) Given a point P not on a line l, there is exactly one line through P parallel to l. I realize that these are probably equivalent and similar in spirit to Hilbert's formulation, but I would like to know whether Krause was merely making his own synopsis of Hilbert's axioms or whether Krause adopted them from some other source in which the real numbers are used more explicitly than they are in Hilbert. 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: Len Smiley Subject: Re: Whose axioms are these? Date: Mon, 11 Sep 2000 20:19:34 GMT Newsgroups: sci.math In article <39BD1309.6556D750@engmail.uwaterloo.ca>, Ray Vickson wrote: > > > Len Smiley wrote: > > > IIRC, these were due to Birkhoff (around 1940). Maybe in the Annals. > > > > Len > > > > (I could be way off) > > > > In article , > > Allan Adler wrote: > > > In the book Taxicab Geometry, by Eugene F. Krause, > > I haven't seen the book, so I can't imagine how it gets its title. Why > "Taxicab" geometry? > > > the axioms for > > Euclidean > > > geometry are stated as follows, there being 13 of them. For > > -- > R. G. Vickson > Department of Management Sciences > University of Waterloo > Waterloo, Ontario, CANADA > > OK, I'll answer RGV's question and then correct my answer to Adler's. In Taxicab geometry the distance function is defined as the shortest distance using only horizontal and vertical "streets" to get from point a to point b. The punchline is that the taxicab plane satisfies everything up to the SAS congruence criterion Axiom, so it's not a "neutral geometry". I was right about Birkhoff, and the Annals, but it was 1932, not 1940! Len Sent via Deja.com http://www.deja.com/ Before you buy.