Talk:Finite geometry

message: It seems that a reference to the website of Alan Offer at Ghent  University, "Finite Geometry Web," , is in order.  I have added an external link to that site. Cullinane 01:59, 9 August 2005 (UTC)
thread_title:  A reference 

message: The 168 symmetries of the Fano plane play a significant role in mathematics.  Hence I have added a paragraph linking to a discussion of these symmetries.  That discussion is my own.  For some background, see other discussions of mine on finite geometry cited at Alan Offer's compilation  of online finite geometry notes at Ghent University. Cullinane 13:47, 9 August 2005 (UTC)
thread_title:  Symmetry group of Fano plane 

message: You are defining finite geometry as the system of finite number of points.How can you define a line with finite number of points while we know that a line is topologically equvalent to the interval[0,1].Threrfore finite geometry cannot hve lines.Please comment.
[B.R.Ivan,BARC,India,3 January 2006]
:One can also define a field with only a finite number of points; i.e. one can add, subtract, multiply, and divide according to the conventional rules, except that only finitely many scalars exist.  See finite field.  When doing plane geometry in which points are pairs (x, y) of scalars from such a field, many of the usual results of plane geometry are still true.  And lines in such geometries have only finitely many points. Michael Hardy 23:41, 3 January 2006 (UTC)
replies: text: Best refrence is MathWorld. Check it out.  —Preceding unsigned comment added by 67.174.157.126 (talk) 19:21, 29 March 2008 (UTC)
thread_title: Finite number of points

message: The article discussed finite plane geometry as though it were the whole of finite geometry.  This introduced errors in the discussion of finite projective geometry, where properties of finite projective planes were discussed as though they were properties of a general finite projective geometry.   I revised the article to correct this.
Cullinane 01:35, 15 February 2006 (UTC)
thread_title: Finite geometry vs. finite ''plane'' geometry

message: The article explained duality as if all projective planes were self-dual. While this is true for the Desarguesian planes, PG(2,q), it is not true in general. The smallest case where we see non self-duality is in the Veblen-Wedderburn Nearfield Plane of order 9. I fixed the statement so that it is at least true, but it is clear that this concept needs to be clarified with an expanded treatment and some specific examples. Wcherowi (talk) 16:38, 25 August 2011 (UTC)
thread_title:  Duality 

message: I have just looked at the section on the axiomatic treatment of projective 3-space. I am very disappointed. After the first paragraph the rest of the section is almost verbatim a copy of Meserve, section 2.1, pp. 26-28. He, in turn, got it from Veblen & Young but left out the essential sections which provided the definitions that you need to make sense of the axioms. Meserve at least attributed this to Veblen & Young. I propose deleting this copyright infringement and replacing it with a more modern treatment (Veblen & Young dates from 1910). There are two ways to do this. One can start with the primitive notions of point, line and plane and give six axioms that define 3-space (G. Eric Moorhouse does this in his 2007 Lecture notes on Incidence Geometry ... unfortunately that source is self-published and thus not an acceptable reference. I would need to find another source, but this may be hard as this approach is no longer popular.) The second approach needs only the primitives point and line because of the very clever Veblen axiom (P-3) which permits planes to be defined. Most modern treatments take this approach, but don't stop at three dimensions ... they just do it for general dimension. Only three or four axioms are needed for this. However, to get just three dimensions another axiom would have to be added which effectively forces this to be true. But that doesn't happen very often these days, so I will have to hunt for some source for this as well. What I am saying is ... this may take me some time, but if someone can come up with such references I would be grateful. Bill Cherowitzo  (talk) 04:20, 16 August 2012 (UTC)
thread_title:  Axiomatic treatment of projective 3-space 

message: I have removed the following recent addition.
The subject of Galois geometry has expanded largely due to the stewardship of J. W. P. Hirshfeld.
While I do believe that James Hirshfeld has significantly contributed to this area, and deserves full recognition of that fact, this statement is way overblown and I am pretty sure that even James would object to it. There is no supporting reference since these are all primary sources and none of them talks about Hirshfeld's stewardship. The books are classics to be sure, so I have placed this here in the hopes that we can come up with something a bit more reasonable and include the references. --Bill Cherowitzo  (talk) 03:07, 8 June 2017 (UTC)
thread_title:  recent addition to history section 

message: If I'm reading this right, the empty subspace has dimension -1, and no lines (but may well have points), and the dimension 0 subspace does have "lines", but each "line" contains only a single point.  If this is correct, it might be worth making explicit, and explaining why the dimension 0 and dimension -1 cases are worth treating separately.
JimJJewett (talk) 17:55, 31 October 2017 (UTC)
replies: text: Not quite. The empty subspace has no points and no lines (dimension -1), it is unique. A dimension 0 subspace consists of one point and no lines (if it had more than one point it would have to have a line by axiom 1); there are as many of these as there are points. Neither of these trivial cases is interesting from a geometrical point of view–they are tacked on just to round out the picture. The one dimensional case is a single line and all the points on it. This barely makes the grade in terms of being interesting. IMHO you need at least two dimensions to get something interesting to talk about. --Bill Cherowitzo  (talk) 19:34, 31 October 2017 (UTC)
thread_title:  axiomatic definition of dimensions