Talk:Finite geometry

A reference

It seems that a reference to the website of Alan Offer at Ghent University, "Finite Geometry Web," [1], is in order. I have added an external link to that site. Cullinane 01:59, 9 August 2005 (UTC)

Symmetry group of Fano plane

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 [2] of online finite geometry notes at Ghent University. Cullinane 13:47, 9 August 2005 (UTC)

Finite number of points

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)

Best refrence is MathWorld. Check it out. —Preceding unsigned comment added by 67.174.157.126 (talk) 19:21, 29 March 2008 (UTC)

Finite geometry vs. finite plane geometry

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)

Duality

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)

Axiomatic treatment of projective 3-space

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)

Introduction: Pixels as a finite geometry

I think the pixels example is misleading because it is so different from the finite geometries usually studied. How would "lines" even be defined without importing all the concepts associated with Bresenham's line algorithm? 72.230.215.230 (talk) 17:22, 17 November 2014 (UTC)

You are quite right in that this example would not hold up well under careful examination. However, given the nature and purpose of a lead to an article, one should be given a little leeway in terms of accuracy. The statement made is not an outright lie and except for computer graphics specialists it succinctly gets across the idea that lines need not have an infinite number of points. I know of no other example that can do a better job than this one, that is accessible to the general public without having to introduce a ton of context. If you know of such an example I'd be very interested in seeing it. Bill Cherowitzo (talk) 18:38, 17 November 2014 (UTC)

To use this example at all, we should mention lines. Different choices of lines produces different geometries, which is utterly invisible in the lead. For example, does every pair of points of the rectangular grid define a line? Also, intuitively this example does not work: the intuition is of a grid of points embedded in the Euclidean plane, where we are not interested in lines defined in the pure geometric sense, but rather in sets of point of various shapes as approximations in a rectangular region (complete with aliasing). We are not generally interested in geometric structure other than that of the Euclidean space being approximated. The lead has an additional problem: it does not adequately define a geometric system.

I'd say that even though the pixel example is familiar, it actually fails to provide even the most basic grasp of what is meant by a finite geometry due to omission of the lines. The example of the finite affine plane of order 2 seems to me to be a better example. We should at least convey the idea of it as a finite incidence structure: what could be simpler than a set of points with specific subsets defined as lines? —Quondum 19:09, 17 November 2014 (UTC)

It's been a while since I have looked at this and you are correct about the role of lines. I'm a bit hesitant about using the affine plane of order 2 since in many regards that example is too simple. It is nothing more than a graph and conveys little of the complexity of the topic. Also, the fact that it can be drawn with straight line segments gives the wrong impression that finite geometries can be embedded in Euclidean space. The affine plane of order 3 (also pictured on the page) would be a better example with respect to these points. However, I am more in favor of fixing up the lead in a different direction. In modern parlance, finite geometries and finite incidence structures are the same thing, the biggest difference being that the blocks in one are called lines in the other. You would not get that impression from this article which is exclusively about finite projective and affine spaces. Only in the **See also** section do you get a hint that there are other topics involved. The lead should convey a sense of this generality and a new section added to make this sense more concrete. The lead needs to have something that directs the reader to consider the possibility of finiteness in the geometric setting. In the AMS article linked to on the page, the finite analogy comes from physics, in particular, quantum considerations. I'd rather not follow those footsteps (it's a turn-off for some readers), so I'd like to fix the computer graphics analogy by calling it such and including a mention of lines. I'd much rather have this nod to finiteness based on a "real world" analogy rather than a (simplistic) abstraction. Bill Cherowitzo (talk) 21:05, 17 November 2014 (UTC)

I agree with your hesitation about using finite projective planes to introduce the topic, since they require quite a mind stretch; they are required to meet properties beyond the basic incidence structure, and as you say, incidence structures generally are the concept to go for.

To develop the pixel idea as a finite geometry, we need to decide what line structure to endow it with. The most regular simple incidence structure that I can think of is with only the vertical and horizontal lines. We could restrict the group of symmetries to the translations and vertical/horizontal reflections (toroidal with the obvious metric). Once we have a clear idea of what the example is in a rigorous sense, we can make it presentable. Does this make sense to you? —Quondum 23:23, 17 November 2014 (UTC)

Not the direction I was headed in. I actually agree with the IP who started this thread, this is not a very good example of a finite geometry and is best when viewed from afar. My response was that even though the example is flawed, it is okay to use it (for a particular reason) in the lead where one may need to be a bit flexible to get a point across quickly. Making the example rigorous, as you suggest, would be reasonable if the example was to go into the body of the article, but IMHO this would defeat the purpose of bringing it up in the lead. What I was thinking of would be something more in the following vein:

On the other hand, consider the graphics on a computer screen where "points" are pixels and what appear to be straight "lines" are just finite sets of pixels. A finite geometry would share this feature, having lines that consist of only a finite number of points.

This phrasing would get around the problem by not calling the example a finite geometry and would still give the reader the impression I'm looking for. Bill Cherowitzo (talk) 18:23, 18 November 2014 (UTC)

Hmm. This sounds like you're just trying to illustrate the definition of an incidence geometry. The typical reader of this article (whom we can assume is somewhat more abstractly inclined than the reader of say Geometry) will understand a finite set of "points" with "lines" being subsets, to which further constraints (axioms) may be added. Using a concrete example in this context has the usual problem of inheriting unintended structure due to the reader's preconceptions/interpretation, such as that there might be some inherent concept of "straight" lines, or that it can be embedded in a "normal" geometry, etc. The crucial properties of interesting finite geometries (e.g. homogeneity / transformation group transitivity of projective planes) are decidedly lacking in this example, further diminishing its value. I think the example is pitched too low in this case. —Quondum 02:24, 19 November 2014 (UTC)

I believe that you are missing my point, or I am not making myself clear enough. I am not calling this an example except on this talk page. The only thing I want to pull out of it is that it is reasonable to talk about lines having only a finite number of points in a quasi-geometric setting. I believe that this is important since I have seen too many grad students struggle with their first encounter of discreteness in geometry. (For example, scroll up this talk page to #Finite number of points.) The single sentence mentioning this that I've written above is all I want to say on the subject. The rest of the lead will be about actual examples of finite geometries. On a personal note, my own first encounter with finite geometries was the diagram of the Fano geometry in a Rota paper I was reading in grad school. I was intrigued by the fact that this finite collection of points and lines was being called a geometry, and I've been hooked on the subject ever since. Bill Cherowitzo (talk) 03:52, 19 November 2014 (UTC)

Well, your first statement seems to go both ways. My "what is a geometry, given that there are finite geometries" answer arrived in the form of Klein's approach of a group action on a space. —Quondum 17:07, 19 November 2014 (UTC)

While I'm a huge fan of the Erlangen Programme, it is precisely in the area of finite geometries that you start to see some of the limitations of Klein's vision. Things are okay for the major examples that have lots of symmetries, but there are many interesting geometries with trivial group actions. Be that as it may, let me work up an alternative lead (in a few days) and present it here for a critique. If you feel that I'm really missing the mark, I'll let the whole thing slide. Bill Cherowitzo (talk) 20:09, 19 November 2014 (UTC)

Sure thing, though it might be more efficient to edit it straight in. This is your area; I'm just an opinionated interested observer.

I know this is off-topic, but I find the side discussion interesting. Your claim of "many interesting geometries" and the implication that they do not fit into the Erlangen Programme suggests a different approach, perhaps synthetic geometry, and hence a different definition of a geometry. Am I on the right track? Is it just the non-Desarguesian geometries? —Quondum 23:31, 19 November 2014 (UTC)

recent addition to history section

I have removed the following recent addition.

The subject of Galois geometry has expanded largely due to the stewardship of J. W. P. Hirshfeld.

References

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)

axiomatic definition of dimensions

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)

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)