, 24 November 2011 (UTC)

My apology if you find a personal attack in my more recent comment above. I have re-read it and can still find none. You "argue that we should make the definition of polygon consistent with what the lay public is likely to encounter and study", while I believe that "the elementary logic of physical objects is best used in the introductory lead". That looks like agreement to me. Where we seem to differ is that you regard this as calling for a Euclidean mathematical definition, while I do not believe that the lay person has sufficient mathematical appreciation for that to be appropriate to the article lead.
As to the actual content, the lead grew historically as a result of several contributors thrashing out our differing approaches. What emerged should not be treated lightly, or substantially edited without prior agreement. I am happy to discuss the rest of your  proposal, and no doubt others will chip in to help reach a consensus. I did restore a little of what you wrote, but in the mean time let us not play ping-pong with the page itself.
On the matter of references, Regular polytopes and Convex polytopes are widely-read works which can hardly be described as specialised: they are indeed precisely what this lay person encountered and studied. Some others which I referenced are original sources - which, out of respect to you as a professional mathematician, I would not pull away from drawing to your attention. Others do move beyond a Euclidean treatment, and were there simply to make the point which begins the subsequent paragraph.
More directly relevant is the issue that most lay people would regard a pentagram as a "star polygon", recognised as such since the days of Archimedes but excluded by the definition you gave, which is more formally understood as a "simple polygon". While mathematicians such as Grünbaum and yourself might choose to frequently omit descriptors such as "simple" or "convex" in treatments of these objects (Grünbaum explicitly, yourself implicitly), a general introduction to the subject can take no such luxury. The article lead should conform as far as is possible to the lay person's preconceptions, whereby polygons may be divided into those which self-intersect and those which do not.
With that proviso, if you have ideas for significant changes to the article lead, I would hope you can share them with us  first. — Cheers,  (Talk) 20:34