Talk:Heptagon

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Look at Talk:Hexagon and Talk:Diagonal. We know the figure for a hexagon is 25, but does anyone know the figure for the heptagon?? In addition, can anyone find a pattern?? Please put the formula in at Talk:Diagonal if you know a formula. 66.32.241.33 00:05, 13 Jun 2004 (UTC)

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[edit] Heptagons can be constructed with an unmarked ruler

http://www.vjecsner.net/presumed%20impossibilities.htm 24.155.175.62 01:40, 9 February 2007 (UTC)

No, it can't. The construction described on that page isn't a compass and straightedge construction. When we say 'cannot be constructed with a compass and straightedge', we mean that it cannot be constructed if the compass and straightedge are allowed to perform only a certain set of actions (create a line passing through two points, transfer a distance given that distance and a center). --Sopoforic 02:18, 9 February 2007 (UTC)
The website linked to for that page is authored by me, and I wish to point out some basic matters concerning compass and straightedge constructions. The actions regarded today as allowed in them according to Euclid, who is held to set the standard, are narrower than Euclid stipulates. He merely speaks (Postulates 1 and 3, Heath translation) of a straight line drawn from “any” point to “any” point, and a circle with “any” center and distance; in current textbooks, “any” is replaced mainly with “given”, a word Euclid uses elsewhere, e.g. for a “finite straight line” in Proposition I.1. In fact, he allows a number of times for points not yet given (whether initially or as derived intersections) through which a line or circle may pass, as in Propositions I.11 and I.12, where such points are “taken at random” for the construction of perpendiculars. In other words, by Euclid there is no restriction to the points—given or not—one can connect with straightedge or compass (straight line or circle).162.83.218.149 18:44, 20 February 2007 (UTC)
That may be true. My Greek isn't good enough to verify quite what was said, and different translations use different words. That's irrelevant, though, to whether it ought to be included in the article. We do (today, of course) accept certain restrictions on what actions are allowable in a compass and straightedge construction. If you can cite published sources for what you say about our present restrictions being narrower than those meant by Euclid, that may be appropriate for inclusion in Compass and straightedge construction, though. --Sopoforic 19:18, 21 February 2007 (UTC)
Since I am new to Wikipedia, I am not sure what particular information is needed, and I welcome advice. The words used in translation of Euclid don’t actually matter much, since the ideas are contextual. To give examples again of his inclusive use of the tools, in I.5 he allows for a point to “be taken at random” (however phrased) on a line, and then a line to be joined to that point, citing Postulate 1 to thereby interpret it that broadly; and as before, in I.12 he allows for a point taken at random in a space, and a circle described through it, citing Postulate 3 for this broad interpretation. I’ll quote some publications concerning narrow present restrictions (I’ll be glad to give more detail as necessary), although Euclid is evidence of absence of restrictions in applying the tools. L. E. Dickson in Construction with Ruler and Compasses: “The straight lines and circles drawn…are located by means of points either initially given or obtained as the intersections of two straight lines, a straight line and a circle, or two circles”. E. W. Hobson in Squaring the Circle, and Other Monographs: “A new point is determined in Euclidean Geometry exclusively in one of the three following ways: (1) Whenever a point P is incident both on [straight lines] (A, B) and (C, D), that point is…determinate…[the author continues for a point P incident on both a straight line and a circle, and on two circles]”. R. C. Yates in The Trisection Problem: “[The] rigid set of rules…permitted: 1. The drawing of a straight line of indefinite length through two given distinct points; 2. The construction of a circle with center at a given point and passing through a second given point”.162.83.148.26 14:26, 23 February 2007 (UTC)Italic text
I'll be happy to offer whatever advice I may. First, may I suggest that you register an account? You don't have to, but it will make it easier for me (and any others) to recognize who we're talking to, along with a few other benefits mentioned at the link I gave.
I can't advise much on which particular information ought to be included in the article, since I'm not familiar with the differences between the current restrictions and those that Euclid may have considered. You'll want to include any information that seems pertinent to you, as long as it's backed up by reliable sources. Specifics on what sort of citations you'll need are at WP:ATT, which, although lengthy, is reasonably clear.
As for the actual text you cited: I wonder if the things that they say aren't equivalent to what Euclid said, though. I'm not in a position to judge that myself, and it should be noted that as far as our content policies are concerned, neither are you. You'll need to cite some other source that has come to the conclusion that these are more restrictive conditions than Euclid used. If you know of some book that discusses this, I'd also be very interested in reading it, so I'd appreciate it if you'd mention it here (or on my talk page) whether or not you decide to use it to expand Compass and straightedge constructions.
If you have any particular questions, then just ask them and I'll try to answer as best I'm able. --Sopoforic 02:23, 24 February 2007 (UTC)
Thank you for your helpfulness. I’ll consider registering an account, and before that maybe I can mention that my above IP numbers begin with 162, and until registered I’ll end my notes with my first name, Paul, supposing there will be more.
Regarding other sources you ask about, I feel that Euclid himself should suffice for his pronouncements. I didn’t quote him as thoroughly as one might, because I understand one should avoid going beyond 100 words, but perhaps this doesn’t apply to articles. The difference between Euclid and most current texts is that he puts no restrictions on the points one can connect with straight lines and circles, whereas others do so, often severely. There are some authors who are more flexible; for instance Robin Hartshorne in Geometry: Euclid and Beyond says, “at any time one may choose a point at random” (page 21), but he doesn’t mention other viewpoints. Paul162.83.148.26 19:09, 26 February 2007 (UTC)
I can't speak about the particulars of this situation, since I haven't made any extensive study of geometry--only high school geometry and what references to geometry occur in mathematics after that point. However, Wikipedia policy is that only very limited synthesis should be done by editors. In order to add info to an article to the effect that current restrictions are narrower than those used by Euclid, you'll need some published source saying precisely that. Basically, Wikipedia reports not on truth, but on what has been published. This is necessary since most editors aren't experts on any particular topic. It also makes articles more useful: 'All metrics are positive-definite (Schechter 2003 p. 44)' is massively more useful than 'All metrics are positive-definite.'
In essence, Euclid will suffice as a source for 'Euclid says that circles are all green,' but not as a source for 'Euclid's circles were greener than circles today.' Euclid and a source on modern geometric axioms will suffice for 'Euclid had seven axioms and nine are commonly used today' but not, necessarily, for 'Euclid's axioms were weaker than the ones currently used today'--that last statement isn't trivial, so you'll need a published source that says so.
I hope that I'm being clear about this. I'd hate to make the editing process seem more complicated that it is, and I fear that I'm doing so. As always, if you have any questions, I'll be happy to answer them. --Sopoforic 04:12, 28 February 2007 (UTC)
Thank you, I understand your point. Perhaps I could add to the article on straightedge and compass that Euclid also used them in such and such a way, which can be referenced. But maybe for the moment I’ll put the matter to rest, while I think about what may be good to do. Paul 162.83.148.26 15:20, 28 February 2007 (UTC)

[edit] Heptagon construction: Eucledian or not?

Dear All,

I need to hear an expert opinion: can all steps of the above construction be considered Eucledian (i.e. straightedge & circle) or is there any of the construction steps that goes beyond Eucledian rules? Thank you. Enrico —Preceding unsigned comment added by 81.182.183.27 (talk) 19:07, 3 September 2007 (UTC)

Simple answer: it is a perfectly fine Euclidean (straightedge-and-compass) construction of a polygon that is not a regular heptagon.
If your circle is the unit circle in the cartesian plane then the point on the right is the point (√3,0) and the other point your line is passing through is (0,3/7). Suppose (x,y) is the point where your straight line between these points meets the circle, then y=√(1-x2). By forming a couple of similar right-angled triangles, you find that x must be a zero of the quadratic 52 x2 - 6√3x - 40 = 0, which means that x = ( 3√3 ± 7√43 ) / 52.
These two solutions are approximately 0.982658117 and -0.782806101. The second of these gives the point with coordinates (-0.782806101,0.622265706), which is very close to (-sin(360/7°),cos(360/7°)) = (-0.781831482,0.623489802) and this is the fact that this construction relies on.
Working out the angle you have actually constructed here prior to joining up the heptagon, arcsin(0.782806101) = 51.51822227° compared with 360/7=51.42857143°.
I am correcting the article to remove this figure. Andrew Kepert 09:25, 5 September 2007 (UTC)

[edit] Alternative formulae for area

What's the formula for the area of a heptagon where the only information you're given is either:

  • the distance from the centre to a vertex, or
  • the distance from the centre to a side.

Thanks. -- JackofOz 01:50, 5 September 2007 (UTC)

Simple trigonometry. Any regular polygon of n sides can be cut into 2n congruent right triangles with hypotenuse equal to the first of your distances (call it a), and a side equal to the second of your distances (call it b). Let θ=180/n° be the angle on each of these triangles at the centre, then b/a=cos(θ) and the third side is b tan(θ) = a sin(θ).
  • In terms of b, the area of the triangle is then 1/2 × b × b tan(θ), and with 2n of these, the area of the polygon is A = nb2 tan(θ) = nb2 tan(180/n°). For n=7, A = 7tan(180/7°) b2 ≈ 3.371 b2.
  • In terms of a, the triangle has area 1/2 × a cos(θ) × a sin(θ) = a2 sin(2θ)/4, and so the total area is A = n a2 sin(2θ)/2. For n=7, A = a2 7 sin(360/7°) / 2 ≈ 2.736 a2.
See also Regular polygon#Area. HTH Andrew Kepert 09:51, 5 September 2007 (UTC)

[edit] How do I add fictional uses?

I mean in what way should I add a fictional use of the heptagon to the article? —Preceding unsigned comment added by Worlder (talk • contribs) 05:17, 28 January 2008 (UTC)