# Talk:Diagonal

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## Article fails to provide basic definition

I'm sorry I have to say this, but this article fails to answer even the basic question "what is a diagonal", moreover someone removed my stub for a section giving that defintion. As they didn't explain their action I'm reverting! --Mike 22:45, 30 October 2006 (UTC)

The objects in your son's homework are polygons. In the Polygon section, in the very first sentence you can read "a diagonal is a line segment joining any two non-consecutive vertices." Gogino 06:47, 31 October 2006 (UTC)

I think the (etymological) dictionary definition is more useful - it was far more useful than the article it said it derives from dia - "through" & gonal? - "angle" (related to knee & agony), therefore I believe it has the meaning of "bi-secting" the angle. I know diagonal also tends to be used for an upward sloping line (which is another use) as in a diagonal mark, or a diagonal brace in engineering (an entire subject in its own to do with strengthening structures), it may also be used in football for forward+side movement.

The use given in the article is such a small percentage of the use of the word that in my opinion it amounts to bias, but the real reason I don't like the article as currently written is that it fails to give a basic definition useable by a school child - who I believe would be the main audience using wikipedia.

As for your quote: "a diagonal is a line segment joining any two non-consecutive vertices." 1. If the line would be outside the angle, then by the common sense definition (through the angle) it cannot be a diagonal because it is not through the angle. 2. I did not spot it. 3. if you don't know what a diagonal is, you are hardly likely to know what a vertice is. 4. Please let me re-emphasis: this encyclopedia will be used by a large number of school children (and parents). Some of whom haven't a clue what a polygon or a matrix, or a polynomial. If they cannot find a sensible definition within the first few paragraphs the article is useless to the 95%? of humanity who aren't doing advanced upper-school or university maths.--Mike 08:31, 31 October 2006 (UTC)

PS. Sorry - I should also say that if I were still (forced to do/) doing advance maths it is a great article! --Mike 08:34, 31 October 2006 (UTC)

PPS. And .... why not put in the classic Harry potter (Diagnonally = Diagon Alley) --Mike 08:37, 31 October 2006 (UTC)

> I think the (etymological) dictionary definition is more useful
It might be so for many people. However, these kind of definitions should be in a dictionary. Luckily, there is wiki-dictionary and you can find there the definition you proposed: diagonal (dictionary entry). There is no etymology there. You can contribute by adding yours there. Here is an example of etymology entry: enemy (dictionary entry).
> it fails to give a basic definition useable by a school child - who
I have noticed this problem and suggested a solution here.
Would you like to help with it?
It wasn't my 11year old that was reading the article, it was me his father and I did mathematics and Universtity! Between us we must represent 99.9% of all wikipedia readers. As an engineer I never use diagonal except to mean: "from corner to corner of a rectangle", therefore I had no idea how to apply it to a rhombus or pentagon - I came to this article and it was useless! However I may have been a bit abrupt in previous comments - I know it is difficult to write an article for the many different readers and I got very annoyed when someone simply removed my request for a basic definition without comment. --Mike 09:51, 6 November 2006 (UTC)
> I believe would be the main audience using wikipedia.
I don't think so. And I, as an adult, don't want to spend time on reading kid's definitions since I have to take care of many things including my kids. I want concise and precise information. Besides, this article is at the level of high school geometry course. That is when you have first complete course of geometry. I understand your frustration. You want to help your son when current school system teaches bits and pieces of "everything at every grade". This is a known problem. Read, for example, here. And consequences of this are published, for example, here. Unfortunately, a solution will not come easily because there are too many opinions and too little research.
--Gogino 02:51, 6 November 2006 (UTC)
You are quite a liberty to change it yourself, but if it is not suitable for the average reader, then I will simply change it back! --Mike 09:33, 6 November 2006 (UTC)
Finally, I found a basic definition of diagonal: Diagonal and Simple Wikipedia is the right place for it. Here should be only a link. The definition which, was here is good for high-school-geometry-class students. I think it is very appropriate. I can understand that the Wiktionary is not the best place, but the Simple Wikipedia was made with simplicity of understanding in mind and should be okay.
--Gogino 02:50, 9 November 2006 (UTC)
I'm sorry Gogino, you don't own this page. It is an encyclopedia not a maths text book. Diagonal is a common term used at all levels both inside and outside of mathematics you can't hog this page for the mathematical use and pretend other uses don't exist. Whether you like it or not the entry should reflect this. Take a look at other pages, this one ought to have sections like "fictonal use of diagonal" "sporting use of diagonal" the mathematics that currently dominates the page completely distorts the article and a short definition only goes a little way to addressing this. --Mike 21:20, 9 November 2006 (UTC)
Mike, I am not supposed to know about other uses. If you know you are free to contribute. I was discussing the basic or simple definition. The following are your statements:
• "... a school child - who I believe would be the main audience using wikipedia."
• "... I did mathematics and Universtity! Between us we must represent 99.9% of all wikipedia readers."
--Gogino 07:37, 23 November 2006 (UTC)

## Proposed opening

Diagonal an upward sloping line or a line crossing a shape that joins two nonadjacent corners. Originally from Greek: διαγωνιος (diagonios) used by both Strabo[1] and Euclid[2] for the longest line across a rhombus or from corner to corner through the middle of a cuboid [3]. Formed from dia- ("through", "across") and gonia ("angle" related to gony "knee.") later taken into latin as: diagonus ("slanting line").

Comment on proposed opening

This is a rare word in Greek with only with 3 greek references in Perseus (all Euclid Proposition 11.28 & propisiton 11.38 "diameter") However, although the greek is not available it is also used in strabo where he is arguing about whether a geographical area akin to a rhombus should be measured by its diagonal or its length:

The fourth section Hipparchus certainly manages better, .... He properly objects to Eratosthenes giving as the length of this section a line drawn from Thapsacus to Egypt, as being similar to the case of a man who should tell us that the diagonal of a parallelogram was its length. ... and a line drawn from Thapsacus to Egypt would lie in a kind of diagonal or oblique direction between them[1]

From the evidence, it would appear the original definition relates to a line across/through a simple shape. Modern use is similar (e.g. in chess queens move diagonally). I've not seen any use outwith the narrow confines of theoretical geometry for it's definition to extend to line entirely outwith a shape so I think it would be wrong to give this as a general definition (although it clearly should be mentioned along with its more precise mathematical definition). --Mike 11:12, 31 October 2006 (UTC)

I've added the paragraph, shortened it a bit and although the greeks seem to use it for the longest length of a rhombus, I've left it as "a line" - not only does it show the origin but it also gives a good example for those who might wonder whether a rhombus can have a diagonal.

I've left in the line through a cuboid as I'm sure I've seen it used in this way in some engineering and it flags up the question of whether a diagonal can be 3-d (N-d?) --Mike 11:35, 31 October 2006 (UTC)

There is a difference between an encyclopedia and a dictionary. What you proposed should be in a dictionary. See my detailed response in section "Failed Article" above.
• I invite others to express their opinions.
--Gogino 03:07, 6 November 2006 (UTC)
I might also say there is a difference between an encyclopedia and a mathemathics text book. Almost all articles start with a basic definition to scope the article. From your response, I can only assume I trod on someone's toes with my size 10 comments - sorry! Ok, lets start again. I don't expect a simple definition for polynonomial or Bessel function, but diagonal is a basic word mainly used outwith mathematics: in sports, literature, road markings, etc. it simply is not possible to have an encylopedic entry that only covers a very restriced use in advanced mathematics! --Mike 10:03, 6 November 2006 (UTC)

References

1. ^ Strabo, Geography 2.1.36-37
2. ^ Euclid, Elements book 11, proposition 28
3. ^ Euclid, Elements book 11, proposition 38

## Challenging Question: 2 for a triangle, 5 for a square, 12 for a pentagon...

We know the pattern for the number of diagonals in an n-gon, but how about the number of regions determined by the number of diagonals??

To make sure you know what I mean, here are some examples:

• A circle determines 2 regions, the inside and the outside.
• A checkerboard determines 65 regions, the squares and the area outside the lines.

A regular hexagon has 25. A general hexagon, assumed to have no more than two diagonals intersecting at any point (other than a vertex) has 26. The general position is already done for us http://www.research.att.com/projects/OEIS?Anum=A027927 . -- Smjg 11:08, 26 May 2004 (UTC)

POLYGON        DIAGONALS     REGIONS DETERMINED BY DIAGONALS
2 Digon          0             1
3 Triangle       0             2
4 Quadrilateral  2             5 (still 5 for a Square)
5 Pentagon       5            12 (still 12 for a regular pentagon)
6 Hexagon        9            26 (but only 25 for a regular hexagon)
7 Heptagon       14           51
8 Octagon        20           92
9 Enneagon       27          155
10 Decagon        35          247
11 Hendecagon     44          376
12 Dodecagon      54          551
13 Triskaidecagon 65          782
20 Icosagon       170       17876
30 Tricontagon    405
40 Tetracontagon  740
50 Pentacontagon  1175
60 Hexacontagon   1710
70 Heptacontagon  2345
80 Octacontagon   3080
90 Enneacontagon  3915
100 Hectagon       4850
1000 Chiliagon      498,500
10000 Myriagon       49,985,000
Uh-oh Googolgon      Extra credit

The numbers in this table come from "the On-Line Encyclopedia of Integer Sequences" http://www.research.att.com/projects/OEIS?Anum=A027927 which lists "a(n) = number of plane regions after drawing (general position) convex n-gon and all diagonals".

Note that this is the "general position convex n-gon", not the "regular n-gon".

The formula is a(n)= 1 + binomial(n,4) + binomial(n-1,2).

(By "binomial()", I mean the binomial coefficient).

If there are any errors, *please* tell the people at "the On-Line Encyclopedia of Integer Sequences" so they can fix it. (Or tell me, and I'll forward it to them).

-- DavidCary 15:49, 26 Jun 2004 (UTC)

What are the number of plane regions after drawing the regular n-gon ?

The formula will depend on whether n is even or odd. We need to look at the number of common intersections of three or more diagonals. Each point where three diagonals intersect loses one region, each point where four diagonals intersect loses three regions, and where d diagonal meet there are ½(d-1)(d-2) regions lost. For the regular hexagon this is simple because the only multiple intersections (other than at the vertices) are the three diagonals in the centre, where just one region gets lost. I'm trying to prove that when n is odd it is not possible for more than two diagonals to intersect at a point inside the polygon, and I'm thinking about a formula for bigger regular n-gons where n is even, but the situation gets more complicated as further symmetries emerge. For example, I think 82 regions are lost for the regular dodecagon, but this reduces to 57 regions lost for a regular 14-sided shape because 14 has fewer factors than 12. There are some experts on the Mathematics reference desk who will soon solve this if you wish to copy it there. Dbfirs 10:02, 10 April 2011 (UTC)

## split diagonal functors

I've made a new section on diagonal functors. These are not very related to other more geometrical notions of diagonals. If I could find enough stuff to write about them to fill an article, I wouldn't mind splitting to diagonal functor (currently a redirect to here). What you guys think? -lethe talk + 23:01, 2 April 2006 (UTC)

I think you have enough for an article. Look at main diagonal - it is much shorter. And you can put diagonal functor under "See also." In your sense, "Diagonal" is an adjective. For example, there are all kinds of diagonal forms not listed here.--Gogino 08:22, 9 April 2006 (UTC)

Yeah, I think you're right. It doesn't belong in this article. I've moved it to its own article, and left a "see also" link. -lethe talk + 04:21, 23 April 2006 (UTC)

## Theatre

The following was removed wholesale. I thought some of it could be saved and put into the "non-mathematical" use, but I couldn't find a general quote that it was a term used by more than one eccentric director. If it is a term in general use in the theatre, then I think it would be worth listing under the "non-mathematical uses" Mike 10:20, 19 March 2007 (UTC)

Naughty words need to be removed from first and third sentences of article. --anon 68.100.190.224 12:32, 3 April 2007 (UTC)

## General Diagonal Formula

This user recently discovered the general use for the diagonal formula. It's pretty insane to think of streaming to dimensions above three.

http://www.mathhelpforum.com/math-help/f13/general-diagonal-formula-191707.html

Scroll down until you see proofs. Is this valid for the article? — Preceding unsigned comment added by Briandiaz (talkcontribs) 18:25, 13 November 2011 (UTC)