# Talk:Kite (geometry)

regarding the latest change, which is the former and which is the latter, it is unclear as you only mention kite flying in the sentence. Can you change it to "the geometric kite was inspired by the shape of the flying object" or whichever - I can't figureout which way it is. Onco_p53 11:09, 26 July 2005 (UTC)
The article is unclear regarding if the kite can be concave or only convex. In Hungary "deltoids" also include concave quadrilaterals having two pairs of equal sides, but of course concave deltoids can't be tangential. Cheers, SyP 20:46, 25 March 2006 (UTC)

In the USA we call non-convex deltoids "darts."  I'm going to see if there's a Wikipedia page for them.... RobertAustin 16:06, 20 November 2006 (UTC)
Can we redirect "Deltoid" to "Kite" after dembuguity because it is also a Muscle?.--Dagofloreswi (talk) 00:03, 18 February 2009 (UTC)
Deltoid is currently a disambiguation page pointing to both the muscle and the shape, as well as to a few other meanings. I don't think we should change that. —David Eppstein (talk) 00:13, 18 February 2009 (UTC)
I have always defined "kite" to be convex, excluding the deltoid, dart, or arrowhead.  Does anyone actually use the term "concave kite"?  Should we modify the definition?  Dbfirs 08:31, 25 April 2009 (UTC)

Is there such a thing as a right kite? I mean, a right triangle reflected over its hypotenuse? What properties does it have?

A right kite is a cyclic quadrilateral.—GraemeMcRaetalk 19:49, 24 June 2009 (UTC)

I want to apologize for some changes I made earlier without consulting this discussion thread. I appreciate the author's respect and interpretation of these changes. The changes I suggested drive toward a general confusion that exists in geometry, albeit mainly at the high school level. Confusion arises when students attempt to classify quadrilaterls using specific properties to create a hierarchical relationship. 
Let me give one such example. If we decide to classify quadrilaterals based on the the parallelism of sides, we end up with three major categories: Kites (no parallel sides), Trapezoids (1 set of parallel sides) and Parallelograms (2 sets of parallel sides.) I teach geometry at the high school level and this is the text's preferred classification. Following these rules, the rhombus and rectangle are special parallelograms and the square is a combination of the two.
The confusion arises when a statement, like the one I edited in this article, reads (to the effect): "any quadrilateral with an axis of symmetry must be either a kite or isosceles trapezoid." Such a statement makes the assumption that the reader is familiar with the classification method being used. It assumes the reader will conclude that rectangles, rhombi and squares can also be considered special Kites and Isosceles Trapezoids. If a student is using the classification method mentioned previously, confusion will occur.
Do the authors of this article feel it would be worth while to briefly describe the classification method used in the article and the subsequent, hierachical relationships? I understand the "correctness" of the article in its current state but wish to clarify some of the assumptions that are made.
Andrewbressette (talk) 19:26, 16 December 2009 (UTC)

The Quadrilateral article has a taxonomy of quadrilaterals.  All references to classification should be made in accordance with that hierarchy.—GraemeMcRaetalk 22:40, 16 December 2009 (UTC)
I did not realize that this issue has been brought up before. "Is a rhombus a kite?" Here, at Wikipedia, the consensus seems to be yes, but in US high school geometry, the answer is no. The quadrilateral classification chart (Euler diagram) on the Wikipedia article shows rhombuses as kites, but excludes darts.  
There are several related questions:
Is a circle an ellipse?
I don't think we have to "decide" for all time, but I do think it is worth mentioning. My daughter would have had several of her geometry test questions marked wrong if she only consulted Wikipedia.
Danieltrevi (talk) 14:15, 27 March 2019 (UTC)

I think the typical usage at the elementary and high school level is more or less as you describe (classifications are exclusive, so squares are not kites) and at any higher level of mathematics is usually inclusive (squares are kites). The reason for using inclusive rather than exclusive categorization is to avoid lots of silly special cases (e.g. with inclusive classifications we can state Varignon's theorem as: the midpoints of every quadrilateral form a parallelogram; with exclusive classifications we would instead have to say that sometimes it's a parallelogram, sometimes it's a rhombus, sometimes it's a rectangle, and sometimes it's a square, and readers would be left wondering why such a vague statement is even a theorem at all). So it bears mentioning that there are these two ways of doing things. But you have repeatedly added value judgements (it is usually one way but that some authors disagree) stated as fact with zero sources supporting those judgements. That is not how Wikipedia does things. Everything needs to be supported by references to reliable publications. So if you want to say that it is usually one way, you need a source that says roughly the same thing. Better would be to say what I have said above (lower level mathematics does it one way, higher the other way, and why) but again that needs a source. —David Eppstein (talk) 18:42, 27 March 2019 (UTC)
British schools usually teach the inclusive definitions.  Is it only American schools that teach the exclusive ones?  (Good teachers try to set unambiguous questions.)  Dbfirs 20:33, 27 March 2019 (UTC)
OK, I will try to find a source. Would a US high school geometry textbook be acceptable? (Pearson or McGraw-Hill). As I stated earlier, the article on the isosceles trapezoid has an almost identical statement, without a source. 
I don't recall ever even hearing about "kites" as a geometry term until this year.
Danieltrevi (talk) 13:48, 28 March 2019 (UTC)

Would a US high school geometry textbook note the existence of both types of definitions and describe which usage is more common at which level of mathematics? That is the kind of source we need. I don't think finding a source that chooses and uses only one of the definitions is helpful for statements about which usage is more common. In particular I think that your addition of "Big Ideas Math" (Texas Ed., p. 401) as a source for this is so far off-topic as to be either incompetent or intellectually dishonest. That page gives a (pictorial) definition of kites but says nothing about the relation between kites and other kinds of quadraterals, let alone clarifying whether the definition should be inclusive or exclusive. —David Eppstein (talk) 18:57, 28 March 2019 (UTC)
I really think your "intellectually dishonest" comment was uncalled for. I think your comment was arrogant. I guess this is your pet page, and you don't tolerate disagreement. Danieltrevi (talk) 20:32, 28 March 2019 (UTC)

I am still waiting for proper sources to appear. —David Eppstein (talk) 20:55, 28 March 2019 (UTC)
This is your page, I guess, and "isosceles trapezoids" is someone else's page, so you have not undone my edits on that page. I think the discrepancies in how certain quadrilaterals are defined between US high schools and universities is worth mentioning. It is not worth abuse. I do not know if there is a source that says explicitly that there is a disagreement. Apparently, for you, a source that is in disagreement with the definition given in this page is not enough. Where is the source for the very first sentence of this article? -
In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.
You could cite the reference "kite definition" in the "External Links" section, except that definition reads:
A quadrilateral with two distinct pairs of equal adjacent sides. A kite-shaped figure. (emphasis added by me.)
And why did you undo the citation to John Page in the link? That is the way he requested for his citations to appear.Danieltrevi (talk) 14:08, 29 March 2019 (UTC)Danieltrevi —(talk) 22:42, 28 March 2019 (UTC)
And as far as you waiting for "proper" sources to appear, I will not be providing them. I have pointed out what I think are several deficiencies on this page. If you, the self-appointed master of this page, do not think they are worth correcting, that is your call. Have a good day, and thank you for all you do for Wikipedia. -Danieltrevi (talk) 14:18, 29 March 2019 (UTC)
Just wanted to point out that the "kite definition" formerly in the external links section (I have removed it) was incorrect. The use of the word distinct is clearly in error. As software engineer John Page explicitly states, two pairs of adjacent sides are "distinct" if they have no side in common! This is confusing the concept of distinctness with a type of disjointedness. Two pairs can be distinct and yet share a common side, consider a triangle with one side designated as the base. The pairs of sides consisting of the base and one of the other sides, are two distinct pairs with a common side. If one views the pairs as sets consisting of two adjacent sides, then what he calls distinct is really the notion that the sets are disjoint (have no common element). Also, even if he used the adjective "disjoint", the case of the rhombus would not be ruled out. The only proper way to use the word distinct in this situation would be to rule out the possibility of considering the adjacent pair of sides A and B as different from the pair B and A. --Bill Cherowitzo  (talk) 19:42, 29 March 2019 (UTC)
Thank you. You must remember, his website is aimed at high school math. This was confusing to me, and probably to many high school math teachers._23:25, 29 March 2019 (UTC)Danieltrevi (talk)
I suspect part of the confusion arises because some of the sources used the wrong word "distinct" rather than "disjoint", perhaps out of a feeling that "disjoint" was too technical. We want to say that a kite is a shape that has two disjoint pairs of equal adjacent sides. That is, the two pairs do not share any edges with each other. The sources that say that a kite has two distinct pairs of equal adjacent sides are wrong, because that definition would also apply to a 1-1-1-2 trapezoid (which I think we can all agree is not a kite). The trapezoid has two pairs of equal adjacent sides (the first 1-1 and the second 1-1), and these pairs are distinct (they are not the same pairs as each other) but they are not disjoint. Once we use the right word in the definition, we can then move on to finer distinctions, such as whether the decomposition into disjoint pairs must be unique or whether it is enough for at least one decomposition to exist. —David Eppstein (talk) 00:09, 30 March 2019 (UTC)
This misuse of the word distinct is very common among students in teacher preparation courses. We often have to start out emphasizing its use to increase the awareness of precision in mathematical language. The lesson is probably learned a bit too well and students begin to use the term indiscriminately because it seems to make their statements sound more precise. Overuse is not penalized as heavily as underuse, so the take-away seems to be "use it whenever in doubt". --Bill Cherowitzo  (talk) 03:49, 30 March 2019 (UTC)
I added some material to the article (based on a source we were already using) distinguishing the two kinds of classification. It explains why one might prefer a hierarchical, inclusive classification (because that is what the source is about) but not what the advantages of a partitioning, exclusive classification might be. It also does not include the statements Danieltrevi wanted to add, claiming that the hierarchical classification is more usual, because I still have yet to find any source that says so. So there's more that could be added here, although we should take care not to unbalance the article with material that is really about quadrilateral classification more generally and not about kites. —David Eppstein (talk) 17:29, 3 April 2019 (UTC)

During decades of teaching mathematics in high schools in the United Kingdom I never once knew of anyone teaching the "exclusive" definitions, but I knew thousands of pupils who believed that they had been taught those definitions. It is highly likely that in some cases they had been taught such definitions, because a large proportion of high school mathematics teaching is done by people who are not mathematicians, and who through ignorance teach stuff that is just plain wrong. (To give just one example, I came across some algebra revision sheets produced by a colleague who taught maths but whose initial training had been in another subject, and she included the identity (x + y)2 = (x2 + y2).) However, it is also certain that some of those who insisted that that Miss X or Mr Y had taught them the "exclusive" definitions were mistaken, because I knew that Miss X and Mr Y absolutely did not teach that. Is it really standard practice to teach the "exclusive" definitions in US high schools? As far as I know it is perfectly possible that it is, but it is equally possible that it isn't, and Daniel thinks it is because that is his personal experience, not typical.
My own view is that a mathematical technical term should be defined in the sense in which it is normally used and understood by mathematicians, which in this case is of course the "inclusive" case. If another meaning is standard in a particular area then it may be a good idea to mention it, but that other meaning should not take precedence.
It is, of course, not for us to decide what words "should" mean; rather we should follow accepted practice, whether we agree with it or not. However, for what it's worth there are excellent reasons why the "inclusive" definitions are preferable (which is why they are the ones accepted by mathematicians). A square is a type of rectangle, and a rectangle is a type of parallelogram, and to define them otherwise would be unhelpful, for numerous reasons, including the one which David Eppstein has described above. The editor who uses the pseudonym "JamesBWatson" (talk) 20:50, 3 April 2019 (UTC)
I was very surprised to find that yes, it is indeed the case, that in US high schools the exclusive definition is the one that is commonly taught. It started with a worksheet she had listing the properties of various quadrilaterals, and you had to choose "always", "sometimes", or "never." For kites, the correct answer to the question "all four sides are congruent" was "never". I asked, "well, what if the kite is a rhombus?" "A kite is never a rhombus," according to her notes, and US high school texts. "A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent." (I am sorry, the links I placed don't seem to be working. Google "US high school geometry textbook" and you can find some free online textbooks.[[ https://bim.easyaccessmaterials.com/index.php?location_user=cchs]] might work. I chose "Texas" and then the kite definition is page 405.
I thought many other homework help sites also used the exclusive definition, but upon review, I see I was confused because of the uses of the words "disjoint" and "distinct". Apparently, US textbook authors made that same mistake. If you Google "quadrilateral family tree"  you will see the exclusive definition predominating.Danieltrevi (talk) 22:41, 4 April 2019 (UTC)

The kites are exactly the quadrilaterals that are both orthodiagonal and tangential. In a tangential quadrilateral, the two line segments connecting opposite points of tangency have equal length if and only if the quadrilateral is a kite.
If this is about tangency to an inscribed circle, that ought to be specified. —Tamfang (talk) 05:26, 12 January 2011 (UTC)

You did read the paragraph immediately above this one, no? —David Eppstein (talk) 05:38, 12 January 2011 (UTC)
Yeah, belatedly (d'oh!). It would be more obvious if the paragraph break were removed. —Tamfang (talk) 05:39, 12 January 2011 (UTC)
My idea of the structure here is that the first paragraph of the section describes properties related to the diagonals, the second paragraph describes paragraphs related to the inscribed circle, and the third synthesizes both of the previous paragraphs by combining the diagonals and the circle into a characterization of kites. But I don't mind if you want to restructure it. —David Eppstein (talk) 17:35, 13 January 2011 (UTC)

The following Wikimedia Commons file used on this page has been nominated for deletion:

Rhombicdodecahedron.jpg
Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 17:53, 18 May 2019 (UTC)

> One of them is a tiling by a right kite, with 60°, 90°, and 120° angles.
> kite with angles π/3, 5π/12, 5π/6, 5π/12.
Some angles are quoted in degrees; some in radians. I prefer degrees. JDAWiseman (talk) 16:08, 9 September 2020 (UTC)

This review is transcluded from Talk:Kite (geometry)/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Ovinus (talk · contribs) 03:54, 27 September 2022 (UTC)
Hopefully some new blood will soon grace WP:GAN#MATH.
GA review – see WP:WIAGA for criteria

Is it well written?
A. The prose is clear and concise, and the spelling and grammar are correct: 

B. It complies with the manual of style guidelines for lead sections, layout, words to watch, fiction, and list incorporation: 

Is it verifiable with no original research?
A. It contains a list of all references (sources of information), presented in accordance with the layout style guideline: 

B. All in-line citations are from reliable sources, including those for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines: 

C. It contains no original research: 

D. It contains no copyright violations nor plagiarism: 

Is it broad in its coverage?
A. It addresses the main aspects of the topic: 

B. It stays focused on the topic without going into unnecessary detail (see summary style): 

Is it neutral?
It represents viewpoints fairly and without editorial bias, giving due weight to each: 

Is it stable?
It does not change significantly from day to day because of an ongoing edit war or content dispute: 

Is it illustrated, if possible, by images?
A. Images are tagged with their copyright status, and valid non-free use rationales are provided for non-free content: 

B. Images are relevant to the topic, and have suitable captions: 

Overall:
Pass or Fail: 


### Spotchecks
(20 random of 36, relative to this version)

[1]: Fine
[3]: Doesn't talk about the naming of the artificial kite? Just the bird.
It's intended as a source for the deepest origin of the word "kite", supposedly based on the sound the bird makes. But I found another source for the connection from the artificial kite to the bird. —David Eppstein (talk) 00:53, 29 September 2022 (UTC)
[5]: Technically correct, but per above I don't see the point; deltoid curves have much broader appeal than as curves determined by some random quadrilaterals
The point is that it's a bad word to use for quadrilaterals because it's too closely related to other things one studies about quadrilaterals. I added a quote from Coxeter explicitly stating that this nomenclature is bad. —David Eppstein (talk) 00:53, 29 September 2022 (UTC)
[7]: Fine
[8]: Fine, although the source mostly uses "dart-shaped" or "dart quadrilateral" and only calls it "dart" once. I don't think this statement needs hard sourcing though.
[9]: Fine
[11]: AGF
[13]: Fine
[15]: Fine, except the source calls it a "partition classification" rather than a "partitional classification". I'd stick to the source here unless it's clear that the latter is the more common name
Isn't this a question of grammar rather than a question of terminology? To describe a classification that might be a partition classification or might be a hierarchical classification (or maybe for parallelism it should be hierarchy classification?) we are saying that we are classifying (some adverb). Do you have an adverbial form of "partition" other than "partitionally" that you think we should use? Also, Usiskin and Griffin use "partition" and "hierarchy" not "X classification". A source we are not currently using, a different paper by the same author as the source here, does use all three of "partition" (as a noun, the result of a classification), "partitional" (as an adjective, describing noun-like classifications), and partitionally (as an adverb, describing verb-like acts of classification). —David Eppstein (talk) 05:20, 8 October 2022 (UTC)
Fine to me. Ovinus (talk) 17:47, 8 October 2022 (UTC)
[17]: Good
[20]: Good
[22]: Can't access
This is Darling? "Pythagoras’s lute. The kite-shaped figure that forms the enclosing shape for a progression of diminishing pentagons and pentagrams, linking the vertices together. The resulting diagram is replete with lines in the golden ratio." —David Eppstein (talk) 01:51, 15 October 2022 (UTC)
Y
[23]: Good
[25]: Fine, but I'd appreciate a page number (probably page 24?)
This is the Crux Math reference? It has a page number. That's what the 241 is. I realize that the Citation Style 1 or Citation Style 2 format for journal citations is cryptic ("VV (NN): PP" instead of "vol. VV, no. NN, pp. PP"), and gratuitously different from magazine citations, but it's the standard style and there's not much I can do about it being that way. It is indeed the 24th page of the linked pdf, but that's not its page number and should not be used in place of the correct page number. —David Eppstein (talk) 01:55, 15 October 2022 (UTC)
Whoops, misread the citation. Ovinus (talk) 02:20, 15 October 2022 (UTC)
[26]: Good
[27]: Good
[30]: Good. It's actually open access, so maybe add link to [1]? A very interesting and fairly accessible topic that I'll have to read in full.
[33]: Fine. Probably an WP:EXPERTSPS.
[35]: Can't access, but AGF
[36]: Good


### Content
"an unrelated geometric object sometimes studied in connection with quadrilaterals" – Why is the stuff after "object" relevant if they aren't strongly connected to kites in particular? Ovinus (talk) 03:54, 27 September 2022 (UTC)
Answered above, but: lots of unrelated things in mathematics have the same names as each other. It's only a problem when they are used in the same context. The "sometimes studied in connection with quadrilaterals" part makes clear that in some cases the context is very close. —David Eppstein (talk) 01:12, 29 September 2022 (UTC)
Fair enough. It's just that I've never heard of deltoids in such a context; I see nothing relevant in the deltoid curve article and seems like a footnote to the subject. But it's harmless. Ovinus (talk) 23:19, 29 September 2022 (UTC)
"The quadrilateral with the greatest ratio of perimeter to diameter is a kite" – Suggest clarifying to "a particular kite" or something like that. It's not trivial that this quadrilateral is unique up to similarity. Ovinus (talk) 03:54, 27 September 2022 (UTC)
I find it curious that you're bothered by this point only in this one sentence, and not for the next sentence about Penrose tiles (which also doesn't say that they are particular kites, only that one is convex and the other concave) or the one after that about polyhedra, all of which require specific shapes rather than just any kite. I would have thought that the fact that only one specific shape of kite was intended would have been implied by the word "The" at the start of the sentence: if it was more than one kite, it wouldn't be the quadrilateral. —David Eppstein (talk) 01:20, 29 September 2022 (UTC)
Who are we writing this article for? Kites are not an obscure topic; many people learned about them in sixth grade. I'd say a good target is mathematically competent high school juniors. Regardless, even if the target is set a bit higher, the difference between the three cases is their intuitiveness. For the tiling, readers may hover over the "prototiles" link and quickly assume, "Hey, it's saying kites can form tilings. Fair enough." For the polyhedra, most readers have seen polyhedra before, and can imagine some fanciful shape. But for this first one, it requires the understanding that such an optimal quadrilateral is unique, rather than that "all [or some] kites have this property". This is an idea many of them will have never seen before. Ovinus (talk) 23:19, 29 September 2022 (UTC)
I agree with your suggestion of likely target audience, except maybe in the dissection, tiling, and billiards sections, which are more advanced. That's why they are later in the article, and also why I was trying to use degrees rather than radians throughout. So if you think this is confusing, I suppose it's likely they would as well. I added some angles to this one to try to make it clearer that they are fixed. —David Eppstein (talk) 04:54, 30 September 2022 (UTC)
Well, as a recently retired high schooler, I have a fairly good idea of what they find confusing. Ovinus (talk) 05:58, 30 September 2022 (UTC)
"These include as special cases the rhombus and the rectangle respectively, and the square, which is a special case of both" Why is this relevant? Ovinus (talk) 05:58, 30 September 2022 (UTC)
This paragraph is about the classification of all symmetric quadrilaterals, and how kites fit into that classification. So as context it's helpful to list the other types of symmetric quadrilaterals. —David Eppstein (talk) 18:55, 1 October 2022 (UTC)
"The right kites have two opposite right angles" How about also say smth like "The sum of their other two angles equals 180°," as this is a useful characterization Ovinus (talk) 05:58, 30 September 2022 (UTC)
Added, but it's maybe a little dubious with respect to WP:SYNTH: we have good sourcing for "right kite = kite and cyclic" and for "cyclic = opposite angles supplementary" but not for the obvious inference "right kite = kite and opposite angles supplementary". —David Eppstein (talk) 19:12, 1 October 2022 (UTC)
Meh, WP:CALC Ovinus (talk) 20:02, 1 October 2022 (UTC)
"actually tricentric, as they also have a third circle externally tangent to the extensions of their sides" is this "tricentic" terminology in wide use? Never heard of it Ovinus (talk) 05:58, 30 September 2022 (UTC)
I don't think it's in wide use, but it's from the source used rather than made-up. —David Eppstein (talk) 19:13, 1 October 2022 (UTC)
Hm. From my Google search it seems to be a neologism; there's this article and mirrors, then there's [2] (2015) which is specifically about kites and says, Thus it could also be called a tricentric quadrilateral in comparison to a bicentric quadrilateral, which only has the first two circles, and then the source you have cited. I'd omit the word "tricentric" unless there's evidence of wider use, as it feels like more of a "pun" than something well-established. Ovinus (talk) 20:02, 1 October 2022 (UTC)
It's from the source. Not only that 2015 article, but the book "A Cornucopia of Quadrilaterals" by Alsina and Nelson. Page 76. The one we use as a footnote for this sentence. I don't think an obscure 2015 journal paper would be enough to establish the terminology but this book seems a lot more mainstream to me. —David Eppstein (talk) 20:36, 1 October 2022 (UTC)
"extending one side of a regular pentagon to a point" Maybe something like "extend two sides ... until they intersect" ? Wasn't sure about the current wording Ovinus (talk) 05:58, 30 September 2022 (UTC)
Replaced description of this construction. —David Eppstein (talk) 19:23, 1 October 2022 (UTC)
Any pictures of ex-tangential circles to a kite? I can make one if need be. Ovinus (talk) 05:58, 30 September 2022 (UTC)
We didn't have one, so I drew one and added it. —David Eppstein (talk) 19:55, 1 October 2022 (UTC)
Lovely! Ovinus (talk) 20:02, 1 October 2022 (UTC)
Incidentally, I'd like to add here that for any two disjoint circles in the plane, of different sizes, the four bitangents to the circles form the sides of both a convex kite and a non-convex kite. But I can't find a source for this observation. —David Eppstein (talk) 20:33, 1 October 2022 (UTC)
How pleasing. I wonder what the area of the kites are in terms of 
  
    
      
        
          r
          
            1
          
        
        ,
        
          r
          
            2
          
        
        ,
        d
      
    
    {\displaystyle r_{1},r_{2},d}
  
. Ovinus (talk) 04:30, 2 October 2022 (UTC)
"while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle" Couldn't this be subsumed under, or at least described with, the ex-tangential part? That way, the difference between the concave and convex be put first, and then the commonality in this ex-tangential extravaganza be explained. Ovinus (talk) 05:58, 30 September 2022 (UTC)
I'm not sure I understand your point here. For the convex kites we have one circle touching all four sides and the other touching four extensions. For the concave kites we have two circles each touching two sides and two extensions. Which of those two do you think is the same as the one touching four extensions, and why? —David Eppstein (talk) 21:42, 1 October 2022 (UTC)
Well, the external, second circle of the convex case. But I suppose they're different enough that it's not helpful to discuss them together. 04:30, 2 October 2022 (UTC)
"One pair of opposite tangent lengths have equal length." Please clarify what "opposite" means in this context" Ovinus (talk) 01:25, 15 October 2022 (UTC)
Clarified in article text. (It means, at opposite vertices.) —David Eppstein (talk) 01:48, 15 October 2022 (UTC)
"congruent kite-shaped facets" Any reason to use "facet" rather than "face"? Ovinus (talk) 02:21, 15 October 2022 (UTC)
"Face" is ambiguous. In more-popular writing about polyhedra in 3d only, it means only the 2-dimensional things. But in more-technical writing about polytopes that may be higher in dimension, it may instead mean all the pieces of boundary: vertices, edges, 2-faces, 3-faces, etc. Even the whole polytope can be a face, for some definitions. Even the empty set can be a face, of dimension −1, in some definitions (needed to make the face lattice be a lattice). "Facets" is unambiguously the ones of dimension one less than the space. —David Eppstein (talk) 05:25, 15 October 2022 (UTC)
Interesting! But is this not pedantry in an elementary article like this one? (Or, to indulge: If faces are a superset of facets, isn't the statement still true?) Ovinus (talk) 06:04, 15 October 2022 (UTC)
Fine, changed to face. —David Eppstein (talk) 21:22, 16 October 2022 (UTC)
Will continue after the weekend. Ovinus (talk) 05:58, 30 September 2022 (UTC)

@Ovinus: are you still waiting for me to do something here, or did you just run short of reviewing time? —David Eppstein (talk) 16:10, 21 October 2022 (UTC)
Sorry, didn't see you responded. Two more questions:
Are the red links in the table appropriate? Ovinus (talk) 18:03, 21 October 2022 (UTC)
What citation(s) corresponds to the two tables? Ovinus (talk) 18:03, 21 October 2022 (UTC)
I was wondering whether you would ask about those. The short answer is that those tables, in the templates {{Deltoidal table}} and {{Trapezohedra}}, are really more like navboxes, used in multiple related articles to connect them to each other, than like an integral part of this one article. I think the redlinks are there to provide a consistent nomenclature rather than just the more telegraphic numeric designations that are visible in the table itself. I didn't edit them at all in preparation for this GA review, and I don't think GA reviews generally concern themselves with the content of navboxes, but I did provide a paragraph of sourced prose before each one explaining what's in the table. So the answer to "what citation(s)" is: the ones in the immediately preceding paragraph. —David Eppstein (talk) 00:40, 23 October 2022 (UTC)
I disagree that this is just a navbox; this is nontrivial explanatory information about kites and it's transcluded exactly once across mainspace: on this article. I suspect at least some of it can be sourced, though. I can't access the citations preceding the table; if they contain most of the information in the table I would appreciate if you could simply put the citation after the explanatory prose or in the table title (probably the latter). If that information isn't present, it might be in:
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
"Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
Perhaps I can get at least one of them myself. (Conway would be nice to have on my shelf, although it's a bit pricey....) If it's ultimately too difficult to source, then that's alright. Ovinus (talk) 18:31, 23 October 2022 (UTC)
What do you think needs sourcing that is not already sourced in the explanatory paragraph above it? We often do not source claims in images and their captions that repeat claims in article body text.
You appear to be correct that the first one is used in only one article, but I am surprised about this. This is part of a big series of templates for big tables of images like this, created by Tomruen, that he has spammed indiscriminately across many polyhedron and tiling articles, so that they all contain large volumes of (usually unsourced) information about all of the other vaguely-related polyhedra, rather than being more focused on the topic at hand. In many of the other tables, much of the content is original research. If anyone tries messing with them, he reverts. It is a problem. But in this specific case, I felt that the two tables were relevant-enough to keep, and could be adequately-enough sourced by the text and its sources that I included in the paragraphs above each table. I didn't attempt to modify the tables (for instance, by omitting the Coxeter diagrams) because I didn't want to get into an edit-war with him. But if it really is used only once, and needs modification, we could subst it into inline material and handle it that way. Then the unused template itself could be taken to a deletion discussion.
The trapezohedron table really is used across multiple articles, though.
As for The Symmetries of Things, see the linked article. There is a lot of good material in it, but also some reason to use it with caution (particular for terminology and history). —David Eppstein (talk) 19:46, 23 October 2022 (UTC)
Gotcha. If you think the linked citations sufficiently support the table entries, then that's alright. Maybe subst it and remove the Coxeter diagrams, but leave the configuration, which are more reasonably routine? Ovinus (talk) 22:18, 23 October 2022 (UTC)
Substed, diagrams removed, and cut down to remove all the empty cells. There's no need to be definitive in listing all hyperbolic tilings (we can't, there are infinitely many) and I think the smaller table size makes the remaining information (especially the pattern of which ones are polyhedra, which ones are Euclidean tilings, and which ones are hyperbolic) easier to pick out. Template taken to TfD but that's not an issue for this GA review. —David Eppstein (talk) 00:31, 24 October 2022 (UTC)
Great! Ovinus (talk) 01:04, 24 October 2022 (UTC)
One last thing: What do you think of collapsing the polyhedra/tiling diagrams by default? At least the second one. On narrow screens it's quite annoying because they take up substantial horizontal space. Ovinus (talk) 18:42, 24 October 2022 (UTC)
I prefer not to per MOS:DONTHIDE. —David Eppstein (talk) 20:14, 24 October 2022 (UTC)
I squished the table, substituted, and removed columns past the (conspicuously missing) nonagonal entry. Ovinus (alt) (talk) 20:30, 24 October 2022 (UTC)
Ok, I think that's better. I trimmed the row headers, made the images a bit bigger (because they weren't what was controlling the cell width) and centered both tables. —David Eppstein (talk) 22:57, 24 October 2022 (UTC)
Brilliant. Passing; nice work, and sorry for the couple delays. College apps exist, so it might be a month before I can do another review. Ovinus (talk) 01:58, 25 October 2022 (UTC)
No problem; real life comes first. And thanks again for the review. —David Eppstein (talk) 04:06, 25 October 2022 (UTC)

A square can be split into two adjacent sides of equal length too, I want to know why it wouldn't be considered as a kite. - S L A Y T H E - (talk) 16:27, 27 January 2023 (UTC)

Why do you think they wouldn't? Our article says, explicitly, "They include as special cases ... the squares, which are also special cases of both right kites and rhombi." —David Eppstein (talk) 18:16, 27 January 2023 (UTC)
It doesn't seem to be included at the "special cases" section though. Thanks anyways :) - S L A Y T H E - (talk) 18:30, 27 January 2023 (UTC)

The word "diameter" appears three times in the article, in the context of "Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles 60°, 75°, 150°, 75°."
Diameter of what? The inscribed circle? If so, then this statement is false, because I can construct a kite with an unlimited ratio of perimeter to diameter (a really long pointy kite with a small circle inscribed).
Or is this restrocted to only equidiagonal kites? If so, then that isn't what the sentence says.
The article should clarify what "diameter" means here. Or clarify that sentence, both in the lead and in the body text. ~Anachronist (talk) 22:21, 11 August 2023 (UTC)

Diameter here obviously means the largest distance between points (diameter = literally "measure across"). Your very pointy kite has ratio of perimeter to diameter of about 2, which is much smaller than the one described for which the corresponding ratio is >3 (for a square the ratio is 
  
    
      
        2
        
          
            2
          
        
        ≈
        2.83
      
    
    {\displaystyle 2{\sqrt {2}}\approx 2.83}
  
). The article gives sources in the relevant section later on. –jacobolus (t) 23:10, 11 August 2023 (UTC)
If you click through the wiki-link to diameter you will find: For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers. –jacobolus (t) 23:19, 11 August 2023 (UTC)
That second definition is correct but more technical than necessary. The definition you quoted earlier, the largest distance between any two points, is simpler. —David Eppstein (talk) 02:00, 12 August 2023 (UTC)
Thumbs up. –jacobolus (t) 02:14, 12 August 2023 (UTC)

@David Eppstein. The article says trapezohedron has kite-shaped faces. But what about the digonal trapezohedron (known as the regular tetrahedron)? It has triangular faces instead of kites. Dedhert.Jr (talk) 13:14, 15 July 2024 (UTC)

@Dedhert.Jr when you call
it a digonal trapezohedron you are really thinking of it's faces as being degenerate kites with 180 degree angles at the two poles of the trapezohedron, at the midpoints of two opposite edges of the tetrahedron. —David Eppstein (talk) 00:09, 16 July 2024 (UTC)
=@David Eppstein Thanks. But what if this is happen to non-mathematical readers after they realized it is not kite-shaped face? What I am pointing here is many people do not understand how does regular tetrahedron being digonal trapezohedron, a degenerate case for trapezohedron? What is degenerate kite means according to them? To put it in plain, writing about the degeneracy of a geometric figure would help readers to understand even more. Dedhert.Jr (talk) 00:42, 16 July 2024 (UTC)
Degeneracy (mathematics) could definitely be improved, with more concrete examples including pictures etc. Anyone who is interested should definitely feel free to work on that. –jacobolus (t) 02:14, 16 July 2024 (UTC)
The only way a reader could see this degenerate example without seeing the discussion in trapezohedron explaining how it is degenerate would be to see the huge multi-screen-wide table of irrelevant data about trapezohedron somehow copied from but not quite the same as in the {{Trapezohedra}} template, which I have just removed as irrelevant. We already have a picture of some trapezohedra (the dice); why do we need so many more? —David Eppstein (talk) 09:37, 16 July 2024 (UTC)
That's another good option. However, when you are saying huge multi-screen-wide table, and the discussion about excessive tables in WT:WPM, it reminds me of how I was trying to restructure and expand the article Bipyramid in which tables were converted into multiple images. I don't think these could be applied in this article in the same way because of GA criteria: images are relevant to the topic and provided with suitable captions. Dedhert.Jr (talk) 14:40, 17 July 2024 (UTC)