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One of the 500 most frequently viewed mathematics articles.

Picture is wrong[edit]

Colleagues, if we are considering a polygon ABCD, then the corners at the picture are named in a wrong way, I mean C and D must be swapped. Anyone who has time, please correct the picture! DrCroco (talk) 05:42, 5 June 2009 (UTC)

This is because there is no picture of a rectangle in Wikimedia Commons with the corners labelled in the standard way. The picture used is of a "Lambert Quadrilateral". Could someone add a correctly labelled rectangle to Commons, then we can correct the error by replacing the picture in Wikipedia? Dbfirs 06:57, 24 August 2009 (UTC)

Cyclic quad[edit]

how do i prove that a quadrilateral is a cyclic quad - Anon

Good question. See cyclic polygon ... which isn't written yet. Wait and see :-) -- Tarquin 00:43 Aug 10, 2002 (PDT)

3 dimensional and n dimensional[edit]

There should be reference to the superclass object that includes all other dimensional "rectangles". I couldn't find what this was but have always thought they were called Boxes. A page for Boxes or n-dimensional rectangles should be made, or if it exists, again should be referenced.

It's called a Cuboid or rectangular parallelepiped. I've added it to "See Also" Btyner 03:42, 29 November 2005 (UTC)

Recent edit[edit]

Few rectangles can be 8 x 4 = 32. What? Does this make any sense at all ? --anon

Yes, that sentence inserted by somebody did not make any sense. I removed it. Oleg Alexandrov 20:33, 27 September 2005 (UTC)


I always thought an oblong was a rectangle standing on its shorter edge, so it's higher than it is wide. For example █ - that would be an oblong but the shape of most flags is not. I'm probably 100% wrong though —The preceding unsigned comment was added by Jodamu (talkcontribs) 15:25, 25 December 2006 (UTC).

. Yep, 100% wrong! :) I'm not sure about the relegation of the term oblong to a colloquiallism: an oblong rectangle is taught to be called an oblong (and a square rectangle a square) currently at UK primary school level. Rcrowdy

Yes, "oblong" is used for a non-square rectangle (usually standing on its longer edge), but, in non-mathematical usage, it can also refer to other shapes that are wider than they are high. This explains why secondary schools prefer the word rectangle. Dbfirs 10:06, 15 February 2009 (UTC)

rectangle is fine, but what about rounded corners?[edit]

it is often useful to have shapes similar to rectangles, but with rounded corners.

but what are these called? the often used "rectangle with rounded corners" is self-contradictory, since a rectangle is defined a something with straight sides which intersect at right angles.

JLW30 10:11, 8 January 2007 (UTC)

If you referred to it as a rounded oblong, that may make sense using the definition of an oblong as something longer than it is wide.

My friend from the Phillipines insists that an oblong is exactly that- a rectangle with rounded corners. In Australia we are taught that an oblong is a non-square rectangle. I think in common usage it may be fine to call a rectangle with rounded corners an oblong, but it is too ambiguous for mathematical usage.

A rectangle with rounded corners is called a "rounded rectangle", per this reference site: — Preceding unsigned comment added by (talk) 19:47, 8 January 2015 (UTC)

Is a square a rectangle?[edit]

Mathematically speaking the properties of a rectangle make it the same as a square but when seen on paper they are different so is a square a rectangle or is it a different shape? I mean seriously people it is very interesting question the needs to be answered.

No a rectangle is part of the squares if Im not wrong. —Preceding unsigned comment added by (talk) 23:38, 7 May 2008 (UTC)

Tex23BM: A Square is in Fact a Rectangle. A rectangle is not however always a square. A rectangle's only defining attribute is that all 4 angles are 90 degrees. It is from this very key fact that all other attributes of the Rectangle are derived. Thus the front page NEEDS TO BE CORRECTED. —Preceding unsigned comment added by Tex23bm (talkcontribs) 18:21, 13 August 2008 (UTC)

All squares are rectangles (because they satisfy the rectangle definition), but this is just a mathematical convention. The word oblong is sometimes used to refer to a non-square rectangle in less-technical usage. Dbfirs 10:11, 15 February 2009 (UTC)

Headline text[edit]

Italic textIs a square a freakn' rectangle or not?????????? —The preceding unsigned comment was added by (talk) 18:59, 31 January 2007 (UTC). i dont think we can say rectangle as square. —Preceding unsigned comment added by (talk) 04:44, 20 January 2008 (UTC) There is another rule for rectangle apart from the corners being 90 degrees. The opposite sides must be parallel and also the same length. If they are not it is possible you have a parallelogram which is not a rectangle! —Preceding unsigned comment added by (talk) 10:06, 5 May 2009 (UTC)

No it's impossible. If two straight lines intersect a third line under the same angle (90 deg. in this case), they are parallel, it's a simple theorem. So, both side pairs are parallel to each other. It is very easy to prove that the opposit sides will be equal in length as well. So, if all the angles are of 90 deg., we definitely have a rectangle :-) DrCroco (talk) 05:39, 5 June 2009 (UTC)


Rectangles are a polygon. —Preceding unsigned comment added by (talk) 03:57, 8 October 2008 (UTC)

Perfect rectangle[edit]

Here's what's wrong with Nemo's 2008-08-18 definition, which I've just replaced. It is not the (in)ability of a rectangle to be partitioned into unequal squares which makes a rectangle (im)perfect, but the (non)absence of two equal squares which makes a squared rectangle (im)perfect. A rectangle can be squared perfectly or imperfectly if and only if its aspect ratio is a rational number. Also, the definition ignores the fact that 'perfect rectangle' has been used to describe tilings by isosceles right triangles (Skinner et al., 2000). Kinewma (talk) 05:41, 16 October 2008 (UTC)

There are many definitions of "perfect rectangle". It is all in the eye of the beholder! Dbfirs 10:11, 15 February 2009 (UTC)

tiger^s prjects[edit]

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5012470 kgkgl 5012470 i ir ity,l0y 5012470 l kt , h 520180 jti jkti m j khtm, vg kg' —Preceding unsigned comment added by (talk) 11:56, 15 February 2009 (UTC)

A perfect rectangle may refer to the golden rectangle, or to a rectangle partitioned into similar polygons all of different sizes. 5012470 —Preceding unsigned comment added by (talk) 13:01, 15 February 2009 (UTC)


definition of a rectangle is too general. Knowledge is power! 23:00, 23 August 2009 (UTC) —Preceding unsigned comment added by Fngosa (talkcontribs)

No, the definition is 100% accurate. See Properties for alternatives. Would you also like a list of alternative (equivalent) definitions? What other improvements would you like to see? Dbfirs 07:03, 24 August 2009 (UTC)

Remove infobox?[edit]

It seems to me that the infobox on this page is unlikely to provide much helpful information for readers of this article. Would anyone mind if I were to remove it? Jim (talk) 22:45, 18 October 2009 (UTC)

Okay, I removed it. Jim (talk) 14:50, 21 October 2009 (UTC)
Sorry, I didn't see the question. I restored the table. Tom Ruen (talk) 06:15, 23 October 2009 (UTC)

Okay, let's talk about this. It seems unlikely to me that the average reader of this article has any idea what a Schläfli symbol or Coxeter-Dynkin diagram (or even a symmetry group) are. Moreover, I expect that those readers who do know what these things are would mostly already know what the Schläfli symbol and Coxeter-Dynkin diagram for a rectangle would be. If you asked an average mathematician to list some basic facts about rectangles, I very much doubt that either the Schläfli symbol or the Coxeter-Dynkin diagram would be among the twenty most cited pieces of information. So what is the purpose of presenting this information so prominently in the article? Jim (talk)

My coming from the side of symmetry of polytopes, included information that connected to that. I considered it valuable in polygons as a connection to the regular and uniform polyhedra. If there's other commonly useful information to include in a stat table for polygons, I'm certainly supportive to include it. Tom Ruen (talk) 08:07, 23 October 2009 (UTC)

definition 2[edit]

Wouldn't "equiangular quadrilateral" be a good definition for both simple and complex rectangles? Wouldn't that avoid "term.. refers to" when what is wanted is a comprehensive definition of the concept of rectangle? How about

In Euclidean geometry, a rectangle is any equiangular quadrilateral. The term "rectangle" normally refers to a simple rectangle, which has four right angles. A complex rectangle, which is self-intersecting or crossed, has the same vertex arrangement as a simple rectangle. Both types have diagonals of equal length.

Also: I think the diagram of the complex rectangle needs to come higher up in article.--JimWae (talk) 07:30, 22 April 2010 (UTC)

You have a good point, but I think that most people would not regard the crossed rectangle as a "true rectangle" (though I agree that mathematicians can define it as they wish). In traditional Euclidean geometry, the rectangle has four right angles, so the existing lead is correct. Dbfirs 07:50, 22 April 2010 (UTC)
... (later) ... but your approach is also correct, so I've added your definition to the second sentence in the lead. Is this a good compromise? Dbfirs 08:04, 22 April 2010 (UTC)

1> That seems sensible. Do mathematicians disagree over whether they "are" true rectangles?

2> "A rectangle that is not simple is complex" is 8 words to say not much of anything. "Shapes called complex rectangles <are also>/<have also been> considered to be rectangles" is 3 or 4 words longer but expresses that this nomenclature is a bit "unusual" (instead of making a proclamation)

3> I will not push this - but has there ever been a debate here about violation of WP:NOTDIC? I mention this because of the use of "term ..refers to" in the 1st sentence --JimWae (talk) 08:20, 22 April 2010 (UTC)

Thanks. Euclid would have disagreed, and probably some classical Euclidean geometers would still disagree. The word itself seems to dictate a right-angle, but modern geometry has moved away from Euclid. Our article on quadrilaterals shows the set of rectangles as a subset of simple convex quadrilaterals, reflecting everyday usage. I agree that "Shapes called complex rectangles are also considered to be rectangles in some branches of mathematics", or similar wording, would be an improvement. In reply to 3>, don't mathematicians always start by definining their terms? Dbfirs 09:06, 22 April 2010 (UTC)

Another shot - trying to talk about rectangles rather than the word "rectangle"--JimWae (talk) 10:03, 22 April 2010 (UTC)

In Euclidean geometry, a rectangle is usually defined as any quadrilateral with four right angles. Mathematicians often categorize such rectangles as simple rectangles, and extend the normal meaning of "rectangle" to include complex rectangles. Complex rectangles are self-intersecting (or crossed) quadrilaterals with four equal angles. A complex rectangle has the same vertex arrangement as a simple rectangle. Both types of rectangles can be defined as equiangular quadrilaterals.
Yes, I think that is a fair summary to provide a simple definition for the beginner and an accurate account of the esoteric for the non-Euclidean mathematician. I'm happy to support the change if you can find references. The statement "An equiangular quadrilateral is a rectangle if convex, and an "angular eight" with corners on a rectangle if non-convex" seems to appear regularly in my Google search. Personally, I would prefer to have the Euclidean definition and properties first with a mention of the "angular eight" non-convex shape (sometimes called a "cross-rectangle") as a note at the end for those who are interested in unusual meanings. This would be in line with standard definitions on most websites and general encyclopaedias, but I will not oppose your change. Dbfirs 11:32, 22 April 2010 (UTC)

1>Is there anything in my proposed edit that altered the meaning? 2>Is there anything that needs referencing that does not already need referencing in the current lede? This is not my field, so [especially without more input] I am not the best person to do any referencing. 3>To help tighten the proposal further: Does one need to be non-Euclidean to include complex rects as rects? 3b> Is "In Euclidean geometry" essential to definition - or is it there to give context to the article topic? 4>angular "eight" (quotes around 8) -- from a resemblance to a figure "8", right? (not 8 angles) (Also an angular "∞") 5>"A complex rectangle has the same vertex arrangement as a simple rectangle" seems very unhelpful if meant as part of a def, compared to "self-intersecting equiangular quadrilateral". Could it be dropped in favor of the latter?--JimWae (talk) 18:50, 22 April 2010 (UTC)

(1&2) No, I was querying the prominence given to the rare crossed rectangle because all the Google hits I find seem to be copies of this article.
(3) I'm trying to find out what sort of geometry uses the "crossed rectangle".
(4) Yes, I was just reproducing the original quotes.
(5) The sentence perhaps belongs in an explanation, not a definition, and I agree that your definition of a complex rectangle is precise and accurate.
As far as I can ascertain, most (or all?) mathematics text books and websites define a rectangle as a quadrilateral having four right angles. I have yet to find a source that considers a complex equiangular quadrilateral to be a rectangle. The situation seems to correspond to the 3-D case where a "hemipolyhedron" (with faces passing through the centre) is not regarded as a true Euclidean polyhedron. In other words, I question whether a cross-rectangle is a rectangle at all. I would suggest that it is just "like a rectangle in its vertex arrangement". I do need to find good references though, because this is just my opinion based on what I have found so far. There is probably a type of geometry where these "degenerate" rectangles are considered "proper", but I have yet to find a text or on-line source. Can anyone help? Dbfirs 21:32, 22 April 2010 (UTC)

This 2009-AUG-26 edit by Kinewma radically changed the lede. I think this was likely a complete misfire. Just because there is a shape that gets the name "crossed-rectangle" does not mean anyone considers that shape to be a rectangle - and there is no source for the claim that anyone does. Crossed rectangles are the shape of a wire-frame rectangle that has been twisted - they are no longer rectangles, just as one folded in half would no longer be a rectangle. The lede needs to be fixed:

In Euclidean geometry, a rectangle is any quadrilateral with four right angles. (plus other stuff before the 2009-AUG-26 edit)--JimWae (talk) 07:50, 23 April 2010 (UTC)
Thanks for checking on this and taking up the matter with the editor who introduced the crossed rectangle. We'll give him a chance to provide a reference before we rearrange the article. I think his information needs to be kept, just out of interest, and because there isn't a lot one can say about a normal rectangle. We have similar information in our polygon articles concerning star polygons. Dbfirs 18:46, 26 April 2010 (UTC)

I think we should also say rectangles are 2-dimensional plane figures (quadrilateral article does not seem to say that either - [but perhaps some include skewed quadrilaterals as quadrilaterals?]). We might also reinsert the "ABCD" way of identifying a rectangle--JimWae (talk) 19:25, 26 April 2010 (UTC) Btw, I also see usage of "skewed rectangle" - But in what ways it remains a rectangle I do not see.--JimWae (talk) 21:26, 26 April 2010 (UTC)

Thanks for your improvements. I agree that the word rectangle means the plane isogonal figure and that "crossed rectangles" and "skewed rectangles" are just "like rectangles in some respects" i.e they are distorted rectangles like a parallelogram is a "pushed-over" rectangle. Dbfirs 07:19, 27 April 2010 (UTC)

moved dubious material from main page + last paragraph about puzzles etc too detailed for lede[edit]

This is a simple rectangle.[dubious ]

A rectangle that is not simple is complex,[dubious ] but more clearly described as self-intersecting or crossed. It is defined as[dubious ] a self-intersecting quadrilateral with the same vertex arrangement as a simple rectangle. An alternative view (covering both simple and self-intersecting shapes) is to define a rectangle as an equiangular quadrilateral.

Rectangles may be used in periodic tilings of the plane. Another popular subject in recreational mathematics is the tiling of rectangles by polygons, ranging from simple puzzles to unsolved problems.

2010 April/May discussion[edit]

Kinewma has found a website by Michael de Villiers (don't know who he is) which talks about crossed rectangles, and claims that they have reflex angles, so we can't say they are isogonal. This seems to differ from the convention for star polygons, so I suppose it is a different "twist" on geometry. I still insist that the shape is a degenerate rectangle (as is the straight line at the changeover), not a true rectangle. I wonder if anyone has a copy of "Some Adventures in Euclidean Geometry" mentioned on Michael de Villiers' website. Dbfirs 06:46, 28 April 2010 (UTC)

According to Stars: A Second Look by Michael de Villiers, "The interior angle sum of the crossed quadrilateral is 720 degrees. This is usually very surprising to children and adults alike." I wouldn't call a 'crossed rectangle' a degenerate rectangle. It's either a true rectangle (like a crossed quadrilateral is regarded as a quadrilateral) or it isn't a rectangle at all (which is what I've reluctantly edited the article to say).Kinewma (talk) 04:29, 29 April 2010 (UTC)

Thanks for the interesting link. Now I know who Michael de Villiers is. I wish I'd read that document when I was teaching LOGO. I would still be inclined to go for "isn't a rectangle at all", but I suppose there is room for discussion in the definition. Most other websites define a rectangle in a way that would exclude crossed rectangles. I agree that the crossed quadrilateral is regarded as a (strange) quadrilateral. It would be helpful if mathematicians could agree on definitions, but "weird" geometries seem to like to use the same words and just extend their meaning. Dbfirs 07:10, 29 April 2010 (UTC)

A crossed rectangle is a self-intersecting instance of a generalized rectangle defined as any quadrilateral with the same vertices as a rectangle.Kinewma (talk) 03:35, 30 April 2010 (UTC)

Yes, I think "generalised rectangle" is a good description, to include the "crossed rectangle", "skew rectangle" and non-Euclidean variants and possibly also the parallelogram. Dbfirs 11:20, 30 April 2010 (UTC)

1> I cannot find any source for any such definition of "generalized rectangle" - but I do find usages that are very unlike the one here.
2> Defining "generalized rectangles" in such a way lends unwarranted credence to the view that these crossed ones really ARE rectangles after all.
3> What is needed for this topic is not a def of "generalized rectangle", but a def of the shape SOMETIMES CALLED a "crossed rectangle".
4> The source for 720 degrees for the interior angles does not mention rectangles at all.
5> The reader should not have to go to the source to figure out which angles are the interior ones - the ones that APPEAR to be interior (at the vertices) ARE equiangular.
6> Instead of saying the (turtle) turns are 135+135-135-135=0, one might be inclined to say they are 135+135+225+225=720 (making the sum of the interior angles 45+45-45-45=0) and their absolute values equal.
7> More explanations and less outright unexplained assertions are needed to make this section meaningful (even to the well-educated) without having to do hours of further research.
8> Perhaps there needs to be an article on "Crossed quadrilaterals" or crossed polygons (no scare quotes)
8a> would they be defined as "self-intersecting quadrilaterals with the same vertices as a quadrilaterl? or
8b> self intersecting polygons with 4 vertices? --JimWae (talk) 03:26, 1 May 2010 (UTC)

If we are going to discuss "crossed-rectangles" (or angular eights) in the lede, we should define it as simply as possible. Defining it in terms of a "generalized rectangle" complicates things by assuming the term "generalized rectangle" is generally accepted term.
Instead of defining "crossed-rectangle" as
a self-intersecting instance of a generalized rectangle, which is defined as any quadrilateral with the same vertices as a rectangle.[citation needed]
a self-intersecting quadrilateral with vertices that could also be the vetrices of a rectangle.'[citation needed]
or perhaps even better (since it does not assume the shape is a "real" quadrialteral)
a self-intersecting polygon with four vertices that could also be the vertices of a rectangle.[citation needed]
--JimWae (talk) 03:42, 1 May 2010 (UTC)

My thought now is that it is a subject better handled under crossed-quadrilateral since there's no dispute in that term, and there a statement can be made of different types, including crossed-rectangle, which alternates pairs of long and short edges like a convex rectangle, but doesn't have right angles. Tom Ruen (talk) 05:19, 1 May 2010 (UTC)

  • <"crossed-quadrilateral" -wiki> gets about 100 google hits. Less dispute, but some will say it's a kind of convex concave hexagon (or "pinched" or "self-contacting" hexagon).
  • <"crossed-polygon" -wiki> gets even less - but also less to disagree with--JimWae (talk) 06:03, 1 May 2010 (UTC)
Same issue with the star polygons, confusion whether intersections are vertices, or not. That's why for clarity I try to make symbolic nodes in the graphics like 10/3 decagram Decagram 10 3.png versus Star polygon 10-3.svg. Tom Ruen (talk) 06:19, 1 May 2010 (UTC)
Yes, your diagram is clearer. I think we ought to clarify this distinction in our polygon article, but my attempt was reverted by a star fan who though everyone ought to understand the concept at first reading. Would you like to make your improvement? I'll support it. Dbfirs 08:30, 5 May 2010 (UTC)

I'm puzzled as to how Michael de Villiers' definition of a rectangle: "Rectangle - any quadrilateral with axes of symmetry through each pair of opposite sides"[1]can allow his "crossed rectangle" as a candidate. Can anyone explain his thinking? Dbfirs 07:52, 3 May 2010 (UTC)

I see that this definition has now been added as a justification for including a weird rectangle. I think that most people would expect the crossed rectangle's two axes of symmetry to be lines of symmetry for the sides. Perhaps a little explanation is needed. Dbfirs 07:03, 9 May 2010 (UTC)
I've added a 'clarification' but it may be deemed superfluous as there's no doubt whatever where the two axes of symmetry of the figure (right-angled or crossed) lie. Each axis of symmetry bisects a pair of opposite sides, but one axis of symmetry of a crossed rectangle bisects both pairs of opposite sides. Kinewma (talk) 01:33, 11 May 2010 (UTC)
On second thoughts I doubt de Villiers intended his definition to include crossed rectangles! Kinewma (talk) 03:41, 11 May 2010 (UTC)
I wasn't sure what he intended, either. His view made an interesting read, anyway, deserves a link. (I'm now having trouble with the angles of skew rectangles (see below).) Dbfirs 05:58, 11 May 2010 (UTC)
My first thoughts were correct - de Villiers DOES regard his definition as including crossed rectangles, because at Cyclic Quadrilateral Incentre-Rectangle he refers to a 'crossed rectangle' which has two axes of symmetry through its two pairs of opposite sides. Kinewma (talk) 01:18, 13 May 2010 (UTC)
OK, it is an alternative definition, though not one that is standard anywhere else. One axis is not a line of symmetry for the sides that it bisects. Dbfirs 07:25, 13 May 2010 (UTC)
How is it not? Each axis of symmetry for the whole figure is also an axis of symmetry for either figure that consists of only one pair of opposite sides. Kinewma (talk) 08:32, 13 May 2010 (UTC) Changed to say it's not an axis of symmetry for either side that it bisects. Kinewma (talk) 00:58, 16 May 2010 (UTC)


From while article:

In American and in Canadian English, whilst can be considered pretentious or archaic.[3][4]

While is used in British English also, so this is not just a matter of Eng-var. I see no reason to use terminology that needlessly makes the article seem archaic. Would anyone object if we dispensed with "whilst" in favour of "while"?--JimWae (talk) 03:00, 3 May 2010 (UTC)

To me, "whilst" has more of an implication of contrast, but I agree that it sounds slightly pretentious. I've no objection to the change if "while" is more acceptable to our North American friends (and some British readers), but even better would be to delete the word and start a new sentence. Dbfirs 07:18, 3 May 2010 (UTC)

I planned a new sentence anyway before reading this! I thought I was logged in. was me (not a permanent url), but not for any previous edits/vandalism!Kinewma (talk) 20:11, 3 May 2010 (UTC)

Thanks, that's definitely an improvement. Dbfirs 19:30, 4 May 2010 (UTC)

Skew rectangle[edit]

  1. I'm having trouble finding cites, but the ones I can find (other than copies from Wikipedia) seem to have skew rectangles in 3-D having all four angles equal, rather than two right-angles. (The definition with two right-angles and two larger angles seems to be a skew parallelogram.)
  2. Photoshop tools use "skew" in 2-D for a range of distortions.
  3. I've removed the false claim that spherical geometry can have a "rectangle" but hyperbolic geometry can't. Why discriminate? I suggest the following: Other geometries sometimes define rectangles as quadrilaterals with opposite sides equal and all angles equal, for example, a "spherical rectangle" ....(etc.) What does anyone else think? Dbfirs 07:57, 5 May 2010 (UTC)

My understanding of the non-planar (or 3D) skew rectangle is that it is the shape that would be formed by folding a rectangular piece of paper along a diagonal. 2 angles are unchanged. The other 2 are equal, but they are not on the same plane as either of the other 2 & have to be measured in 3 dimensions. The only way I see they could still be right angles is if you fold the paper so the planes are perpendicular - but that is a special case.--JimWae (talk) 08:22, 5 May 2010 (UTC)

We need sources to prove or disprove my assumption that a skew rectangle is the result of folding a planar rectangle along a diagonal (I didn't think it could be anything else). If ABCD is a skew quadrilateral, with opposite sides equal and DAB and BCD right angles, then ABC and CDA are acute angles. (All angles are in 2D as any 3 points are coplanar.) Spherical rectangles exist (google 968 hits), but mathematicians do not regard any figure in hyperbolic geometry as a 'rectangle' - see Hyperbolic Geometry. —Preceding unsigned comment added by Kinewma (talkcontribs) 23:54, 5 May 2010 (UTC) In spherical geometry a spherical rectangle is divided by a diagonal into two similar spherical triangles, but in hyperbolic geometry (according to Mathworld) there are no similar triangles. Kinewma (talk) 00:22, 6 May 2010 (UTC) My mistake - hyperbolic geometry does have similar triangles, but just as in spherical geometry they are always congruent. Kinewma (talk) 00:34, 6 May 2010 (UTC)

"Skew rectangles" seems to also be a nomenclature used in planar computer graphics. In non-planar geometry, the choice is folding on a diagonal or elsewhere (presumably a perpendicular somewhere?). From what I can see, nobody maintains any of these ARE rectangles, & should probably all go into a separate article with a link from here.--JimWae (talk) 00:39, 6 May 2010 (UTC)
Most of the rectangles that we mention in the article are not rectangles by the standard definition. I was thinking of a skew rectangle created by twisting a rectangular frame. This differs from your diagonal-fold rectangle only in allowing the possibility of two obtuse and two acute angles, but is it then a skew parallelogram rather than a skew rectangle? Is the computer graphics skew rectangle the same as the Photoshop one? Dbfirs 21:40, 6 May 2010 (UTC)
I get even more Google hits for "spherical rectangle", and the first one says that it doesn't exist! I also get 321 hits for "hyperbolic rectangle" and only your Tiffany Choi of Stuyvesant High School (Hyperbolic Geometry cited above) claims that it doesn't exist. I don't think we should favour one geometry over the other. Dbfirs 21:56, 6 May 2010 (UTC)

When mathematicians (not just source I gave) say there are no rectangles in hyperbolic geometry I think they just mean that there are no rectangles with 4 right angles. My skew rectangles are just bent rectangles! Skew rectangles having four equal angles (not right angles) sounds a plausible definition. Are the edges of a skew rectangle straight lines or arcs? If they are straight lines (AB, BC, CD, DA) then they and the rectangle's diagonals (AC and BD) are the edges of a tetrahedron in which AC and BD do not intersect so a saddle rectangle (a skew rectangle whose diagonals intersect) does not exist. Or are the edges straight lines but the diagonals arcs? Kinewma (talk) 05:56, 7 May 2010 (UTC)

I do not think we can determine a single definition for a "skew rectangle". It seems to mean different things in different geometries. All we can do is note how the term might be used in various geometries - planar, polyhedral, spherical, hyperbolic, whatever. They are all increasingly remotely connected from the main topic of this article - the plane rectangle--JimWae (talk) 06:06, 7 May 2010 (UTC)
I agree that there doesn't seem to be one universal definition for the skew, and once we depart from four right-angles we seem to be in Humpty Dumpty land. I think the spherical and hyperbolic rectangles are consistently defined elsewhere. Should we mention other less-common geometries? Dbfirs 07:11, 7 May 2010 (UTC)
Isn't a saddle rectangle an example of a skew rectangle? If so, then we need to remove the requirement for two right-angles. Dbfirs 07:14, 9 May 2010 (UTC)
The saddle rectangle could not tile a polyhedron - does not have flat faces. But do we have a source for the flat-faced one, yet?--JimWae (talk) 09:10, 9 May 2010 (UTC)
Yes, I see what you mean, though I would question whether any skew rectangle could be described as having a flat interior. If you divide any interior into separate regions then you could make each one flat. Isn't a (weird) rectangle defined by its vertices and edges, not by the shape or curve of its interior? Dbfirs 10:03, 9 May 2010 (UTC)
... and can we say anything at all about the angles of a skew rectangle? If it is made by folding a normal rectangle along a diagonal then it has two right angles and two acute angles. A saddleback skew rectangle has four acute angles (since each diagonal is shorter than a Pythagoras diagonal), and the ones I create by twisting a rectangular frame can have two acute and two obtuse angles. Dbfirs 06:03, 11 May 2010 (UTC)


Regarding the question of whether the images shown in the Tessellations section of this article are isogonal: The article isogonal figure says: That implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. With the exception of the third image, it looks to me like that criterion is met. What am I missing? Duoduoduo (talk) 20:05, 12 April 2011 (UTC)

I've been scratching my head over this, too, because the rectangle is clearly an isogonal figure, but I think the point is that an isogonal tiling has to have four vertices together, a practice that is frowned on by professional bricklayers. Dbfirs 20:18, 12 April 2011 (UTC)

Perhaps the original list was isogonal and then more were added? The basket weaves" can be seen as 2-isogonal, with two types of vertices, with 3 or 4 edges connecting them. The first, second and last are all isogonal. Tom Ruen (talk) 20:31, 12 April 2011 (UTC)

1-isogonal 1-isogonal 2-isogonal 2-isogonal 1-isogonal
Stacked bond.png
Stacked bond
Wallpaper group-cmm-1.jpg
Running bond
Wallpaper group-p4g-1.jpg
Basket weave
Basketweave bond.svg
Basket weave
Herringbone bond.svg
Herringbone pattern
Tom Ruen is correct. "Isogonal" means "1-isogonal" -- all vertices are the same. "2-isogonal" means there are two types of vertices, so the two basket weave patterns are not isogonal. If you have trouble seeing this for the basket weave patterns, look how you can see four lines coming together for certain vertices and for other vertices only three lines coming together. Another way to see this is by noticing that some vertices are surrounded by four rectangles and some by three. If you omit the basket weave patterns, you can say the others are isogonal. I think it would be better to leave the basket weave pattern and just avoid discussion of isogonal because it might confuse readers. --seberle (talk) 20:53, 12 April 2011 (UTC)

"The two diagonals are equal in length and bisect each other. Every quadrilateral with both these properties is a rectangle."[edit]

This fact is given under "miscellaneous" but does it not belong as a "characterization"? This seems rather inconsistent to me. TheGrappler (talk) 16:39, 17 February 2017 (UTC)

This characterization is already given in another form: a parallelogram with diagonals of equal lengths. Circlesareround (talk) 09:59, 20 February 2017 (UTC)