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Are triangles with curved sides still considered triangles, in Euclidean space? So the sum of their angles can be bigger (or smaller) than 180 degrees? I can imagine triangles consisting of three curved, connected line pieces. Are they considered in mathematics?

As is suggested in a comment below, a Reuleaux triangle is a triangle with curved sides. It has very specific sides though. Why not introduce general triangles of which the "normal" triangle and the Reuleax one are specific cases?

Don't curved triangles (of which the circle would be a special case, like all cone intersections) have any value? Like the distorted images of circles in cone intersections?

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  • $\begingroup$ @Gae.S. But there are three angles involved. $\endgroup$ – Methadont Jun 14 at 10:39
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    $\begingroup$ As far as the Euclidean plane goes, triangles have three line segments as sides (for the record, while the name is tri-angle, the definition is always given in terms of sides). In a generic Riemannian, or pseudo-Riemannian, manifold there is an argument for having a predilection for curves that are geodesics for the metric tensor (which might not be straight lines when you embed the manifold in $\Bbb R^n$). However, the machinery at work in that context is way more convoluted. $\endgroup$ – user239203 Jun 14 at 10:42
  • $\begingroup$ Aren't curved line segments line segments too? $\endgroup$ – Methadont Jun 14 at 10:44
  • $\begingroup$ If by "curved lines segments" you mean arcs of a curve, then no. Segments in the context of polygons are line segments. $\endgroup$ – user239203 Jun 14 at 10:46
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    $\begingroup$ @Methadont We do consider them. We just don't call them triangles... $\endgroup$ – 5xum Jun 14 at 10:49
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There's a deeper point addressed in the comments (especially by 5xum in their answer), but that based on Methadont's comments is perhaps the heart of the question: A thing and the name of a thing are distinct. (I've heard this called The Fundamental Theorem of Semiotics.)

Natural language is flexible, and that flexibility gives us idioms and metaphors, which can be useful (or telling).

Mathematics generally strives for precision. Despite this, words do not map bijectively to concepts, and there is no practical way to adjust language to form a bijection.

In Euclidean geometry, terms such as segment, triangle, and circle have specific and rigid (heh) meanings.

In differential geometry, the same terms have wider meanings involving geodesics and geodesic distance, though they do reduce to the Euclidean sense when we view Euclidean geometry from a differential-geometric viewpoint.

In differential topology, the terms have even wider meanings that do not reduce to the Euclidean sense. Topologists frequently speak of segments when they mean connected smooth $1$-manifolds with boundary; triangles when they mean smooth images of a standard $2$-simplex; circles when they mean compact, connected $1$-manifolds, and so forth.

In other fields, a segment might be a primitive concept; a triangle might mean a three-element set; a circle might mean the solution set of an equation $x^{2} + y^{2} = r^{2}$ or some such.


Strictly speaking, it's not well-posed to ask a terminological question as if it's a question about reality, it's only meaningful to ask in some mathematical context.

"Are triangles with curved sides still considered triangles, in Euclidean space?" (Emphasis added.) As 5xum's answer says, "no". In the context of Euclidean geometry, a triangle has three (probably non-collinear) segments as sides.

As JPC's answer notes, a triangle in spherical geometry means something else.

Mathematicians do sometimes speak of (curvilinear) triangles in the sense of the question, but only in settings such as topology, where it's clear the term is not restricted to its Euclidean meaning.

As the joke goes, "Democritus called [the fundamental substance of the universe] atoms. Leibniz called it monads. Luckily the two men never met, or there would have been a very dull argument."

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Definitions:

A triangle is a polygon with three edges and three vertices.

A polygon is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain (or polygonal circuit).

From those two definitions, it should be clear that no, a "curved triangle" is not a triangle.

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  • $\begingroup$ But what is the reason to define them as consisting of three straight lines? I could squeeze the polygon so a new curved polygon appears. $\endgroup$ – Methadont Jun 14 at 10:41
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    $\begingroup$ @Methadont That's not a mathematical question, that's a philosophical one. If you ask me, we define them as such because such a definition is useful. Even if we defined "triangle" to mean something else, the collection of concepts that is currently encompassed by the word "triangle" would still be useful, and we would need a new word to describe that set. Which just invites tremendous amounts of confusion for zero benefit. $\endgroup$ – 5xum Jun 14 at 10:43
  • $\begingroup$ There are triangles in the geometry of the surface of a sphere. Here the sides are geodesics, i.e. paths which are shortest connecting two points. In usual 2-space, geodesics can be proved to be straight line segments. $\endgroup$ – coffeemath Jun 14 at 10:45
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    $\begingroup$ @Methadont In other words, if you want "triangle" to mean something else, then we need a new name for a polygon with three edges and three vertices. For example, we could call them threedoodles. Now, every theorem in every book that currently refers to triangles is wrong. It should instead be referring to threedoodles. When you read a book, you now need to check if it was written pre-2021 (in which case, when it says "triangle", it actually means "threedoodle") or post-2021. And that's just a whole world of messy. $\endgroup$ – 5xum Jun 14 at 10:48
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    $\begingroup$ @Methadont That was a tounge-in-cheek name, but my point is absolutely not meant as a joke. $\endgroup$ – 5xum Jun 14 at 10:50
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In Euclidean geometry, a polygon with curved sides is sometimes called a curvilinear polygon. For example, the MathWorld definition of Reuleaux polygon defines a Reuleaux polygon as a particular kind of curvilinear polygon.

If a curvilinear polygon has only three sides then it is a curvilinear triangle. Searching for "curvilinear triangle" I found some curvilinear triangles in the Euclidean plane, but other than the Reuleaux triangle the examples I found had only one curved side. (Thanks to someone else's comment, however, I can also say that an arbelos is another special type of curvilinear triangle.)

I don't think there are a lot of interesting things you can say about curvilinear triangles in general, the way you can say interesting things about a triangle in general (such as the sum of its interior angles, or the sine and cosine laws). A curvilinear triangle is also kind of closed curve; there is a lot to be said about closed curves, but not so much about the kind with three "sides" in particular that would be different from one with some other number of sides.

So yes, you can form a figure by connecting together three curves to make a single closed curve on a Euclidean plane, but when people do this they usually have a very specific curvilinear triangle in mind and will draw conclusions about that curvilinear triangle that do not translate to other curvilinear triangles.

If you go beyond Euclidean plane geometry there are all kinds of variations in three-sided figures, but then you have things that are meaningful only in the context of whatever specific kind of geometry that you are doing.

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  • $\begingroup$ So all curved triangles are in fact normal ones? $\endgroup$ – Methadont Jun 14 at 13:32
  • $\begingroup$ It is not completely clear what you meant in that comment, but I guess you are asking again, "Are triangles with curved sides still considered triangles, in Euclidean space?" The answer to that question is an emphatic no. If you connect together three curves in Euclidean space, you need a word like "curvilinear" or "Reuleaux" in front of the word "triangle." Otherwise people will assume you mean a triangle with straight sides, and they will react unfavorably when they find out you mean something else. $\endgroup$ – David K Jun 14 at 22:57
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I'm pretty sure that a "curved triangle" in a 3d euclidian space would just be a section of a sphere, but in a 2d space I'm not so sure what the correct nomenclature would be, but it would not be a triangle.

btw here is a video about noneuclidean geometry, it does not go very deep on the subject but it is very entertaining. At the beginning tho, it kind of talks a bit about spherical geometry and euclidian tiling, which could be composed of "spherical triangles". https://www.youtube.com/watch?v=zQo_S3yNa2w

edit: I just googled it and the nomenclature for a "curved triangle" on a 2d space would be a reauleaux https://en.wikipedia.org/wiki/Reuleaux_triangle#:~:text=A%20Reuleaux%20triangle%20%5B%CA%81%C5%93lo%5D%20is,other%20than%20the%20circle%20itself. Math can be strange

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    $\begingroup$ A Reuleaux triangle is one kind of object defined by three curved sides. There are infinitely many other ways to connect three points. $\endgroup$ – David K Jun 14 at 11:05

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