Are curved triangles still considered triangles?
By Sebastian Wright
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?
$\endgroup$ 94 Answers
$\begingroup$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."
$\endgroup$ $\begingroup$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.
$\endgroup$ 7 $\begingroup$In Euclidean geometry, a polygon with curved sides is sometimes called a curvilinear polygon.For example, the MathWorld definition of Reuleaux polygondefines 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.
$\endgroup$ 2 $\begingroup$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".
edit: I just googled it and the nomenclature for a "curved triangle" on a 2d space would be a reauleaux . Math can be strange
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