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.Īs the joke goes, "Democritus called atoms. In the context of Euclidean geometry, a triangle has three (probably non-collinear) segments as sides.Īs JPC's answer notes, a triangle in spherical geometry means something else. "Are triangles with curved sides still considered triangles, in Euclidean space?" (Emphasis added.) As 5xum's answer says, "no". 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. 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 $x^$ or some such. 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 differential topology, the terms have even wider meanings that do not reduce to the Euclidean sense. 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 Euclidean geometry, terms such as segment, triangle, and circle have specific and rigid (heh) meanings. Despite this, words do not map bijectively to concepts, and there is no practical way to adjust language to form a bijection. Mathematics generally strives for precision. Natural language is flexible, and that flexibility gives us idioms and metaphors, which can be useful (or telling). (I've heard this called The Fundamental Theorem of Semiotics.) 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.
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