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Squares are special types of rectangles.

Are circles special types of ellipses/ovals?

Are cones special types of pyramids? I guess the answer is no because of the 2D basis: circles are not special types of however you categorise the bases of pyramids. However, cubes are cuboids/(right?) rectangular prisms because

  1. Why/why not to all of the above? How do we know if a certain thing is a special type of another thing?

  2. Who decides? Or is there a way to determine logically? If someone decides, should this be on some official list on the state where this someone has authority?

Maths is supposed to be about logic. If someone decides that squares are rectangles (that is defines the words to be that way) rather than that there is some logical way to deduce such, then these all seem so arbitrary. I think it's similar to how English says 'The plants are tall' not 'The plants is tall' whereas in French or Spanish, I think, the linking verb may contain an 's' if the noun would contain an 's'.

  1. Why isn't there a term for equilateral triangle but there is a term for equilateral rectangle?

  2. Why does the word rectangle exist? Why don't we just use oblong? Is this an English thing? Or a maths thing? Or what?

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    $\begingroup$ For what it's worth: I would say a pyramid is a special case of a cone. The most common cone is a "right circular cone" but the generic word "cone" includes others, too. $\endgroup$
    – Gerald Edgar
    Commented Mar 27, 2018 at 11:52
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    $\begingroup$ I'm voting to close this question as off-topic because it is really a question about language, and this topic has been covered exhaustively in a half-dozen other questions. $\endgroup$
    – Chris Cunningham
    Commented Mar 27, 2018 at 12:35
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    $\begingroup$ Since downvoting suggests I should comment with ways to improve the question, here goes: will any answer satisfy your curiosity on this topic? Have you asked yourself if you actually have an answerable question, or maybe you'd rather take this over to the chat page? $\endgroup$
    – Nick C
    Commented Mar 27, 2018 at 16:13
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    $\begingroup$ For the record, I voted to close this as "too broad" (if I'm not mistaken) and not as "off-topic." $\endgroup$
    – JRN
    Commented Apr 2, 2018 at 10:04

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The question, as you pose it, is a bit too generic.

What does it mean to be "A special case"? It means there is a definition of a class of math objects such that this object belongs it. A square is a special case of rectangle (a rectangle with equal sides) and a special case of quadrilateral, and a special case of closed simple curve in the plane and so on...

Are cones special cases of pyramids? It depends on your definition of pyramid. If you intend pyramid as a cone over a closed simple plane curve than yes, while if you insist that the base is a polygon then no.

Is there an authority accepting formally definitions? No. It is the community of mathematicians that decides what should be an accepted definition of a concept: so it has to do with "sociological issues" as well as to logical ones.

For years the definition of parallel lines was "with no common points". Now it is usually said that two lines are parallel if there exists a non trivial vector to which they are both parallel. This is because we want to stress the fact that being parallel is an equivalence relation and therefore we want to say that any line is parallel to itself. Two centuries ago maybe equivalence relation and equivalence classes were not used as ubiquitously as now. This may explain the difference.

Of course it is a matter of conventions: but such conventions are not certified by authorities but by the community of mathematicians as a whole. Historical reasons enter the scene, here.

If a concept has a constant widespread use it will sooner or later find its name...

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  • $\begingroup$ Thanks Nicola! 1. So, is there a universally accepted definition of pyramid? 2. What do you say to isosceles or trapezoid ? $\endgroup$
    – BCLC
    Commented Mar 27, 2018 at 7:56
  • $\begingroup$ "For years the definition of parallel lines was "with no common points"." I'm skeptical of this claim, because it seems unlikely that skew lines were ever considered to be parallel. $\endgroup$
    – ryang
    Commented Oct 11, 2021 at 9:50

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