What is the square of a triangle?

1 Answer

Answer :

The only way to answer the verbal form is by carefully being sure of the meanings of each word. I used to be very strong in math, but I’m rusty. When I hear square I think of a 2D shape, or the concept of multiplying a number (a one-dimensional value) by itself. There is a spatial metaphor for that, but I don’t know if it’s accurate to extend that to say (other than metaphorically, or in some other more precise way) that the square of a line segment is a square, or especially that the square of a square is a tesseract, per se, although it does make metaphorical sense… but I’m not sure there’s formal agreement on that point. I would also think that if a square of a square is a tesseract, then the square of a circle would be more likely to be the metaphorical tesseract equivalent of something with one more dimension than a sphere. I got pretty far in math but it was decades ago, and I don’t know that there was agreement on what 4+ – dimensional shapes would be conceived as. Calling for help from Google, I see there is also an existing expression for squaring a square which seems to be something else. Similarly cubing a cube, which is not really talking about the same spatial concept you seem to have been going for. Squaring is also the inverse of finding the square root, and I believe it applies to one-dimensional values, not to n-dimensional shapes, even though there is a nice analogy in the case of length of segments (L = L1), area of squares (L x L = L2), and volume of cubes (L x L x L = L^3). And I would say you can actually say that a segment – square – cube corresponds to segment, circle, sphere and to segment, (right?) triangle, cone (not pyramid). But if there are accepted four-dimensional shape concepts & terms, I don’t know what they are. Hopefully we have someone with less rusty (and outdated?) math than mine. @stanleybmanly I think it’s really interesting trying to understand mathematics by spatial analogy. Seems like an intelligent creative approach, as opposed to the usual one based on concise verbal definitions, which seems to be the weak point.

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