What is the square of a triangle?

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 = L^{1), area of squares (L x L = L}2), 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 Hopefully we have someone with less rusty (and outdated?)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.

It’s all quite simplex really.

0-simplex is a point
1-simplex is a line
2-simplex is a triangle
3-simplex is a tetrahedron
k-simplex is a k-dimensional polytope

https://en.wikipedia.org/wiki/Simplex

A tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square.

https://en.wikipedia.org/wiki/Tesseract

The Windmills of Your Mind – Noel Harrison
https://www.youtube.com/watch?v=WEhS9Y9HYjU

@Pinguidchance Ok but @RedDeerGuy1 seems to have been asking about extending the pattern he perceived to spheres and triangles beyond three dimensions. Cube is to tesseract as sphere is to what and as cone is to what, if anything?

http://mathworld.wolfram.com/Hypersphere.html

https://en.wikipedia.org/wiki/Hypercone

Your terms are all wrong. What do you mean by “a square of” in this context? A “square of a circle“ is impossible and a tesseract exists in four dimensions but a sphere in three.

To try to answer your main question this may help…”The triangle and the triangular pyramid have higher-dimensional analogues, known as simplexes. Three points in a plane, not lying in a line, determine a triangle, also called a two-simplex. Four points in space, not lying in a plane, determine a triangular pyramid, also called a three-simplex or tetrahedron. The n-simplex is the smallest figure that contains n + 1 given points in n-dimensional space and that does not lie in a space of lower dimension.

If we analyze the numbers of edges, triangles, and other simplexes in an n-simplex, we find a pattern that occurs over and over again in algebra and in the study of probability. The reappearance of such patterns is one of the most beautiful aspects of mathematics.”

See here

If by “square of a triangle” you mean a shape with triangles on the top and bottom having corresponding vertices joined by segments then you are talking about triangular prisms. This can be generalized in an obvious way. A cube is a square prism.

Instead of getting hung up on the names for things, here is something profound to contemplate. Take any polyhedron without any holes in it. Let F = number of faces, E= number of edges and V = number of vertices. Then F – E + V = 2. For a cube this gives 6 – 8 + 12 = 2 and for a tetrahedron it is 4 – 6 + 4 = 2. The result is due to the great mathematician Leonhard Euler and is one of the fundamental results of the branch of mathematics called topology, which he created.

Here are a bunch of other names and concepts to contemplate.

Two parallel copies of a shape can also be joined to form an antiprism:
Link

Then there are the 5 beloved Platonic solids, where all the faces, edges and vertices are the same.
Link

Finally, there are the Archimedean solids, which have identical vertices. The most familiar of these is the soccer ball shape formally known as a truncated Icosahedron:
Link

@all thanks