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Comment by honkhonkpants | Hacker News Reader

What is the geometric meaning of a squared square?

It's more how the information is presented, and the style of deduction used by mathematicians of the time. For over 2000 years, Euclid's Elements (13 incredibly dense volumes of geometric deduction) were the foundation of mathematical education in academia:

https://en.wikipedia.org/wiki/Euclid%27s_Elements

They aren't really books of equations, just lots and lots of verbose axioms and theorems and conclusions about properties of points/lines/triangles/circles/etc.

A square (the squaring operation that is, not the shape) in this context still means x * x, but it would be described (and reasoned about) in terms of being the surface area of a shape or the length of a line, and logically used in comparison to the surface areas of other shapes or the lengths of other lines. A squared square would still mean x^4, but it would be presented in terms of some quantity (x^2) which was shown to be the square of some other quantity (x), which also happens to be the square root of yet another quantity (x^4) according to the logic of whatever property the proof was trying to establish.

An example of a geometric property of the squaring operation could be a simple statement along the lines of, if the side of one (geometric) square is shorter than the side of another square, then the area of the first square is guaranteed to be less than the area of the second square. Which all sounds painfully trivial, to the point of not even being worth talking about, but in the context of Euclid it would have been a meaningful relationship that could be exploited as part of some much more elaborate proof.

At no point would any of this be expressed in terms of modern mathematical notation, it would all be semiformal verbal descriptions of the relationships between 2D geometric primitives. In the world of philosophizing about 2D geometry, squares and square roots have a conceptual primacy that other exponents do not have (likewise for cubes and cube roots in 3D geometry).

What little I've read of this stuff comes across like the machine language of mathematical deduction, where everything is built up from a very long series of very simple statements that (given enough statements) can eventually produce very sophisticated results. The way math is taught now (at least to non-math majors) comes across more like a very high-level language where we're simply taught rules like x^y * x^z = x^(y + z), and we aren't required to understand it in terms of low-level geometric proofs.

Volume of a hypercube?

That's sort of what I figured, that it was necessarily hyperdimensional. But then I don't see why geometers of this era would have been willing to ponder hyperdimensional solids, but aren't willing to admit arbitrary exponents. Path-dependent historical attitude, I guess.

I think it was probably the first hints of polynomials that could stand on their own without a sane geometric interpretation. As in, "this makes no geometric sense, but I can use the same geometry-based techniques I used on squares and cubes, and it all works out, so let's just go with it."

Another way to interpret it could be, "okay, take this cube with a volume, take that volume and pretend it's actually the width of another special mega-cube and never mix mega-cubes and regular cubes".

Volume of a hypercube? I swear, there's something about maths that makes your brain work in the strangest ways.

A square of a square is a square made up of squares :)

(like a Sudoku puzzle)

Not really. A sudoku puzzle is a set of squares arranged inside another, bigger square. If it were a square of squares, it would mean that you square the initial square's volume including the unit. But in this case, you're only multiplying with a count (or rather, a squared count, since we're talking about two dimensions).

? We're not talking about any units, we're talking about numbers. There are 3^4 squares in a Sudoku puzzle. You don't need a hypercube or any unit of measurement to see that's true.

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