> I never understood there is a relationship between quadratic equations and some kind of underlying mathematic geometric symmetry.

In a polynomial equation, the coefficients can be written as symmetric functions of the roots: https://en.wikipedia.org/wiki/Vieta%27s_formulas - symmetric means it doesn't matter how you label the roots, because it would not make sense if you could say "r1 is 3, r2 is 7" and get a different set of coefficients compared to "r1 is 7, r2 is 3".

Since the coefficients are symmetric functions of the roots, that means that you can't write the roots as a function of the coefficients - there's no way to break that symmetry. This is where root extraction comes in - it's not a function. A function has to return 1 answer for a given input, but root extraction gives you N answers for the nth root of a given input. So that's how we're able to "choose" roots - consider the expression (r1 - r2) for a quadratic equation. That's not symmetric (the answer depends on which one we label as r1 and which we label as r2), so we can't write that expression as a function of the coefficients. But what about (r1 - r2)^2? That expression IS symmetric - you get the same answer regardless of how you label the roots. If we expand that out we get r1^2 - 2r1r2 + r2^2, which is symmetric, which means we can write it as a function of the coefficients. So we've come up with an expression whose square root depends on the way we've labeled the roots (using Vieta's formulas you can show it's b^2-4c, which you might recognize from the quadratic equation).

Galois theory is used to show that root extraction can only break certain types of symmetries, and that fifth degree polynomials can exhibit root symmetries that are not breakable by radicals.

> What is this [Greek notation] about?

The Greeks "notation" was a diagram full of points labeled by letters (Α, Β, Γ, ...) with various lines connecting them and a list of steps to do with an unmarked ruler and a compass, some of which added new points. But those tools alone can't be used to describe cube roots of arbitrary numbers (or, equivalently, trisections of arbitrary angles).

> What is this [statisticians' function]?

The integral of the bell curve (normal distribution) is called its cumulative distribution function (CDF). The CDF of the normal distribution is closely related to a special function called the "error function" erf(x).

> Is there a good intro to [the symmetry interpretation of the quadratic formula]?

There's some discussion at Wikipedia's article about the quadratic formula: https://en.wikipedia.org/wiki/Quadratic_formula#By_Lagrange_...

> However, I have questions: "Turns out the quantity they needed exists, but couldn't be described in their notation" What is this about? Sounds interesting.

There are hierarchies of numbers (quantities) in mathematics, just as there are hierarchies of patterns (formal languages) in computer science, based on how difficult these objects are to describe. The most widely accepted hierarchy is actually the same in math and CS: rational, algebraic, transcendental.

In math, a rational number is one that can be described by dividing two integers. In CS, a rational pattern is one that can be described by a regular expression (regex). This is still "division": Even when we can't do 1-x or 1/x, we can recognize the pattern 1/(1-x) = 1 + x + x^2 + x^3... as "zero or more occurrences of x", written in a regex as x*.

In math, an algebraic number is one can be found as a root of a polynomial with integer coefficients. The square root of 2 is the poster child, solving x^2 - 2 = 0, and "baby's first proof" in mathematics is showing that this is not a fraction of two integers.

In CS, an algebraic pattern is one that can be described using a stack machine. Correctly nested parentheses (()(())) is the poster child here; we throw plates on a stack to keep track of how deep we are. The grammars of most programming languages are algebraic: If the square root of math is like nested parentheses, then roots of higher degree polynomials are like more complicated nested expressions such as "if then else" statements. One needs lots of colors of plates, but same idea.

In math, everything else (e, Pi, ...) is called trancendental. CS has more grades of eggs, but same idea.

One way to organize this is to take a number x and look at all expressions combining powers of x. If x^3 = 2, or more generally if x is the root of any polynomial, then the list of powers wraps around on itself, and one is looking at a finite dimensional space of expressions. If x is transcendental, then the space of expressions is infinite.

So where were the Greeks in all this? Figuring out where two lines meet is linear algebra, but figuring out where a line meets a circle uses the quadratic formula, square roots. It turns out that their methods could reach some but not all algebraic numbers. They knew how to repeatedly double the dimension of the space of expressions they were looking at, but for example they couldn't triple this space. The cube root of 2 is one of the simplest numbers beyond their reach. And "squaring the circle" ? Yup, Pi is transcendental. Way out of their reach.

When you have a hammer you see nails. When you have a circle you see doubling.

Yes, this is all Galois theory.

What is so unbelievably frustrating about math education is that these interesting questions are not even hinted at until far, far down the line (and before people make the assumption I was educated outside the US).

I avoided math like the plague until my PhD program. Real analysis was a program requirement so I had to quickly teach myself calculus and get up to speed—and I found I really, really liked it. These high level questions are just so interesting and beyond the rote calculation I thought math was.

I hope I can give my daughter a glimpse of the interesting parts before the school system manages to kill her interest altogether (and I would welcome tips to that end if anyone has them).