In R formulas, why do I have to use the I() functi

I'm trying to get my head around the use of the tilde operator, and associated functions. My 1st question is why does I() need to be used to specify arithmetic operators? For example, these 2 plots generate different results (the former having a straight line, and the latter the expected curve)

x <- c(1:100)
y <- seq(0.1,10,0.1)

plot(y~x^3)
plot(y~I(x^3))

further, both of the following plots also generate the expected result

plot(x^3, y)
plot(I(x^3), y)

My second question is, perhaps the examples I've been using are too simple, but I don't understand where ~ should actually be used.

The issue here is how formulas are interpreted. In a formula the tilde separates the left hand side from the right hand side. In formulas the ^ operator is for constructing interactions so that x = x^2 = x^3 rather than the perhaps expected mathematical power. If you had typed (x+y)^2 the R interpreter would have produced (for its own good internal use), not a mathematical: x^2 +2xy +y^2 , but rather a symbolic: x + y +x:y where x:y is an interaction term.

?formula

The I() function acts to convert the argument to "as.is", i.e. what you expect. So I(x^2) would return a vector of values raised to the second power.

The ~ should be thought of as saying "is distributed as" or "is dependent on" when seen in regression functions. It implies an error term in model descriptions which will generally be labelled "(Intercept)" and the function context and arguments may also further determine a link function such as log() or logit().

In plot()-ting functions it basically reverses the usual ( x, y ) order of arguments that the plot function usually takes. There was a plot.formula method written so that formulas could be used as a more "mathematical" mode of communicating with R. In the graphics::plot.formula, curve, and 'lattice' and 'ggplot' functions, it governs how multiple factors or numeric vectors are displayed and "facetted".

I learned later that ~ is actually an infix (or prefix) primitive function that creates an R 'call' which can be accessed with list extraction operators. All of that is hidden from the typical user, but it can be a facility used by more advanced function authors.

The overloading of the "+" operator is discussed in the comments below and is also done in the plotting packages: ggplot2 and gridExtra where is it separating functions that deliver object results, so it acting and as a pass-through and layering operator. The aggregation functions that have a formula method use "+" as an "arrangement" and grouping operator.