lol I see how this shower-thought can seem obvious.

What lead to the shower-thought was thinking about dimensions in linear algebra. If you want to represent a function with more parameters, you need more dimensions.

For example, two parameters could be represented by ax + by = c where a, b, and c are constants and x and y are real numbers. Note that this equation describes a 2-D plane. Three parameters would require an additional variable and an associated constant: ax + by + cz = d, where d is an additional constant and z is an additional real number. Note that this equation describes a 3-D space.

Can you see how if you wanted to represent four parameters, you would need four dimensions?

However, facet plots seem to override this need for more dimensions in a particular way: splitting up axes, like cutting up a cake. If you have four parameters (in which two of them can only take up discrete values), instead of requiring four dimensions, you can split up two dimensions in discrete chunks, like a cake, and represent four parameters in two dimensions. That was interesting for me to realize.

I guess for cake-cutters, this post is silly and trivial. But for someone trained to think “more parameters = more dimensions in the sense of going from ax + by = c to ax + by + cz = d”, it was surprising to realize facet plots break that rule.