For those of you who have seen the documentary "Gigantic: A Tale of Two Johns", this was one of the things they did during the five-year period after they left Elektra as a means of reminding fans that they still existed. It's mostly a collection of rarities that, for some reason or another, didn't make it onto other albums, but there are a few stand-out tracks. One of my favorites is a little ditty called "Lullaby to Nightmares".
It has a nice tune with a nice arrangement and performance by Klezmer trumpeter Frank London. And the lyrics have a lot of great somewhat-morbid, somewhat-reassuring imagery. But my favorite lines are the following:
Underneath a shady tree,
A shadow sitting next to me
And we stare at the sun.
At first, it seems like a pretty scene, just me and my shadow sitting under a shady tree and staring at the sun. But once you start thinking about it, you realize something's not quite right.
How can a shadow stare at the sun?
How can you stare at the sun if you're underneath a shady tree?
If you're underneath a shady tree, why is the shadow next to you?
The three lines may paint a nice picture, but none of the lines logically makes sense in the context of either of the other two lines. It's like, in some vague figurative sense, the three lines are mutually orthogonal.
Or are they?
On further inspection I realized that these three lines are, in fact, not mutually orthogonal in some vague figurative sense. They are mutually orthogonal in an explicit literal sense.
Consider the following diagram:
The three vectors in the picture point to where the sun has to be for each of the lines. In order to be "underneath a shady tree" the sun has to be above you, so that the crown of the tree is between you and the sun. In order for a shadow to sit next to you, you have to be between the sun and the shadow, or the sun so the sun has to be on your side. In order for you and to shadow to stare at the sun the sun has to be in front of you or else one of you will block the other's view or the tree will block both of your views.
So the vectors going from you to the sun in each line are mutually orthogonal. If you try to reconcile any two lines of the song by taking their dot product, you get nothing. It just won't work.
On the other hand, if you try to reconcile two of the lines by taking a cross product, you'll get either the third line or negative the third line depending on what order you take the cross product.
Assuming the shadow is sitting to your left, then those three lines of the song, in the order in which they appear, form a right-handed system resulting, so the first line cross the second line is the third line. So in this sense, the lines make perfect sense.
And that's why "Lullaby to Nightmares" is utterly brilliant.