Essays/A Fine Line

A fine line

A good way of thinking about frames and cells is, when looking at the shape of a noun (for example 4 3 2 7 6), to put a line somewhere in it.

The frame-shape is the numbers to the left of the line, and the cell-shape is the numbers to the right.

Examples

So, for example, a rank 2 verb will put the line 2 axes in from the right of the shape:

   rank  2:   4 3 2 | 7 6

Similarly:

   rank  1:   4 3 2 7 | 6
   rank  0:   4 3 2 7 6 |
   rank  _: | 4 3 2 7 6

Now, whereas a verb with a nonnegative rank N will put the line N axes in from the right of the shape, a verb with negative rank -N will put the line in N axes from the left of the shape:

   rank _1:   4 | 3 2 7 6
   rank _2:   4 3 | 2 7 6
   rank __:   4 3 2 7 6 |

Consequences

So, nonnegative ranks let you specify the rank of the cells on which you'll operate; negative ranks let you specify the rank of the frame you want to produce.

That's why we say nonnegative ranks are absolute and negative ranks are relative. A verb with a nonnegative rank will never see a noun with rank greater than that; a verb with negative rank could see a noun of an arbitrary rank. A cute insight is that this maxim is inverted for the infinite ranks: a verb of (positive) rank _ could see a noun of arbitrary rank, whereas a verb of (negative) rank __ will never see a noun larger than an atom.