Fine-tuning,
made visible.

Two chemicals diffuse and feed on one another on an even field. Nothing here decides to become a leopard’s coat, a coral, or a dividing cell. The same two lines of arithmetic run at every point, and the pattern is what those lines must produce.

And only just here. The whole of this teeming variety lives inside a thin, curved sliver of the feed/kill plane. Turn either dial off its narrow band and the field stops. It settles to a flat, sterile uniform, chemistry with the life subtracted out.

Scatter fine sand on a metal plate and draw a bow across its edge. At almost every pitch the grains only shiver and wander, with no picture in them, just agitation.

Then you find a note. The plate divides itself into still lines and trembling fields, and the sand walks itself into a figure of perfect symmetry. A hair sharp or flat and it dissolves. The plate will hold a pattern only at its exact resonant frequencies, a ladder of single numbers standing in a continuum of silence.

Pick a single complex number c. For every point of the plane, ask whether repeatedly squaring it and adding c keeps it bounded. The points that stay form an intricate shape, and the same squaring rule writes its detail at every depth you care to zoom.

Now move c. While c stays inside the Mandelbrot set the shape holds as a single connected body. The instant c steps over the boundary, the body disintegrates into a dust of infinitely many disconnected pieces. The change is total, and it turns on a boundary of zero width. On the real axis that crossing falls at exactly one quarter.

One equation for a population that grows and competes with itself. Raise the single parameter r and the steady level splits in two, then four, then eight, each doubling arriving sooner than the last, the forks crowding toward a wall at r∞ = 3.5699, where the period becomes infinite and order gives way to chaos.

The crowding has a measure. Each gap is about 4.669 times the next, and that ratio, Feigenbaum’s constant, is the same for a dripping tap and a stirred chemical reaction, which know nothing of this equation. A number sits underneath all of them.

Every carbon atom in you was assembled inside a star, three helium nuclei at a time. That reaction would be hopelessly slow, but for one thing: the carbon nucleus carries an energy level sitting almost exactly where the heat of the star can reach it. Fred Hoyle reasoned that the level had to exist, on no evidence but the fact that carbon does. It was found at the energy he named.

Shift the strength of the strong nuclear force by a single percent and that level slides off the narrow thermal window. The reaction rate, riding the same exponential that sets every fusion rate in every star, falls by orders of magnitude, and the carbon is never made. Push the force the other way and the carbon that forms is burned on to oxygen. A star makes both, the two atoms a chemistry needs, only inside a narrow band of one constant. Hoyle, who began an unbeliever, wrote that it looked as though a superintellect had monkeyed with physics.

A growing tip lays down its seeds one at a time, and each new seed sits a fixed turn around the centre from the seed before it. A sunflower does this, and a pinecone, and the scales of a pineapple. The whole look of the finished head, whether it fills evenly or breaks into coarse spokes, is settled by that one angle and nothing else.

Set the angle to a simple fraction of a turn and the seeds fall onto a handful of radial arms with bare ground between them. Every fraction does this, each at its own count of arms, and those angles lie scattered the length of the dial. One value alone slips past all of them, because the number that names it, the golden ratio, is the hardest of all to reach with any fraction. There, near 137.508 degrees, the seeds pack the tightest and leave no seam.