Explore. Look closer
The lake from Chapter 2, a cubic kilometer of water, carried the way an economical machine would carry it. Wherever you look, the machine works out the detail you can see, and only there.
Most of the lake can be carried coarsely. Its depths can be a field of slow currents and temperatures, its far shallows a handful of numbers, and its surface computed finely only where light, wind, and a skipping stone make demands on it.
The lake
Drag to move across the lake, and pinch, double tap, or use the buttons below to look closer. With the keyboard, the arrow keys move and the plus and minus keys look closer or farther.
- What you are looking at, carried at the grain of your looking
- The rest of the lake, carried as summaries that grow coarser with distance
- Each fish, carried in full with its halo, whether or not anyone is looking
What it must compute is everything any of them could reach, carried at the grain of the reaching, for as long as anyone is there to reach.
As you are looking at it, the water costs about 200 million operations each second. Carried in full at molecular grain, the whole lake would take about 3 × 10to the power 54 each second.
The lake
Described at the grain of fluid flow, the lake becomes cheap. Its water is treated as a continuous medium with a velocity, a pressure, and a temperature at each point, governed by the equations of hydrodynamics on a mesh of cells, and a single modern workstation can produce a surface of rings and reflections that most observers would accept as water.
Its surface
The light that breaks on the lake must be as fine as the boy’s eyes can resolve, and the far stars must hold as much detail as the best telescope of his descendants can find.
Where the fish swim
Around every fish there is therefore a halo of finely computed water reaching as far as the fish can sense, moving with it as it swims. The halo thins with distance, since a sound from across the lake reaches the fish as a pressure wave that a coarse description carries faithfully, while the water at its gills must be carried molecule by molecule.
Ripples
Whatever the inhabitants actually perceive and investigate must also be preserved, at the grain at which they perceive and investigate it.
A drop
When a fish takes in water or a native turns a lens on a drop of sea, the organisms the machine supplies must be lawful draws in the sense already set out for molecules, and they must carry no record of experiences that never occurred.
Finer than a lens can show
A machine that answers each act of one person with the sensations it would bring must compute only what she probes, at the grain at which she probes it and consistently with everything it has shown her before, so that nothing she does gives the machine away.
Molecules of water
Described at the grain of molecules, the same lake becomes a finite problem. Each molecule is treated as a small body moving under forces from its neighbors, and a machine advances them all together in steps of about a femtosecond, a millionth of a billionth of a second, which is the time scale on which molecules vibrate.
The floor
A physics can be built so that close enough inquiry reveals its floor, or so that the floor stays hidden as far down as inquiry can go.
The floor
As the designer, choose what lies at the bottom of this world, then look all the way down.
A lattice. A careful inhabitant can find this floor, and its rows point in preferred directions.
A regular lattice, with points spaced evenly in rows, is the cheapest to carry, since the machine can lay out its memory the same way, and it gives the bottom of the world a set of preferred directions.
A random scattering. A careful inhabitant can find this floor, and it favors no direction.
A random discreteness in the manner of causal sets would keep the world free of preferred directions at every scale, at the cost of an irregular structure that is harder for a machine to store and to update.
Refined wherever anyone looks. No one reaches this floor. Each closer look finds new structure, consistent with every look before it.
If the machine refines its description of matter only when someone probes deeply enough, then every deeper probe must find structure consistent with every shallower result, across every laboratory and every century. The promise has no end date.
How the count is made
The count follows the chapter’s rule for molecules, a hundred operations for each cell at each step, and everything else in it is an assumption of this page. The lake is taken to be about twenty meters deep, so its cubic kilometer covers some fifty square kilometers. The water you are looking at is carried on a mesh fine enough for what the view can show, about 256 cells across and as deep as the view is wide or the lake is deep, with each step lasting as long as sound takes to cross a cell. Once those cells would be smaller than a nanometer, the machine carries the water molecule by molecule, in steps of a femtosecond. The rest of the lake is a field of currents and temperatures on cells a hundred meters across. The boy and every fish are carried in full with their halos whatever you look at, and the count leaves them out.