Drawing the States Instead of the Clock

Last timeNudge It, and See Whether It Comes Back

Plot the two numbers of a state against each other instead of against time. The clock disappears, a history becomes one curve, and the fact that curves cannot cross decides a great deal.

So far every picture has had time running along the bottom. That is the obvious

choice and it has a cost: a system of two numbers needs two curves, and the thing we

actually want to know, how the two numbers depend on each other, is not drawn at all.

This lesson throws the clock away.

FIG 1A state is a point, a history is a curve

: the state after k steps, which was a list of two numbers and is now a location
: the two numbers read as coordinates. For the valley, rabbits across and foxes up
: is a point of the plane. The notation says only that two numbers make a location, which is the whole idea this lesson rests on

The two numbers of the state are read as the two coordinates of a point. Every situation the system can be in is somewhere in this plane, and every point in the plane is a situation. Advancing the rule moves the point, so a history is a sequence of points, and joining them gives a curve. The step number k labels the points but is not plotted.

The state as a point

Take a weight on a spring, which is the simplest two-number system there is and the

one case where the answer is known exactly in advance. Its state is where the weight

is and how fast it is moving. Plot those against each other.

FIG 2A weight on a spring, drawn as a state picture

total energyunder 0.001% of it gone by the end, at every setting of every slider

220 of 220 steps

step220

how far it is pulled out before releasea = 1

stepped by: v → v - 0.03*2.0*x x → x + 0.03*v
each rule reads: whatever the rules above it have already produced
started at: v = 0.0, x = a
held steady: total energy = 0.5*v*v + 0.5*2.0*x*x
drift: under 0.001% over the run, against 0.20% allowed; swings 4.2% and returns

Press run and the weight slides back and forth, while the chart beside it draws not the motion against the clock but the position across and the speed up. The weight starts pulled out and at rest, which is the point on the right where the curve meets the horizontal axis, and goes clockwise. It is at its fastest as it passes through the middle, which is the top and bottom of the curve, and momentarily at rest at each end of its travel. The whole history is one closed curve, and pulling it out further with the slider draws a larger curve around the same centre rather than a differently shaped one. The figure was kept only because the total energy stayed fixed across the entire run, at every setting, to within a part in a million.

Everything about the motion is in that single loop, and the clock is nowhere in it.

Going round once is one complete swing out and back. The slider is the size of the

first pull, and moving it does the one thing worth seeing here: the loop grows and

shrinks about a fixed centre without ever changing shape. Pulled all the way in to

nothing, which is the resting state, it would draw a single point.

The lesson stops here

5 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

Drawing the States Instead of the Clock

The state as a point

5 more paragraphs to go

This is the reading half

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