From: Bill Wood (william.wood3 at comcast.net) Date: Mon Mar 20 09:28:15 EST 2006 Previous message: [plt-scheme] 3rd-8th Grade Next message: [plt-scheme] 3rd-8th Grade Messages sorted by: [date] [thread] [subject] [author]

```On Mon, 2006-03-20 at 05:51 -0800, Gregory Woodhouse wrote:
. . .
> Okay, maybe just one (or two) other thoughts before I head off to the
> office. Even with all the nice geometric intuitions we might have
> into Lie algebras, once we pass to n-dimensions (i.e., try and make
> it scale), we end up reasoning about certain pseudograpahs known as
> Dynkin diagrams. The human (or at least, this human) mind doesn't do
> well with n dimensions. An interesting fact about geometry in two
> (complex) dimensions is that it is easy to "algebraize" by passig to
> the quaternions, but you hear little about the Cayley numbers
> (octonians) these days, primarily because they don't even form an
> associative algebra. What may seem a natural correspondence in low
> dimensions may be difficult to generalize.

When we were developing courses on Dijkstra's program development
methods for presentation to engineer-programmers we used a visual model
where a closed curve in the xy-plane indicated the set of states
characterized by a predicate, and evolution of the program step-by-step
was described as a trajectory (up the t-axis).  A loop invariant formed
a tube within which the state sequence was constrained, and the
termination function defined a kind of cone.  As the computation
progressed, the individual states could dance pretty irregularly about
the surface of the cone, but they were drawn inexorably to the vertex, a
circle characterized by the termination condition.  Notice that the
combination of the loop invariant and the termination function acts a
lot like a Lyapunov or potential function.

Engineers appreciated the metaphor, but it doesn't scale at all.  Real
soon you just have to abandon the pictures and rely on the math.

-- Bill Wood

```

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