[plt-scheme] 3rd-8th Grade

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