Jason (jcreed) wrote,
Jason
jcreed

Got to the bottom of a bug at work which turned out to have a totally different cause from what I expected.

I keep having this recurring thought about polyhedra and linear algebra which goes like this:

Suppose I ask you to pick two real numbers that have to add to zero. How many ways you can you do this? Obviously, infinitely many, but the linear algebraist in you says that, aha, there's one degree of freedom, there's one *dimension*'s worth of ways that you can do that: (r, -r) for any r.

Think of one of these possible (r,-r) pairs
*      *
r     -r

like a labelled edge with "r amount of stuff flowing to the left"
*--<--*
   r   


Imagine now a polygon with a bunch of sides, like a square maybe:
*----*
|    |
|    |
*----*

and I ask you how many ways you can label the edges with flows,
such that every vertex "satisfies Kirchoff's law", i.e. has
a (in-flow + out-flow) of zero. Again, infinitely many, but
the linear algebraist on your shoulder says there's *one dimension* many
ways to do it. Here's one example:
     r
  *-->--*
r ^     |
  |     v r
  *--<--*
     r

Now if I had some configuration of edges that *wasn't* a polygon, I wouldn't have exactly one dimension to wiggle around in.
Something with dangling edges like
  *-----*
        |
        | 
  *-----*

this graph doesn't admit any nonzero flows without some vertex being a source or a sink.
And a graph with more than one loop would admit *more* than one dimension of Kirtchoff-satisfying flows:
     r     s
  *-->--*--<--*
r ^     |     ^ s
  |     v r+s |
  *--<--*---->*
     r       s

So anyway this is nice that the polygons are *characterized* as the graphs for which the Kirchoff-satisfying flows, the flows that sum to zero at every vertex, form a one-dimensional vector space.

But we can turn the crank some more, and redraw
     r
  *-->--*
r ^     |
  |     v r
  *--<--*
     r

as
  *-----*
  |  r  |
  |  @  |  
  *-----*

(where that @ sign is meant to stand for a little clockwise pointing
swirly arrow) and investigate arrangements of polygons that are allowed to share edges.
We can try to choose amounts of swirly-flow going around each polygonal face
on its edges, such that for every edge, the flows coming from all the faces
it connects to add up to zero.

And my intuition says that these are exactly closed oriented (but not necessarily simply-connected) polyhedra, and things start feeling all Stokes'-theoremy.
Tags: math, work
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