> btw can you tweak it so your comments dont fall off the page
Tricky to find anywhere else to put them. But if you just move the
cursor until a particular comment of interest is the active cell,
you'll be able to read its entire contents in the formula-entry text
box at the top of the screen.
> i am so pleasantly surprised also to see you opening here below to
the
> possibility that a line which is nonphysical in the first place
might
> need to disconnect from itself in order to pass over an also
> nonphysical point that is actually part of that line etc etc while
it
> could easily within nonphysical reality simply cross itself no
problem
> so unless such an elegant solution to what you call your odd
problem
Did I say that? I think the boat you're pushing out isn't the one I
launched. I'm happy to wave as you recede into the mist, though.
As I say in my most recent 24-hr delayed mail, I think of borders as
line segments that connect the dots of tri- and higher points. (Can I
coin "polypoint", at least for the restricted purposes of this
posting, to mean "tri-point or higher"? I think you've already
used "multipoint" for other duties.)
So in my boundaries spreadsheet, a border segment is either a closed
loop or a line with a polypoint at each end (that may be the *same*
polypoint, in the case of Jungholz, and the three other binational
quadripoints.)
If you're making a border tour, you march along a boundary segment
until you come to a polypoint, and then you can choose a *new*
boundary segment to follow, from the 2 or more on offer at that
point. So the lines don't cross the polypoints because, as Bill says,
there's a different borders on the other side of the point - clearly
so in the case of a tripoint, but not so convincingly in the rare
case of a binational quadripoint, I admit.
Now, you can (and do, with some relish) choose to make your border
tour along the German/Austrian border and around Jungholz in a
crossed loop. Me, I like to think of the border segments as having a
*direction* imposed on them by any border tour: a vector defined by
some statement like "I'm walking with Germany on my left and Austria
on my right"; equivalent to "I'm going round Austrian territory
clockwise." So at the quadripoint I either turn left and loop
clockwise around Jungholz, or right to continue my tour of "mainland"
Austria, either way maintaining the handedness of my tour. You steam
straight ahead, unfazed by the sudden switch of Austria from your
right to your left hand, doing a sort of interdimensional jig
analoguous (I just realise) to the outside-becomes-inside
interpenetration of the single surface of a Klein bottle.
But I think you quite enjoy doing an interdimensional jig.
Grant