what i do all day II
previously on implicity.. i got as far as explaining how to measure the size of some, smooth-edged shapes. as mercedo pointed out, this is basically the integral calculus. and as mikesheffler observed, smooth is all good and well, but what if the shapes you're measuring aren't smooth?
the first way of resolving this problem is both straightforward to carry out and conceptually plausible: if you have jagged edges, just zoom in until they look smooth. in other words, use smaller and smaller grids to measure the shape until relatively few squares contain pieces of the boundary. this will mean your inner and outer estimates get closer and closer together, until they're close enough for your measurement purposes, whatever they are.
there's two problems with this: one is that, in the case of the stochastic processes mentioned by mike and similar mathematical objects (shapes), no matter how much you zoom in, there are always still smaller zigzags. in fact the big W himself, Karl Weierstrass acheived immortality by, among other things, coming up with something like this, a line which is continuous (joined up) but zigzags up and down at any scale, however small. imagine taking a big zigzag, and replacing each line that makes it up with a smaller zigzag. then take all the smaller lines in the smaller zigzag, and replace them with even smaller zigzags. this may seem like an obvious thing to invent, but the early nineteenth century was rather cautious about such things, and the poor man had to do rather a lot of serious mathematics to prove it would all work.
the second proviso is that while smoothness of boundaries is a surmountable problem. rather more insidious is if we have the same kind of strange-details-at-any-level-no-matter-how-small in the actual fabric of the shape. for example, every piece that looks like solid colour turns out to be filled with tiny and tinier holes all over. or an area that looks like it doesn't contain any of the shape actually has lots of tiny blobs that are part of it when we zoom in.
this is the classic example of the sort of thing we're thinking about
now imagine trying to measure something like this using the techniques from part one. if we have an object which is 'full' of small holes all over the place, we're not going to count a single square as being filled by the shape. so we'll give something an area of zero which seems to be more substantial than that. conversely, when we're counting all the squares which contain some part of our shape, however small, we only need a shape which has tiny pieces sprinkled all over the plane, and we'll overestimate its area as being the same as the whole plane. we could even have a shape which combines these two tricks, so we don't know whether it covers all of the plane or none of it.
(remember that just to have to worry about this sort of shape, we have to accept that we can zoom in forever on the plane. if we tried to do this with a real rather than a mathematical space, we'd find that we soon reach, if not a point where there are no more smaller details, at least a scale where we can't see them any more, no matter what we use to look at them.)
also, the sierpinski triangle pictured above is deceptively simple for solving this kind of problem, because it has the same structure at each scale. what we're really worried about is having to work out the area of something which has different detail at every level. perhaps like this julia set:
these kind of 'difficult' shapes have a whole theory dedicated to them and how to measure them in way which produces 'sensible' answers, ie ones that fit in with the measurements of the simpler rectangle and circles we already know the areas of. my own research has been directed towards examining an alternative way of measuring these awkward shapes, and trying to find out if it works, and works as well as the classical way of doing things, for the whole assortment of 'measurable' shapes dealt with by the classical theory.(to be continued..)