steveb (03/11/83)
Ever hear a claim like "Podunk County has more miles of beaches than the Atlantic Ocean"? This leads me to ask: Take a perfectly circular lake with diameter 1 / pi and thus a circumference (shoreline) of 1. Now deform the shoreline by filling in parts of the lake (i.e. the new lake may have any shape whatsoever as long as it lies entirely within the boundaries of the old lake). What is the maximum shoreline you could obtain in this way? Can an algorithm be specified that tells one how to deform the original so as to get to the maximal? What is its shape? It seems to me that posting responses to the net is appropriate, but if you wish to mail me something probably not of general interest, I am: ucbvax!teklabs!tekecs!steveb or decvax!tektronix!tekecs!steveb (DONT use header address)
leichter (03/12/83)
The Question was: What's the maximum shoreline a lake, all of whose water is inside a 1/pi radius circle, can have? (I paraphrased, but I think this is the intent.) The answer is: Unless you put some restrictions on what the shoreline can be, there is NO maximum; you can get as large a shoreline as you like. Any space- filling curve, in fact, will give you an infinite shoreline, I think - although I'm not sure (space-filling curves may not have definable arclengths, in which case the "length of the shoreline" is meaningless.) You can probably build shorelines of any finite lengths using the constructions in fractal geometry. A couple of years back, there was an article giving the algorithms in "Software- Practice and Experience". For their first year or so, they had a column of "Computations Recreations" (or something like that) by "Aleph 0". One of their articles gave programs that would plot (approximations to) space-filling curves. (If you are interested, it should be pretty easy to find - S-P&E was a quarterly then, and the column only ran for about 2 years or so.) -- Jerry decvax!yale-comix!leichter
ss (03/14/83)
I have not worked this out, so I may be shooting my mouth off, but it seems to me that the maximum shoreline would be obtained by filling in the lake with two "comb" structures. ie --------- | ------- -------- | | ------- etc. the total length of the shoreline would depend upon how thin you want to make the water and land areas. Sharad Singhal.
berry (03/16/83)
#R:tekecs:-61200:zinfandel:9600004:000:619 zinfandel!berry Mar 14 13:05:00 1983 If you take a circular lake with circumference 1, and 'deform' the shore by filling in, How long can you make the shore? The answer is, there is no limit. fill in so that the shoreline follows a mapping of any space-filling curve (Peano curves spring to mind) and there you are: Infinite shoreline and finite area. I am reminded ofwhat you get if you rotate a canonical hyperbola about the x-axis: a sort of cone thing with finite volume but infinite surface area. Sou you can fill it with paint but never wet the entire surface! Berry Kercheval Zehntel Inc. (decvax!sytek!zehntel!zinfandel!berry) (415)932-6900