I'm not sure what "at the limit the areas disappear and everything is boundary" means exactly. But if each of the six regions is in the limit space-filling the very same space, then isn't this a lot like the sets A_n and B_n in the reals?: Where A_n = union of [k/2^n,(k+1)/2^n] for odd integers k, B_n = union of [k/2^n,(k+1)/2^n] for even integers k. both sets having the common boundary {k/2^n | k in Z}, but for any real x as n -> oo, dist(x,A_n) -> 0 and dist(x,B_n) -> 0. ----- I agree the illustrations with the Wikipedia article are unhelpful. Here's a picture of the second stage: 4 cyclically linked solid tori going around one, and in each of those 4 there are four smaller cyclically linked tori going around *it*: < http://onionesquereality.files.wordpress.com/2011/07/antoines-necklace.jpg >. (There are a few extraneous lines, which should be ignored.) Yes, three should work. In fact any sequence of numbers of tori at various stages would work -- as long as the diameters of the tori in stage n approach 0, as n -> oo. (I wonder what happens in the case where there is only one torus in each stage, pulled longitudinally around the previous torus and made to link with itself.) --Dan RWG wrote: << Robert M. wrote: << . . . . . . But the necklace one bothers me. That transition from linked set of tori to unlinked set of points seems impossible.
A similar thing happens with increasingly fine polygonal approximations to spacefilling functions. In http://www.tweedledum.com/rwg/rad2fill.htm the upper figure defines the iteration. For any finite order, six of these equipartition the area near the center of the lower figure. But at the limit, the areas disappear and everything is boundary. I don't get the many-torus illustrations in http://en.wikipedia.org/wiki/Antoine%27s_necklace . Doesn't the text say four tori? Shouldn't three work? (Bogus Borromean rings.)