Paul wrote: << http://www.georgehart.com/bagel/bagel.html _______________________________________________
As a grad student I wondered whether there was any way to fill up a positive volume of 3-space with disjoint congruent unit circles (just the curves, of course). Couldn't figure it out, but in 1992 I found that each usual torus of revolution whose core circle was of unit radius could be filled up by disjoint unit circles. By then considering all the tori of revolution having the same core (unit) circle, the union of all of their disjoint unit circles solved the original problem . . . filling up a volume of 2*pi^2. This volume consists of all points of space that are < 1 unit from the core circle. (Turns out these circles filling up a torus of revolution were discovered by French mathematician Villarceau in 1848. This is the same kind of circle that George Hart's bagel decomposition is based on.) I was able to prove there is an *upper bound* to the volume of any connected open set in space that is continuously filled up with disjoint congruent unit circles. Though it's easy to come up with *an* upper bound that works, I don't know what the *least* upper bound is (aka the sharpest one). I suspect 2*pi^2 is in fact the sharpest upper bound, but haven't been able to prove it. --Dan P.S. In a 1964 paper of (J.H.) Conway and Croft, it is stated that all of 3-space can be filled with disjoint congruent unit circles, using the Axiom of Choice. I'm not sure I fully understand their argument, which is only sketched briefly. The statement above about an upper bound to the filled volume applies only to a *continuous* family of disjoint unit circles. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele