Thank you Dan -- I get it now! The two copies of T change size, and are "pushed" apart from each other radially. I was imagining the two copies of T (or T_0 in your earlier description) remaining the same size and being pushed apart from each other purely by translation, with no rotation, shear, resizing, or anything else. Which makes it hard to avoid having them intersect each other (-: On Thu, Dec 2, 2010 at 03:29, Dan Asimov <dasimov@earthlink.net> wrote:
The simplest example of this induction step is showing that T^2 embeds in R^3, assuming that T^1 embeds in R^2.
Let T := T^1 be embedded in R^2 as the circle of radius 2 about (0,0).
At time s, for -1 <= s <= 1, let
P_s := { p in R^2 | distance(p, T) = sqrt(1-s^2) }
P_s is one circle for s = +-1. Otherwise P_s is the union of two circles, of radii 2 +- sqrt(1-s^2). [...]
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