🚀 go-pugleaf

In article <1992Nov10.000228.14551@samba.oit.unc.edu> Bruce.Scott@launchpad.unc.edu (Bruce Scott) writes:
>What about the one-dimensional helix embedded in R^3. Is the helix not a
>1-manifold? Note that if you compact it along its axis you get a circle
>(also a 1-manifold--is this S^1?) embedded in R^2. But a circle is a
>degenerate case of a helix ("stretching" along the axis having vanished)
>and is not in general a helix.

That projecting the helix along its axis yields another manifold (S1)
is something of a coincidence.  If you tilt the projection slightly you
no longer get a manifold, since where the coils intersect is not
homeomorphic to R1.  If you keep tilting until the intersections (and
the cusps) disappear, you get a 1-manifold again, this one
diffeomorphic to the helix itself (and to R1), but now embedded in the
plane.

Up to diffeomorphism the helix in R3 *is* R1.
-- 
Vaughan Pratt