# [FOM] Removing Deep Pathology 1

katzmik at macs.biu.ac.il katzmik at macs.biu.ac.il
Sun Aug 23 08:22:59 EDT 2015

```Colin, the Hopf fibration starts out as a fibration of the 3-sphere, but you
can easily translant this to 3-space using stereographic projection.  The
circles remain true round circles (though of course of different radius)
except for the one passing though the origin which becomes a line.  That's why
it does not solve the problem of partitioning 3-space into circles
(literally), as noted before by one of the participants.  MK

On Fri, August 21, 2015 14:25, Colin McLarty wrote:
> The circles of the Hopf fibration are just topologically circles, while I
> think those intended by the example
>
>> 4)  The partition of 3-space by (infinitely long) non-parallel lines.
>> 4a)  "      "     "     "     "  non-parallel circles.
>
> are metric circles.  Anyway the fibers of the Hopf fibration do not lie in
> non-parallel planes, since they do not lie in planes at all.
>
> More to Harvey's point, while the Hopf fibration had a surprising
> consequence in topology (non-triviality of higher homotopy of spheres), it
> is not pathological in any logical sense.  It is explicitly definable by
> degree 4 polynomials.
>
> Colin
>
>
> On Thu, Aug 20, 2015 at 4:36 PM, Hendrik Boom <hendrik at topoi.pooq.com>
> wrote:
>
>> On Thu, Aug 20, 2015 at 12:36:45AM +1200, W.Taylor at math.canterbury.ac.nz
>> wrote:
>> >
>> > >just what does nonparallel mean for circles?
>> >
>> > Lying within non-parallel planes.
>>
>> My spatial visualisation is challenged here, but with this definition,
>> it's possible that the Hopf fibration satisfies the nonparallel
>> constraint.  Anyone know for sure?
>>
>> It does have one straight line -- that's the circle of infinite radius
>> I mentioned.  Perhaps that's a deal-breaker.
>>
>> -- hendrik.
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```