That's pretty neat but who is Alexander and why is it a sphere?
david sharp wrote:
> A rendering of Alexander's "Horned Sphere".> What you are supposed to see is how the 'horns'> recursively intertwine.>> [Image]
From: david sharp
Subject: Re: Horned Sphere
Date: 18 Jun 2001 19:43:19
Message: <3b2e9217@news.povray.org>
"j charter" <jrc### [at] aolcom> wrote in message
news:3B2EA763.D679D92E@aol.com...
> That's pretty neat but who is Alexander and why is it a sphere?
James Alexander was a topologist, and his 'sphere' is only topologcially a
sphere. You could continuously retract all those horns and smooth out
the result to get an actual sphere.
A little more:
Alexander's Horned Sphere is famous probably mostly because it is a
counter-example to the obvious.
That is, the inside and the outside of a (topological) sphere are both
simply connected (any loop can be continously deformed to a single point).
This is obviously true for a real sphere, and for (most) rocks and other
topological spheres. Any loop of string outside a beach ball (or one inside
the ball)
can be retracted to a point without cutting through the ball. Its not true
for a coffee
cup if you make the loop go around the handle before tying it.
And a loop around one of the horns of the horned sphere can't be
continuously
deformed to a point without passing through the horned sphere!
com> wrote in message
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