The Alexander Sphere

Back



A path that is homoemorphic to a circle devides a compactified plane into two pieces (inside and outside). Arthur Schönflies proved in 1906 that in this situation the inside and outside are homoemorphic. To prove a similar statement in 3 dimensions was an open problem for many years. It was solved by James Alexander in 1928 who constructed the Alexander "Horned" Sphere, as illustrated in this video. The Alexander horned sphere is a topological space which is homeomorphic to a sphere, but inside and outside are not homeomorphic. This proves that there is no analog of Schönflies Theorem in three dimensions. This Video was produces for a topology seminar at the Leibniz Universitaet Hannover. http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1 This animation was #1 on our geometric animations advent calendar: http://www.calendar.algebraicsurface.net

Channel: Howto & Style
Uploaded: November 9, 2006 at 1:50 pm
Author: bothmer

Length: 0:00:38
Rating: 4.59
Views: 38,161

Tags: topology mathematics schönflies alexander sphere inside outside iterative fractal bothmer cg geometry advent calendar

Video Thumbnail #1:




Video Thumbnail #2:




Video Thumbnail #3:




Video Url:


Embed Code:


Video Comments:
TQCKyle (Tuesday 12th of August 2008 02:40:28 PM)
Unless you're a graduate mathematics student, you're not supposed to!
shortyskater524 (Monday 11th of August 2008 06:50:34 PM)
i don see the point in this?
waffle12996 (Thursday 5th of June 2008 11:28:21 AM)
This is a neverending fractal.(That may sound redundant but it's not;really) Color fractals aren't infinity because eventually you will get down to only 1 color and that's it. For The Alexander Sphere the two "arms" will never touch so it can get smaller and smaller and smaller and smaller until infinity.
Dragozine (Wednesday 9th of July 2008 01:29:39 PM)
actually, they will touch...when an arm grows from an arm, it gets closer ad closer to the other side doesnt it?
waffle12996 (Wednesday 9th of July 2008 01:48:53 PM)
But each time another arm grows it will get smaller and so will the space between them but they still won't touch.
TQCKyle (Tuesday 12th of August 2008 02:38:36 PM)
There's an important theroem from analysis, the Archimedean Principle I believe, that says given any two points in R^n (or R^3, 3-d space in this case) you're guaranteed to find a point along the "line" between them. This implies infinite divisions and thus the arms could be made to never touch.
OMGLOLOLOLOLOLOLOLOL (Friday 25th of July 2008 01:21:43 AM)
Color fractals are still "infinite" fractals, limitations of computer processings should not strike you as direct output, such an offending opinion, color fractals are still fractals! there will always be a hole left to generate upon!
saha1994 (Wednesday 4th of June 2008 10:55:09 PM)
at risk of sounding dumb, i dont get the significance of this figure what makes it special? i read the info, but I dont get it.
PurpleLightsaberAlex (Monday 19th of May 2008 12:40:09 PM)
Wow, if you bend them apart, will it break?
DividedAkhiloth (Tuesday 29th of April 2008 03:20:08 AM)
It doesn't actually have to be a circle.. You could just use 2 tubes or a torus. And also, that's a one sweet fractal.