The Alexander Sphere

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

Category:

Howto & Style

Tags:

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

[bothmer]

There used to be sound to these videos, but we had to erase them due to copy-right problems.

[kaengogyoubodi]

what does homeomorphic mean?

[bothmer]

A and B are homeomorphic if a continuous bijective map f: A->B exists whose inverse f^-1 is also continous. Less formally it means that the two objects can be deformed into each other without cutting or gluing. In the movie you can see a visualisation of the homeomorphism between the usual sphere and the alexander sphere. The arms grow out of the sphere with no cutting or gluing.

[chengtsu]

can someone please illuminate me, so i understand the arms will never meet, not even in infinity, my question is: is there any application of this figure, or is it just another example of never ending, repetitive pattern of fractals?

[bothmer]

It shows that a certain topological Theorem which is true in 2 dimensions is false in three dimensions. Click on the explanations for this video to get a little more detail.

[Greeneyeskater93]

i can do that with my fingers

[teckforce37]

Just like eigenvalues and eigenvectors, this is pointless...

[RyanKaufman]

Looks like one heavy kettle bell.

[osheaad]

THAT IS AN IMPOSSIBLE SHAPE, YOU CANNOT SHOW WHAT THE FINISHED PIECE LOOKS LIKE, YOU ONLY MAKE-BELIEVE THERE IS AN INFINITUM ITERATION. YOU CANNOT PROVE THE INFINITUM EXISTS, or can you?

[Aeleita1]

FRACTAL!!!

[Nulono]

What's the point of that thing?

[97kevinhuanle]

Copyright can eat an inside out square! >:(

[tigershark132]

My name's Alexander! XD