Alexander Horned Sphere Matthew Roberts

Motivation & definitions A curve in the plane is a trace of a moving point, if the starting point of the curve coincides with the terminal point of the curve, the curve is called closed. A curve is called simple if the curve is non-self-intersecting. Jordan Curve Theorem: States that a simple closed curve C divides then plane into two domains an “interior” and an “exterior”. Jordan Curve Theorem

Motivation & definitions continued Homeomorphism & A. Schoenflies Two domains are homeomorphic if there exists a bijective map of one onto the other such that both the map and its inverse are continuous. Example: Interiors of a circle and square are homeomorphic. The annulus is not homeomorphic to either of them. A. Schoenflies proved that any simple closed curve in the plane one chooses, its interior will be homeomorphic to a usual circle, and its exterior will be homeomorphic to a plane with a round hole in it. (1908).

Generalizations What about spatial generalizations? Are there theorems in spatial geometry similar to the Jordan & Schoenflies theorems? If so, closed curves would be replaced by closed surfaces. What about spatial generalizations? While closed curves are homeomorphic, closed surfaces are different. There exist various types of surfaces: spheres, tori, spheres with handles., etc.

Generalization What about spatial generalizations? We will only consider surfaces that are obtained by continuous deformations of a sphere. We are attempting to obtain similar results with surfaces to that of the Jordan & Schoenflies.’ theorem. The spatial analog of the Jordan Curve Theorem turns out to be true. A surface divides the space it lives in by two parts an “interior” & an “exterior”. It turns out that the same is true for spheres with handles which can be generalized to n dimensions.

That is good and all but.. Is there an analog for the Schoenflies Theorem for closed surfaces? The spatial analog for the Schoenflies Theorem should state that the interior and exterior domains of the closed surfaces are homeomorphic to that of a usual sphere and to the complement of a closed ball. In 1924 the young American mathematician James Alexander II proved that however likely it may seems this conjecture is true, that it is actually false.

Horned Sphere Construction of the horned sphere Alexander presented a deformed construction of the sphere which divides the space it lives in by non-standard parts. To begin: Take two disjoint small disk inside a larger planar disk and pull out two fingers such that the end of the “fingers” come close together but do not meet. As shown below:

Horned Sphere Construction of the horned sphere The ends of “fingers” remain planar discs. We perform the pulling of the fingers simultaneously from parallel discs. The four fingers will form a “lock”.

Horned Sphere Construction of the horned sphere Next we will describe the whole process. Begin with a round sphere, then pull out from two disc on the sphere two fingers. These two fingers should almost touch each other. As shown below:

Horned Sphere Construction of the horned sphere The result is that the ends of the two fingers are two parallel planar disc that are close to each other. From these two disc we pull out four fingers from each disc. Then we “lock” them. This process continues infinitely, which we will describe in detail next. It is worth noting the distance between two points on the sphere in their final distance is not less than 1% of their initial distance.

Horned Sphere Construction of the horned sphere There are disc of sizes 1,2,3,4,…n. n is an element of the naturals. Disc size is the step of the iteration. There are 2^n disc of size n, and each disc of size n contains exactly two disc of size n+1. disc size 0: contains 1 disc and each disc has 2 disc of size 1 (2 disc) disc size 1: contains 2 disc and each disc has 2 disc of size 2 (4 disc) disc size 2: contains 4 disc and each disc has 2 disc of size 3 (8 disc) disc size 3: contains 8 disc and each disc has 2 disc of size 4 (16 disc) disc size 4: contains 16 disc and each disc has 2 disc of size 5 (32 disc . .

Distance General Principle of Distance: One fundamental property of Construction: On the n-th step of the construction we only pull together points from the same disc of size n-1. General Principle of Distance: Let n be the greatest number such that point A and B belong to the same disc of size n. Then the distance between them remains unchanged under steps 1 through n. If neither A or B belongs to a disc of size n+1, the then the distance between them remains unchanged under all following steps of the construction. If only one of the points belong to a disc of size n+1 then after all subsequent steps of the construction the distance between the two points decreases no more than thrice. If points A and B both belong to disc if size n+1, then under the n+1-st step the distance between them decreases no more than ten times.

Distance General Principle of Distance: n+2 case If neither if the points belong to discs of size n+2, then the distance between them remains unchanged after the n+1-st step. If exactly one these points (A or B) belongs to a disc of size n+2 then the distance between them may decrease thrice on the n+2-nd step but not change significantly after all subsequent steps. Lastly if both points A and B belong to discs of size n+2, then the distance between them decreases at most , say, ten times on the n+2-nd step and is not changed significantly after all subsequent steps. In all cases the distance between A and B decreases not more than one hundred times. The Alexander horned sphere is a sphere, it is “homeomorphic” to a sphere.

Exterior of the horned Sphere The interior of the horned sphere is homeomorphic to the open ball. (we shall not prove this). The exterior of the horned sphere is not homeomorphic to that of a usual sphere. The sketch of the proof is “simple” but interesting due to its connection to topological flavor. The exterior & the interior of the usual sphere is simply connected: Every closed curve can be continuously deformed to a point.

Exterior of Horned Sphere The exterior of the horn sphere is not homeomorphic to the exterior of the usual sphere: Homeomorphic domains are simply connected simultaneously: which means if one domain is connected, the other should also be simply connected. The exterior of the horned sphere is not simply connected. A closed curve enclosing the handle of the weight can not be continuously pulled out of the handle. In order to pull the curve through the handle we would have to carry the curve or “rope” between a pair of close parallel disc of any size. Which means in the process of deformation “rope” would become arbitrarily close to the horned sphere which means it would touch the sphere at some point. This is prohibited because the deformation should be performed in the exterior of the sphere. This means that the exterior of the horned sphere is not simply connected and that it is not homeomorphic to the exterior of the usual sphere. This shows that the conjecture of the spatial version of the Scheonflies Theorem is false.

Exterior of horn sphere Exterior of horn sphere is not simply connected nor is it homeomorphic to the exterior of a usual sphere.

Interesting note Variety: We could pull the horns inside the sphere so that the exterior of the horn sphere is homeomorphic to the exterior of the usual sphere but not the interior. There exist a variety of possibilities of the horned sphere and this is the genius of Alexander’s proof.

Alexander Horned Sphere Animation:

Credits Credits: Mathematical Omnibus: Thirty Lectures on Classic Mathematics Dmitry Fuchs & Serge Tabachnikov pg: 361-368 Picture: Simon Fraser (Picture with John Conway of Alexander Horn Sphere growing from his head) Youtube video: http://www.youtube.com/watch?v=d1Vjsm9pQlc Thanks!