Logic and Language
Load the menuLoad the menu


Copyright   James R Meyer    2012 - 2020 https://www.jamesrmeyer.com

Note: Full functionality of this website requires JavaScript to be enabled in your browser.
 

Alexander’s Horned Sphere

The Alexander Horned Sphere isn’t really a sphere at all - in fact, there isn’t any physical thing that is an Alexander Horned Sphere. It’s just a hypothetical notion. The notion revolves around the idea that you start off with a stretchy rubber-like sphere that you can stretch and bend and squeeze any way you like. You then shape it so that it is like a torus (the shape of a standard finger ring) but with a bit cut out if it, like in this picture:

 

Split Torus

 

Now, you add to each cut end, two arms that almost but don’t quite touch together. And there’s an important bit here. The arms from either end of the cut go around each other, but since the arms don’t touch together, and since the whole thing is completely elastic, you can still separate the arms, and so by a bit a judicious squeezing and stretching, you can still return it to its original shape (as in the above picture).

 

The Horned Sphere notion is “defined” as the result of doing this adding of quasi-interlinked arms to each existing arm end infinitely many times. And this is where the problems arise. By vaguely stating that the process is continued infinitely many times, you aren’t making a clear definition at all. This is because, for this process, the continuation of the process never finishes, regardless of how many times it is repeated. And if all you have is a vague definition that talks about repeating the process infinitely many times, then you don’t have a clear definition of anything at all. Since the repetition never completes, then since the object is perfectly squeezable, then it must always be possible to separate the quasi-interlinked arms and return to the original shape. The picture below gives an idea of what is happening.

 

Horned picture 1

 

Given the “definition” of the horned sphere, analyzing what is actually happening is complicated by the fact that at each repetition, more arms appear, and the object becomes increasingly hard to visualize. But we can concentrate on what is happening by ensuring that the number of arms does not increase at each repetition. This gives us a situation somewhat like the picture below:

 

Horned picture 2

 

Now it is easy to see that as the added arms get smaller and smaller, it might be possible to define a limiting case where the arms converge to a single point. Now, if the definition is such that this single point is the same for both ends of the left arms, and similarly for both ends of the right arms, and if the definition is that there is no break between these single points and the defined arms, then, clearly, the arms are properly interlinked, and cannot be separated by any amount of squeezing or reshaping.

 

And if the limiting case for the infinite repetition is that the limit points for the left arms and the right arms are separated, then the arms are not fully interlinked, and can be separated by a bit of squeezing and reshaping.

 

The only difference for the case where the number of arms increases at each repetition is that there is then infinitely many limiting points. But the question of whether the arms are inseparably linked or not is only dependent on whether each limit point of the left arms coincides with the corresponding limit point of the right arms.

 

And that’s all there is to the Alexander horned sphere. There’s nothing difficult or strange about it at all if you analyze it logically.

section divider

 

 

As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Frivolous, irrelevant comments.
Comments devoid of logical basis.
Derogatory comments.
Long-winded comments.
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical terms are correctly formatted.


Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous usernames for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it wrong and rewrite it again - still erroneously, or else attack something else on my site - erroneously. After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.


Difficulties in understanding the site content are usually best addressed by contacting me by e-mail.

 

Note: a password enables editing of comments, an email enables notification of replies

HashOver logoBased on HashOver Comment System by Jacob BarkdullHashOver logo

Copyright   James R Meyer   2012 - 2020
https://www.jamesrmeyer.com