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Paper chain cutting

An acquaintance approached me with the following starter: 

"You know, it's interesting what happens when you glue together a paper chain, and then cut it along its center lines..."

and then he demonstrated the effect with two rings. Yes, interesting--the two rings somehow turned into a square...!

Then, we proceeded to contemplate what would happen using more rings. 

Later, he demonstrated what happens when using paper chains with 1-8 rings; the results were quite fascinating. 

You can see some (Japanese) articles about this effect, by following the top Google search results from the links below--just stare at some of the pictures, and perhaps the video, and you'll probably get the point...

Now, I will try to explain what is happening, in layman's terms. 
(I won't try to do the proper Topological analysis here... hopefully someone will fill in)

From here, I will continue on, expecting the reader to have seen the above articles, or have actually tried this out. 

 the Japanese counterpart will probably be ahead of this page in terms of content...]