How to Make a Sphericon
If you imagine cutting a double cone with each apex 90° (fig a) down a plane which includes both apexes (fig b) and rotating one half 90° (fig c) and sticking them together again, you would make a sphericon:
 
 
 
 

In order from top: Steps a, b, c1, c2, and the result.

A far easier and more practical method, however, is to cut out a print-out or drawing of the following template (each angle is approximately 127.28 degrees) and sticking together the two highlighted ends before sticking the corresponding edges:

Sphericon Template

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For more information, contact: paul.roberts99@ntlworld.com