Could you create this drawing starting and finishing in the same place, without ever taking the pencil off the page, and without the path ever crossing itself? It's possible according to the Jordan curve theorem.
Show us your Jordan curve theorem creations using Scoot & Doodle on Hangouts or Scoodle Jam for iPad. scootdoodle.com+Richard Green
Michelangelo and the Jordan curve theorem
This picture by Robert Bosch, which is based on a detail from Michelangelo's famous painting The Creation of Adam, is remarkable because it is made from a simple closed curve. That means that it is possible (in principle) to draw this entire picture with a pencil, starting and finishing in the same place, without ever taking the pencil off the page, and without the path ever crossing itself!
A result in mathematics called the Jordan curve theorem proves that any simple closed curve drawn in the plane (such as the one in the picture) splits the rest of the plane into two connected pieces: an inside, which has finite area, and an outside, which has infinite area. This means that it is possible to colour in the “inside” of this picture so as to form a connected shape with finite area (and no holes).
I think this picture should be printed on children's menus in restaurants, with instructions to colour in the inside. That would keep them busy for a while.
This picture comes from the artist's website: http://www.dominoartwork.com/tspart.html
In three dimensions, the analogue of the Jordan curve theorem is false and things are much more complicated. Here's a post by me about that: https://plus.google.com/101584889282878921052/posts/GqSA4ePLt4G
I am unable to look at this picture without thinking of the credits of The South Bank Show: http://youtu.be/duNjQelqKkQ?t=7s
(Found via +Patrick Honner.)