This node is about creating M.C. Escher style periodic space filling patterns. That is not to say that anything you or I can create will be comparable to Escher's work but the principles are the same. There is a whole raft of mathematics devoted to this subject but that is beyond the scope of this write-up (but not this node, hint, hint!). This is a simple procedure that can create pleasing if not spectacular examples of periodic space filling.

Escher used all kinds of shapes: squares, rhombuses, hexagons, pentagon-hexagon tessellation. Take a look at his "Metamorphosis" for an example of all of those and more. The starting point for this node is simply a tesselating rhombus because it makes for simple ASCII.

                      /\      /\      /\      /\     
                     /  \    /  \    /  \    /  \
                    /    \  /    \  /    \  /    \
                   /   1  \/   1  \/   1  \/   1  \
                   \      /\      /\      /\      /
                    \    /  \    /  \    /  \    / 
                     \  /    \  /    \  /    \  /
                      \/   2  \/   2  \/   2  \/
                      /\      /\      /\      /\
                     /  \    /  \    /  \    /  \
                    /    \  /    \  /    \  /    \
                   /   1  \/   1  \/   1  \/   1  \
                   \      /\      /\      /\      /
                    \    /  \    /  \    /  \    /
                     \  /    \  /    \  /    \  /
                      \/      \/      \/      \/

Next simply change one of the shapes a little and replicate the change on all of the shapes. It should be noted that for these purposes vertical layers that are next to each other need not be the same. In fact unless you want the same figure for all the tiles you should consider the two layers separate but interacting. In the case of the above diagram tiles labelled (1) need not be the same as tiles labelled (2).

                        
                      /\      /\      /\      /\ 
                     /  \    /  \    /  \    /  \  
                    /   /\  /   /\  /   /\  /   /\   
                   /    \_\/    \_\/    \_\/    \_\
                   \      /\      /\      /\      /
                    \    // \    // \    // \    //
                     \  //   \  //   \  //   \  // 
                      \/      \/      \/      \/
                      /\      /\      /\      /\
                     /  \    /  \    /  \    /  \
                    /   /\  /   /\  /   /\  /   /\
                   /    \_\/    \_\/    \_\/    \_\ 
                   \      /\      /\      /\      /
                    \    // \    // \    // \    // 
                     \  //   \  //   \  //   \  // 
                      \/      \/      \/      \/

The two shapes are now different but because they are just modifications of a pattern that we know tessellates they are guaranteed to also tessellate.

The next stage is to add more changes and remove the construction lines that we put there to guide us at the beginning.

                        __      __      __      __ 
                      /  /_   /  /_   /  /_   /  /_  
                     /   _/  /   _/  /   _/  /   _/  
                    /   /   /   /   /   /   /   /   
                   /    \__/    \__/    \__/    \__
                   \      /\      /\      /\      /
                    \     / \     / \     / \     /
                     \  _/   \  _/   \  _/   \  _/ 
                      \/__    \/__    \/__    \/__
                      /  /_   /  /_   /  /_   /  /_
                     /   _/  /   _/  /   _/  /   _/
                    /   /   /   /   /   /   /   /
                   /    \__/    \__/    \__/    \__ 
                   \      /\      /\      /\      /
                    \     / \     / \     / \     / 
                     \  _/   \  _/   \  _/   \  _/ 
                      \/      \/      \/      \/

Now we are almost done. All that remains is to add details to the tiles. Again, make sure the details are the same for each layer of tiles but not necessarily the same overall.

                        __      __     __      __ 
                      / _/_   / _/_   / _/_   / _/_  
                     / ___/  / ___/  / ___/  / ___/  
                    /  _/   /  _/   /  _/   /  _/   
                   /\o  \__/\o  \__/\o  \__/\o  \__
                   \/   __/\/   __/\/   __/\/   __/
                    \  ___/ \  ___/ \  ___/ \  ___/
                     \  _/   \  _/   \ __/   \  _/ 
                      \/__  o/\/__  o/\/__  o/\/__
                      /__/_  \/__/_  \/__/_  \/__/_
                     / ___/  / ___/  / ___/  / ___/
                    /  _/__ /  _/__ /  _/__ /  _/
                   /\o  \__/\o  \__/\o  \__/\o  \__ 
                   \/   __/\/   __/\/   __/\/   __/
                    \  ___/ \  ___/ \  ___/ \  ___/ 
                     \  _/   \  _/   \  _/   \  _/ 
                      \/      \/      \/      \/

It is also possible to use this procedure to create simple metamorphoses. The examples above can be combined to give a simple example:

                      /\      /\      /\      /\
                     /  \    /  \    /  \    /  \
                    /    \  /    \  /    \  /    \
                   /      \/      \/      \/      \
                   \      /\      /\      /\      /
                    \    // \    // \    // \    /
                     \  //   \  //   \  //   \  /
                      \/      \/      \/      \/
                      /\      /\      /\      /\
                     /  \    /  \    /  \    /  \
                    /   /\  /   /\  /   /\  /   /\
                   / o  \_\/ o  \_\/ o  \_\/ o  \_\
                   \      /\      /\      /\      /
                    \    // \    // \    // \    //
                     \  //   \  //   \  //   \  //
                      \/__  o \/__  o \/__  o \/__
                      /  /_   /  /_   /  /_   /  /_
                     /   _/  /   _/  /   _/  /   _/
                    /   /   /   /   /   /   /   /
                   / o  \__/ o  \__/ o  \__/ o  \__
                   \      /\      /\      /\      /
                    \     / \     / \     / \     /
                     \  _/   \  _/   \  _/   \  _/
                      \/__  o/\/__  o/\/__  o/\/__
                      / _/_  \/ _/_  \/ _/_  \/ _/_
                     / ___/  / ___/  / ___/  / ___/
                    /  _/__ /  _/__ /  _/__ /  _/
                   /\o  \__/\o  \__/\o  \__/\o  \_
                   \/   __/\/   __/\/   __/\/   __/
                    \  ___/ \  ___/ \  ___/ \  ___/
                     \  _/   \  _/   \ __/   \  _/ 
                      \/      \/      \/      \/

Much of Escher's work contains space filling of one kind or another. I would recommend particularly "Metamorphosis", "Angels and Devils", "Day and Night", "Fish" and "Reptiles".


Sources:
Bruno Ernst, "Magic Mirror of M.C. Escher". ISBN: 886155003.

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