THE CONSTRUCTION:

Here is the tiling we constructed before as an example of a self-similar tiling.  It will serve as our test subject.

From such a tiling it is possible to extract many new tilings called  derived Voronoi tilings.  The process goes in several stages.

• Select a finite configuration of tiles, called a patch P of tiles.
• Create the locator set for P by marking each place the patch P arises in the tiling.
• Create the Voronoi tiling for the point set P as follows:  each point p in P  will get its own tile, and that tile will be composed of all points closer to p than to any other point in the locator set.

A SPECIFIC EXAMPLE:

• Let the initial patch P be simply a single white tile.
• In this case the locator set looks like this.  Compare it to the tiling above to see the correlation. 
• The Voronoi tiling is shown here superimposed upon the locator set. 

First off, one should realize that tilings can be used as mathematical models for the atomic structure of crystals and other crystalline solids.  In fact, the discovery of a quasicrystalline alloy with 10-fold rotational symmetry (not previously considered possible by crystallographers) was preceeded by the discovery of a tiling with that symmetry.   Tilings can also model cell growth and help with problems like the control-tower problem:  given a certain configuration of control-towers,  which one should you contact from any specified location?
Voronoi tilings are useful for this problem.  I have even seen an application of Voronoi tilings to artificial intelligence---they were used to help a robot decide how to get around without hitting things.

My application of the Voronoi construction---the derived Voronoi tiling---produces not only beautiful images, but useful ones too.   In my dissertation, I showed that one can detect whether the structure of a tiling has certain kinds of hierarchy using derived Voronoi tilings.  And these tilings give information about the occurrences of patches on the large scale.   If you are at a patch in the tiling, and you want to know where the next one is,  or at least the nearest ones, you should look at the derived Voronoi tiling.  If your tiling was modelling a crystal, the derived Voronoi tiling could tell you where all of the molecules of a certain type are located and which ones are next to each other.
 

AN ASSORTMENT OF DERIVED VORONOI TILINGS:

To get more interesting pictures, I have iterated the substitution one more time, and then created the tilings for several different choices of P.   I will show the tilings without drawing in the dots from the locator set.

Here is the iterated tiling and some of the derived Voronoi tilings that come with it:

Here are the derived Voronoi tilings which find the locations of single tiles:
 
 

locate 0 = white
locate 1=light purple
locate 2=dark purple 
 

Here are some of the tilings given by locating two horizontally adjacent tiles:

locate 00=two white tiles

 locate 11=two light purples
 
 

locate 22=two dark purples
 
 

Here are some obtained by locating three horizontally adjacent tiles:
 

locate 000=three whites
locate 111=three light purples
locate 222=three dark purples
 
 

This really neato one came from locating the patch  given by :
*0*
000
*0*
where the * denotes a wildcard entry.
 
 

There are infinitely many choices you can make, and only some of them are pictured here.  It is interesting to experiment, trying to find tilings which are radically different from one another.  If you are interested is obtaining the MATLAB code which makes these pictures, along with the fourier transforms and self-similar tilings, you can email me at nafrank@vassar.edu.