Gummelt, in 1996, described a single decagon tile that covered the plane aperiodically.  These covers were in one-to-one correspondence with cartwheel covers of the Penrose’s tiling by kites and darts.  This work describes two different approaches to constructing a single structure that tiles the plane aperiodically.

The first approach involves identifying fixed points in the Robinson tiles exposed through decompositions of the cartwheel patch.  This tile results in a tiling that is dense everywhere in the plane.  The tile, however, has zero measure.

The second approach makes use of a Cantor-like construction of porous Robinson tile.  Instead of removing a fixed percentage of tiles at each stage, we remove an amount that decreases at each iteration.  This generates a structure with positive measure. A process called refilling allows us to add parts of the fractal construction back into the set in order to generate a tile that is well-mixed.

The latter technique results in tiles with non-uniform measure.  We can correct this through a bin-packing approach that attains equal distributions of points between tile colors with an arbitrary degree of accuracy.

Resources:

1. • Feng Zhu, Search for the Aperiodic Tile, an undergraduate thesis, May 2002. pdf
   
   

2. • Duane A. Bailey and Feng Zhu, A Sponge-like (Almost) Universal Tile, extended abstract of above.  pdf
   
   

3. • Duane A. Bailey and Feng Zhu, A Porous Aperiodic Decagon Tile, Gathering for Gardner X, March 2012.  pdf
           Tiling Kit for G4GX, pdf.  Print on transparencies using a color printer.
           The beginnings of a tiling by porous decagons, pdf

Duane A. Bailey, Williams College
Feng Zhu, University of Southern California