Completing the Tileset

The decagons made from two hexagons and one boat look like this:

We deliberately replace these three piece decagons by one tile which has the same vertex colour placement

This completes our set of tiles.

If we do the replacement in a tiling, the number of hexagon and boats left is relatively small.
I show on the next page that the relative frequencies for decagons and stars are 1 / √5 and φ-1 / √5 . φ is the the golden ratio.

The relative area which is covered by the decagons is furthermore φ-1 and the relative area of the stars is φ-3, so only less than 15% of the area is covered by boats and hexagons.

This tiling with our tileset and the underlying Penrose tiling with darts and kites are mutually locally derivable. We can replace unambiguously all decagons by one boat and two hexagons. The remaining hexagons, stars and boats then are replaced - also unambiguously - by darts and kites and the result is a Penrose P2 tiling.

 Hexagon, star and boat partitioned into darts and half kites


On the other hand, from a given Penrose tiling with darts and kites we can create a tesselation with decagons, hexagons, boats and stars but be aware that the substitution of 2 hexagons and 1 boat by a decagon is not always unique. In the valid configuration below with 3 hexagons and 2 boats there are two choices for the replacement by a decagon.

After all the theoretical considerations here you can see a 5-fold symmetric example laid out on the table.Should you wish to physically create the puzzle pieces, for private use, you may download the following pdf-files:


Here you see another example which on first sight may look like a 10-fold symmetric example. But there is only one vertical symmetry axis because it is impossible to fill the space between the inner ring of ten decagons and the center decagon in a rotational symmetric way.

Many thanks to Craig S. Kaplan who gave me some insight into the design of the tiles in his PhD thesis (2002)
Computer Graphics and Geometric Ornamental Design.

I also recommend reading
Cromwell, Peter R. (2009), The Search for Quasi-Periodicity in Islamic 5-fold Ornament, The Mathematical Intelligencer 31, no. 1:36-56


The computation of the relative frequencies of the tiles is not really easy but the results are of surprising simplicity and beauty. So you should consider to take a look at the next page . . .

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© 2019  Herbert Kociemba