After my
last post, I have sent its link to some people whom I thought might be
interested. Mr. Dirk Frettloeh from Bielefeld University of Germany was
the only person kind enough to reply. He stated in his reply that these nice
pictures won't mean anything unless there is a matching rule between the tiles.
That was something I have tried before but it wasn’t possible to apply similar
approach like the matching rules of Penrose Tiles with 2 prototiles, So, this
was a big challenge for me and I couldn't succeed.
After
receiving his reply, I have tried different approaches and you can see one of
them below. This matching rule requires each corner of the prototile to have 3
different orbit like bonds (shown with black numbers). Each bond has a different color. A matching
color for each different color type is located in another corner of the prototile. Each bond
gets stronger while the orbit number increases. So a 2nd orbit bond
with same colors is stronger than the ones with 1st orbit bonds
and the 3rd orbit bonds are the strongest ones. I didn’t apply any mathematics to define the exact strength of the orbits but this can be done easily if required (like an
association between the bond strength and the square of the orbit number etc.)
Below you
can see the prototile with the orbits and their bonds shown with different
colors. Matching colors will create the bonds that I have explained above.
And you can see below, the first steps of the progress with these rules (you can click on the picture to see it larger)
The scaled up versions for 7 vertex neighbourhoods and their corresponding vertex points in penrose rhomb tiling are given below.
So, this is what I could do so far. Don't know if this satisfies the need for a matching rule since all the penrose tiling vertices above shall be made of infinite number with the same prototile. Hope this means something.
