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Topic: Aperiodic Tiling
Replies: 4   Last Post: Dec 30, 2010 10:41 AM

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Mary Krimmel

Posts: 629
Registered: 12/3/04
Re: Aperiodic Tiling
Posted: Dec 29, 2010 11:46 AM
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> Consider the following 2 tiles:-
> 1.Tile ABCD has angles A=80, B=80, C=40 and D=160
> degrees.Segment BC has length 1-2sin(pi/18).
> The other sides have length 2sin(pi/18).
> 2.Tile PQRS has angles P=20, Q=200, R=60 and S=80
> degrees.Segment SP has length 1-2sin(pi/18).
> The other sides have length 2sin(pi/18).
>
> Can above 2 tiles be made aperiodic by giving some
> matching conditions?
>
> I had given this problem on this site in 2003
> also.That time John Conway had also shown little
> interest
> in this problem.Nothing much of interest came out
> after lot of discussions.I hope Mary Krimmel remember
> all that.I am giving it again with the hope that new
> readers may find it interesting and this
> time something special comes out.
> Reason behind my belief on this tiles is my already
> expressed view that 2sin(pi/18) seems as interesting
> as Golden Ratio.
> In 2006, I knew about Heesch tiling which is again
> very interesting topic.Using these I formed a single
> tile which were of Heesh no.1 and 2.


Is there any way that we can access some of the discussion from 2003? It seems to me that the 2 tiles above are not the same as the ones I remember. At that time you were able to transmit some diagrams. I enjoyed very much "playing" with the tiles, made many models.

Apparently more is now known about the Heesch number than was known in 2006. I believe that you proposed another, somewhat similar, number which could be a characteristic of a tile.



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