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Aperiodic Tiling

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Sujeet Kumar

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Dec 26, 2010, 7:50:34 PM12/26/10
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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.

Mary Krimmel

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Dec 29, 2010, 2:55:11 PM12/29/10
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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.

Steve Brian

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Dec 30, 2010, 8:37:33 AM12/30/10
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All of the old posts are on http://mathforum.org/kb/forum.jspa?forumID=129&start=0

I don't know if any of the drawings are available.

The tiles appear to be the same.

Steve Brian

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Dec 30, 2010, 8:37:37 AM12/30/10
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A correction. My posts are there, as are yours, but not Kumar's, other than direct replies to one of our posts. All his diagrams were, of course, in his original posts, which seem not to have been archived.

NB: See, for example, http://mathforum.org/kb/profile.jspa?userID=53630
- the list administrators

Sujeet Kumar

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Dec 30, 2010, 12:17:12 PM12/30/10
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>From long time I haven't searched about Heesch Tiling if you say much development has taken place.
I am going to search about it.I hpoe that I find and do something more interesting this time.

Regards , Sujeet

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