> 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.