[seqfan] Re: L-reptiles
Benoît Jubin
benoit.jubin at gmail.com
Tue Mar 30 20:09:01 CEST 2010
If the small tiles are required to have the same size and the sides of
the original tile have commensurable lengths, then the number of tiles
is a perfect square. This is because of rational independence
considerations, see for instance
http://qchu.wordpress.com/2009/07/02/square-roots-have-no-unexpected-linear-relationships/
Benoit
On Tue, Mar 30, 2010 at 8:11 AM, Tanya Khovanova
<mathoflove-seqfan at yahoo.com> wrote:
>
> Hello all,
>
> Carolyn Yackel gave a talk at G4G9 on L-reptiling: tiling of letter L with the same shape. (Letter L means a 2 by 2 square with one of the squares removed).
>
> I we want to have identical tiling pieces then the shape can be tiled with 4, 9, 16, pieces:
> http://mathworld.wolfram.com/Rep-Tile.html
>
> Is it true that any square number will work?
>
> Now suppose we do not require them to be the same size. Then if we divide the L-shape into 4 pieces, we then can divide only one of them into 4 pieces. Continuing that we get 1, 4, 7, 10 as the number of possible pieces. If we combine that with dividing some pieces into 9 smaller pieces, we can get 1, 4, 7, 9, 10, 12, 13 as the number of possible pieces.
>
> Carolyn presented a ceramic L-shaped thingy tiled into L-shapes.
>
> Is the number of possible pieces a good sequence to submit? Is it true that the compliment of the above sequence are impossible to achieve?
>
> Is L-shape interesting enough to submit the sequence?
>
> For example, an isosceles right triangle can be tiled into two smaller triangles, and hence, into any number of similar triangles if we do not require the size to be the same.
>
> Tanya
>
>
>
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