[seqfan] Re: L-reptiles
Benoît Jubin
benoit.jubin at gmail.com
Mon Apr 12 20:49:35 CEST 2010
After several attempts, I think there is no replicate tiling of the
"L" shape with 8 pieces, but I have no proof of this. Actually, I
have no proof of this impossibility for 3 and 5 pieces neither. How
do you prove such things ? Here is a proof for 2 pieces (which is
geometrically obvious...), but this method would give lengthy and
inelegant proofs (if any) for more pieces.
If the tiles have shorter sides a and b in (0,1), then considering the
area and a short side, we have respectively:
a^2+b^2 = 1,
there are natural numbers m and n such that am + bn = 1.
Now if m=0 or n=0 then a or b would be 1, which cannot be. So both m
and n are strictly positive, so a+b <=1, which contradicts a^2+b^2=1.
Also, I wrote:
>> 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/
I still think it's true and uses the result in the link I provided,
but I didn't come up with a proof...
Benoit
On Sat, Apr 10, 2010 at 10:35 AM, Tanya Khovanova
<mathoflove-seqfan at yahoo.com> wrote:
> Thank you.
>
> The case of 11 is easy. You take a 6-tiling and divide one of the tiles into 6 smaller tilings. So the only case left to settle is 8.
>
> Tanya
>
>
> --- On Tue, 3/30/10, Benoît Jubin <benoit.jubin at gmail.com> wrote:
>
> From: Benoît Jubin <benoit.jubin at gmail.com>
> Subject: [seqfan] Re: L-reptiles
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Tuesday, March 30, 2010, 5:44 PM
>
> In the case of the "L" tile, any perfect square is the number of tiles
> of a rep-tiling by tiles of equal size -- and no other number works,
> from my previous email. Indeed, say we want to tile an L of side n
> (and area 3.n^2) with L's of side 1, and reason by induction. If n is
> even, then we are reduced to tile 4 "L" tiles of side n/2 (using the
> tiling with 4 tiles). If n>3 is odd, then we tile the "central" "L"
> of side n-3, and we are left with a surface that we can tile by
> cutting it into 2x3 rectangles and one L of side 3 at the bottom-left
> corner (I hope you can follow this: draw a picture!).
>
> If we allow different sizes, then there is a tiling with 14 pieces
> (find one!), so every number at least 14 is realizable. Also, there
> is one with 6 pieces. Since there is no tiling with 2, 3 or 5 tiles,
> the only open cases are 8 and 11 (which might not be too hard to
> settle).
>
> Benoit
>
> 2010/3/30 Benoît Jubin <benoit.jubin at gmail.com>:
>> 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
>>>
>>>
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
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