「無限に広い部屋の畳の敷き方」 ：
以前書き込んだ「畳数列」のトピックに、ＢＡＢＹＭＥＴＡＬ掲示板における、めめさんという音楽ファンでありながらスーガクもお好きなかたとの議論のコピーを記しましたが、 めめさんが４３３．の書き込みで、「無限に畳を敷く方法は、煉瓦型か羽根型のどちらかだけか」と疑問を呈されています
コウ博士のその疑問に対する答えは、６月２３日の日記に、この４個がたぶん全てじゃないだろーか、と書き込みました
いまいち、どう証明したらいいのかが分かっていませんでしたが、仏人のスーガク者のＭａｘｉｍｉｌｉａｎ君が詳しい証明をメールしてくました
コピーします
うーむ、むずかしい、しかもえーごやし
今度よーく読んで理解して、日本語で解説します
−−−−−−−−−−
----- Original Message -----
From: M. F. Hasler
To: Yasutoshi Kohmoto
Date: 2020/8/16, Sun 04:36
Subject: Re: Tatami
Dear Yasutoshi,
On Sat, Aug 15, 2020 at 3:44 AM <zbi74583_boat@yahoo.co.jp> wrote:
Here is the archive of Number Theory ML
https:/
The mail "Taxicab" is my posting
Thank you for this NMBRTHRY link !
I am the first Mathematician who told Tatami sequence to people of the world
I found your Tatami sequence on OEIS
Thank you for noticing these. It was not my idea but an exercise on ProjectEuler which I did in that time (more than 10 years ago).
It inspired these sequences to me. I have almost forgotten them.... :-)
One theorem exists
T. Tatami Tatami Tiling of infinite room is these four
I see only L1 (Tomoe / Tomok ? -- uses a half tatami, and a bit trivial)
and L2 (castle type, more interesting).
Both have the purpose of making square and (only for L2) near-square rectangular tilings.
What 2 others do you mean ?
Oh, now I read "blade type" and "brick type".
There is no information but I can guess brick type P2 is:
...A A B B C C...
.... D D E E F F...
...G G H H J J ...
and then I guess "blade type P1" must be
...A B B H G ...
...A C D D G...
.... C E F F ...
There are also the "flipped" variants : axial symmetry for L1 and 90° rotation for L2.
But notice that there is also another "blade type" :
...A B B G G ...
...A C D D H H ..
.... C E F F I I ...
I hope you can see / understand what I mean.
Although it seems at first related to P1 it is fundamentally different from P1.
Actually it is maybe closer to P2 :
it can be can be seen as one diagonal half-plane filled with horizontal brick P2 and the other diagonal half-plane filled with vertical brick P2'.
And it is easy to see that at othr parallel diagonals we can switch the direction again. for exemple:
... A A|B B|C|D|E|F|H H|I|J|K K|L L|M M ...
......P P|Q Q|D|R|F|G|T T|J|U|V V|W W|... (yellow = "vertical" bricks)
....... X X|Y Y|R|S|G|Z|A A|U|C|D D|E E ...
Type P1 is the special case of this when there is always in alternance 1 diagonal layer of horizontal and 1 diagonal layer of vertical bricks.
L1 can be extended to make infinite strips of given odd width w = 3, 5, 7, ... ; 1 is trivial because it uses a half-tatami every w unit lengths.
P2 and L2 can be extended to make infinite strips of any given width.
(for example: = | = | = | = ... or :
A B B F G G K K
A C C F H I I L
D D E E H J J L ...)
But yes, I see no other bi-directional infinite tatami tilings than the above.
Is it correct ? Could you prove it ?
Yes, I think that I can prove that there no other possibilities.
For L1 it is easy to see that a half-tatami can only be extended infinitely with full-tatamis in one way (plus the symmetric variant).
Concerning the other ("true") tatami tilings, you can start with one tatami , say horizontal,
and add below :
1) one parallel tatami with common side of length 2 : it will lead to the L2 tiling, you have no other choices for all subsequent tatamis which touch those already there..
2) two "vertical" tatamis touching each 1/2 of the first tatami's side: It will also lead to the L2 tiling (rotated by 90°), again you will never have a choice for all subsequently attached tatamis.
3) two tatamis, one vertical and one horizontal. This will define one diagonal "blade" direction extending in the direction of the "outer" angle.
At each time, you don't have a choice and must put a tatami at the "outer angle" where the two meet.
But in the "inside angle", you have 2 choices :
3a) continue with the "blade", or
3b) put a tatami completely parallel to the other "inside tatami".
This will make the "kernel" of an L2 pattern and determine all other tatamis of the whole plane.
It is the same as if you started here with (1).
If you never make the choice 3b), then you have an infinite diagonal of "blade type", and you can at both sides of the infinite diagonal blade decide to go on with a "brick" pattern of the same direction, or to switch the direction of the brick pattern. But you have no other choice.
4) Two horizontal tatamis. This can
4a) either start a horizontal brick pattern ; we know these can extend as far as we wish and / or switch at a diagonal to vertical direction.
4b) or it can also be just below the "core" of an L2 pattern: If you put a vertical tatami left *and* right to the initial tatami:
then there is no other choice.
(If you put a vertical tatami only to the left or only to the right of the initial tatami, then this vertical tatami and the adjacent horizontal tatamis we added at step 4 will define the diagonal border line separating
a horizonal brick tiling below and a vertical brick tiling above.
This diagonal can go on forever as in (3a) (and then there can be parallel diagonals)
or it can stop with an L2 core (then the rest of the plane is determined by the L2 pattern).
Do you agree that this proof is complete? (Although some minor details remain to be written written out explicitly.)
Best wishes,
Maximilian
以前書き込んだ「畳数列」のトピックに、ＢＡＢＹＭＥＴＡＬ掲示板における、めめさんという音楽ファンでありながらスーガクもお好きなかたとの議論のコピーを記しましたが、 めめさんが４３３．の書き込みで、「無限に畳を敷く方法は、煉瓦型か羽根型のどちらかだけか」と疑問を呈されています
コウ博士のその疑問に対する答えは、６月２３日の日記に、この４個がたぶん全てじゃないだろーか、と書き込みました
いまいち、どう証明したらいいのかが分かっていませんでしたが、仏人のスーガク者のＭａｘｉｍｉｌｉａｎ君が詳しい証明をメールしてくました
コピーします
うーむ、むずかしい、しかもえーごやし
今度よーく読んで理解して、日本語で解説します
−−−−−−−−−−
----- Original Message -----
From: M. F. Hasler
To: Yasutoshi Kohmoto
Date: 2020/8/16, Sun 04:36
Subject: Re: Tatami
Dear Yasutoshi,
On Sat, Aug 15, 2020 at 3:44 AM <zbi74583_boat@yahoo.co.jp> wrote:
Here is the archive of Number Theory ML
https:/
The mail "Taxicab" is my posting
Thank you for this NMBRTHRY link !
I am the first Mathematician who told Tatami sequence to people of the world
I found your Tatami sequence on OEIS
Thank you for noticing these. It was not my idea but an exercise on ProjectEuler which I did in that time (more than 10 years ago).
It inspired these sequences to me. I have almost forgotten them.... :-)
One theorem exists
T. Tatami Tatami Tiling of infinite room is these four
I see only L1 (Tomoe / Tomok ? -- uses a half tatami, and a bit trivial)
and L2 (castle type, more interesting).
Both have the purpose of making square and (only for L2) near-square rectangular tilings.
What 2 others do you mean ?
Oh, now I read "blade type" and "brick type".
There is no information but I can guess brick type P2 is:
...A A B B C C...
.... D D E E F F...
...G G H H J J ...
and then I guess "blade type P1" must be
...A B B H G ...
...A C D D G...
.... C E F F ...
There are also the "flipped" variants : axial symmetry for L1 and 90° rotation for L2.
But notice that there is also another "blade type" :
...A B B G G ...
...A C D D H H ..
.... C E F F I I ...
I hope you can see / understand what I mean.
Although it seems at first related to P1 it is fundamentally different from P1.
Actually it is maybe closer to P2 :
it can be can be seen as one diagonal half-plane filled with horizontal brick P2 and the other diagonal half-plane filled with vertical brick P2'.
And it is easy to see that at othr parallel diagonals we can switch the direction again. for exemple:
... A A|B B|C|D|E|F|H H|I|J|K K|L L|M M ...
......P P|Q Q|D|R|F|G|T T|J|U|V V|W W|... (yellow = "vertical" bricks)
....... X X|Y Y|R|S|G|Z|A A|U|C|D D|E E ...
Type P1 is the special case of this when there is always in alternance 1 diagonal layer of horizontal and 1 diagonal layer of vertical bricks.
L1 can be extended to make infinite strips of given odd width w = 3, 5, 7, ... ; 1 is trivial because it uses a half-tatami every w unit lengths.
P2 and L2 can be extended to make infinite strips of any given width.
(for example: = | = | = | = ... or :
A B B F G G K K
A C C F H I I L
D D E E H J J L ...)
But yes, I see no other bi-directional infinite tatami tilings than the above.
Is it correct ? Could you prove it ?
Yes, I think that I can prove that there no other possibilities.
For L1 it is easy to see that a half-tatami can only be extended infinitely with full-tatamis in one way (plus the symmetric variant).
Concerning the other ("true") tatami tilings, you can start with one tatami , say horizontal,
and add below :
1) one parallel tatami with common side of length 2 : it will lead to the L2 tiling, you have no other choices for all subsequent tatamis which touch those already there..
2) two "vertical" tatamis touching each 1/2 of the first tatami's side: It will also lead to the L2 tiling (rotated by 90°), again you will never have a choice for all subsequently attached tatamis.
3) two tatamis, one vertical and one horizontal. This will define one diagonal "blade" direction extending in the direction of the "outer" angle.
At each time, you don't have a choice and must put a tatami at the "outer angle" where the two meet.
But in the "inside angle", you have 2 choices :
3a) continue with the "blade", or
3b) put a tatami completely parallel to the other "inside tatami".
This will make the "kernel" of an L2 pattern and determine all other tatamis of the whole plane.
It is the same as if you started here with (1).
If you never make the choice 3b), then you have an infinite diagonal of "blade type", and you can at both sides of the infinite diagonal blade decide to go on with a "brick" pattern of the same direction, or to switch the direction of the brick pattern. But you have no other choice.
4) Two horizontal tatamis. This can
4a) either start a horizontal brick pattern ; we know these can extend as far as we wish and / or switch at a diagonal to vertical direction.
4b) or it can also be just below the "core" of an L2 pattern: If you put a vertical tatami left *and* right to the initial tatami:
then there is no other choice.
(If you put a vertical tatami only to the left or only to the right of the initial tatami, then this vertical tatami and the adjacent horizontal tatamis we added at step 4 will define the diagonal border line separating
a horizonal brick tiling below and a vertical brick tiling above.
This diagonal can go on forever as in (3a) (and then there can be parallel diagonals)
or it can stop with an L2 core (then the rest of the plane is determined by the L2 pattern).
Do you agree that this proof is complete? (Although some minor details remain to be written written out explicitly.)
Best wishes,
Maximilian
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