Self-integers tiling the
plane
Ron H. Hardin’s
sequence (http://oeis.org/A197049) has given me the
idea to tile the plane with integers in such a manner that any digit of the plane
would describe the quantity of copies of itself it orthogonally touches.
In the hereunder box,
for example, all zeroes have 0 copies of 0 as “ortho-neighbor”,
all ones have 1 copy of 1 as neighbor, all twos touch two other “2”:
2 2
2 2 2
2 0 1 1 2
2 2
2 2 2
4 is the smallest
integer tiling the plane (as 44 does):
...
4 4 4 4
... 4 4 4 4
4 4 4 4 ...
4 4 4 4
...
Is 233 the next one?
...
2 3 3 2 3 3 2 3 3
2 3 3 2 3 3
2 3 3 2 3 3
2 3 3 2 3 3 2 3 3
... 2 3 3
2 3 3 2 3 3 2 3 3 2 3 3
2 3 3 2 3 3
2 3 3 2 3 3 2 3 3 ...
2 3 3 2 3 3 2 3 3
2 3 3 2 3 3
2 3 3 2 3 3
2 3 3 2 3 3 2 3 3
...
323 and 332 tile the
plane too, of course; and 233233, 323323, 332332, etc.
30333 does the job:
3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3
3 0 3 3
3 3 0 3 3 3
3 3
3 3 0 3 3 3 3 0 3 3 3
3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0
3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3 3
0 3 3 3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3
3 3
3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3
3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3
3 0 3 3
3 3 0 3 3 3
3 3
3 3 0 3 3 3 3 0 3 3 3
3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0
3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3 3
0 3 3 3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3
3 3
3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3 3 3 3 0 3 3
3 3 0 3
3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3
3 0 3 3
3 3 0 3 3 3
120212 is a
candidate:
1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2
0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2
1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2
1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2
0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2
1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2
1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2
0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2
1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2
1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2
What would be the
sequence of “self-integers” tiling the plane? Does S start like this?
S = 4, 44, 233, 323, 332, 444, ...
Best,
É.
__________
[Remark #1]:
I think the huge
integer 1133133333133303331333331331133333333 tiles the infinite plane – is
thus part of S – because of the hereunder tile:
3 3 1
3 3 3 0
3 3 1
3 3 3
3 3 3 3 1 1
1 1 3 3 3 3
3 3 3 1 3 3
3 3 3 1 3 3
Make a copy of this
tile and stick its left side against its right side – under the zero. Repeat.
The same happens with
133033133 and the tile:
1 3 3
0 3 3 1 3 3
1 3 3
1 3 3 0 3 3
0 3 3
1 3 3 1
3 3
[Remark #2]:
Some infinite tilings are impossible to build with a single integer:
0 1 1
0 1 1 0
1 1 0 1 1 0 1 1 0 1 1
0 1 1 0
1 1 0
1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1
1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1
0 1 1
0 1 1 0
1 1 0 1 1 0 1 1 0 1 1
0 1 1 0
1 1 0
1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1
1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1
0 1 1
0 1 1 0
1 1 0 1 1 0 1 1 0 1 1
0 1 1 0
1 1 0
1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1
1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1
0 1 1
0 1 1 0
1 1 0 1 1 0 1 1 0 1 1
0 1 1 0
1 1 0
1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1
1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1
0 1 1
0 1 1 0
1 1 0 1 1 0 1 1 0 1 1
0 1 1 0
1 1 0
... or this one:
1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2
2 2 1 2 2
1 0 1 1 0
2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2
2 2 0 1 1 0
1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2
1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2
1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0
0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1
1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1
2 1 2 2
0 1 1 0
1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0
2 1 0 1 1 0
2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2
2 2 1 2 2
1 0 1 1 0
2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2
2 2 0 1 1 0
1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2
1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2
1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0
0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1
1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1
2 1 2 2
0 1 1 0
1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0 1 1
0 1 2 2 1 2 2 0
2 1 0 1 1 0
2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
0 2 2 1 2 2 1 0 1 1
1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2
... or that:
2 2 2
0 2 2
2 0 2 2 2 0 2 2 2 0 2 2 2
0 2 2
2 0 2 2 2
2 0 2 1 1 0 1 1
2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2
2 2 2
0 2 2
2 0 2 2 2 0 2 2 2 0 2 2 2
0 2 2
2 0 2 2 2
1 0 1 1 2 0 2 1 1 0 1 1
2 0 2 1 1 0 1 1 2 0 2 1 0 1 1
2 2 2
0 2 2
2 0 2 2 2 0 2 2 2 0 2 2 2
0 2 2
2 0 2 2 2
2 0 2 1 1 0 1 1
2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2
2 2 2
0 2 2
2 0 2 2 2 0 2 2 2 0 2 2 2
0 2 2
2 0 2 2 2
1 0 1 1 2 0 2 1 1 0 1 1
2 0 2 1 1 0 1 1 2 0 2 1 0 1 1
2 2 2
0 2 2
2 0 2 2 2 0 2 2 2 0 2 2 2
0 2 2
2 0 2 2 2
2 0 2 1 1 0 1 1
2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2
2 2 2
0 2 2
2 0 2 2 2 0 2 2 2 0 2 2 2
0 2 2
2 0 2 2 2
1 0 1 1 2 0 2 1 1 0 1 1
2 0 2 1 1 0 1 1 2 0 2 1 0 1 1
[Remark #3]:
Some integers might
tile the plane in more than one way.
Best,
É.