1) Heesch-tiles based on n-gons
Referring to the book "Tilings and Patterns" from Prof. Grünbaum and Prof. Shephard (Pag 155), let define a Heesch-tile as a tile T which can be surrounded r times by tiles congruent to T , but not r + 1 times. ( r = Heersch-number)
Heesch-tiles with r = 1, can be constructed based on n-gons, with n odd and n > 3.
So the smallest n-gon we can use for this construction is the pentagon. Inside the pentagon we first draw the penta-star.
Using the dark area as a tile results in a complete tiling of the plane.
The dark area is now used twice to construct a new tile:
Now we can surround this tile only once with congruent tiles:
The same construction can now be used for n > 5 and one can prove that this method works for all odd n > 3 (see proof):
In this case we must construct the n-star with the smallest possible angle in the arms of the star. So the first star is ok, but not the second one.
f.e. n = 7
n = 9
n = 11
The number of tiles needed to surround the central tile is n+2. Using the odd numbers , starting with 5, the number of tiles needed are: 7, 9, 11, 13, ......
There are several other variations possible:
f.e. for n = 7
For n = 5 it was necessary to use the dark area twice, but this is no longer the case for n > 5.
This is even no longer necessary to use odd n as long as n > 6.
f.e. n = 7 (we can so this in several ways: symmetrical and asymmetrical)
The number of tiles needed to surround these sort of tiles in an asymmetrical way is n + 5. So using the odd number > 5, the numbers of tiles needed are 12, 14, 16, ......
We can ask the question: "Does there exist for all N > 1, a tile which can be surrounded by N congruent tiles in order to form a Heesch-tiling with Heesch-number 1?
Untill now we have not yet a positive answer for: 2, 3, 4, 5, 6, 8, 10.
For n = 8, we find a solution with N = 10. And the original Heesch-tile had a N = 8.
Mr. Andrew Crompton found a tile with N = 6:
This leave the open question: "Does there exist a Heesch-tile with Heesch-number r = 1, with N = 2, 3, 4, 5 ?"
(see further part 3)
Tilings and Patterns
Prof. Grunbaum and Prof. Shephard
W.H. Freeman and Company, New York, 1987
2) Further Heesch tilings
Example of Heesch-tile based on a pentagon:
Example of Heesch-tile based on a hexagon: probably there are many of this kind.
Other example based on a hexagon: all tiles of the second and third column are surrounded.
3) Heesch Tiles with Heesch-number = 1 and N = 3 and 4
Erich Friedman (Math Department, Stetson University, DeLand, USA) proved that Heesch Tiles with Heesch-number = 1 and N = 3 and 4 exist.
He also conjecture that there is no such tile for N = 2.
The solution for N = 4 is a 5x7 rectangle with three 1x1 additions and one 1x1 hole.
The solution for N = 3 is more complicated.
This leaves us with the question: Is the Erich Friedman-conjecture true or not?
Ref: Erich Friedman, Heesch tiles with surround numbers 3 and 4, Geombinatorics, Volume VIII, April 1999, Issue 4, 101-103
For more info about Heesch tiles: see also:Marc Thompson
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