# [seqfan] Heesch numbers

Fri May 8 15:02:21 CEST 2015

```Hello sequence fans, let S be any shape in the plane (2-dimensional space).
Then the maximum number of layers of congruent copies of S that can
http://en.wikipedia.org/wiki/Heesch%27s_problem).

Let P be a finite or infinite patch of adjacent copies of S. If P is
infinite, it is usually called a 'tiling' or 'tessellation'. Two specific
shapes S_1 and S_2 that are part of P are said to be in the same
'translation class' t, if S_1 can be turned into S_2 via translations only
(not rotations or reflections).

Let T be the set of all t in which all tiles S_1, S_2, S_3, .... S_i of P
occur.

I am interested in the following sequence:

a(n) is the minimal possible cardinality of T when S has Heesch number n.

Some known upper bounds:

a(1): Erich Friedmann has constructed a tile with Heesch number 1, for
which |T| = 3 (see Figure 1 at
http://www2.stetson.edu/~efriedma/papers/heesch/heesch.html)
a(2): Casey Mann gives a tile with Heesch number 2 with |T| = 6 (see
http://math.uttyler.edu/cmann/math/heesch/heesch.htm)
a(3): Casey Mann gives a tile with Heesch number 3 with |T| = 4 (see
http://math.uttyler.edu/cmann/math/heesch/heesch.htm)
a(4): Casey Mann gives a tile with Heesch number 4 with |T| = 6 (see
http://math.uttyler.edu/cmann/math/heesch/heesch.htm)
a(5): Casey Mann's tile with Heesch number 5 has |T| = 10, as can be seen
in the illustration given at
http://math.uttyler.edu/cmann/math/heesch/heesch.htm

So sequence based on known results would start 3, 6, 4, 6, 10

I don't think those values are proven to be minimal, though.

Best regards
Felix Fröhlich
```