You should add "allowing tiles to be turned over, reflected, rotated, etc." to the rules. If not, the 2D answer changes. Another likely 3D issue is whether you require that the tiling be physically assembleable, without requiring some temporary interpenetration. Rich ----- Quoting Veit Elser <ve10@cornell.edu>:
It's easy to show, for p>2, F(p)<= 2 3^(p-1) - 1.
-Veit
On Jun 13, 2013, at 2:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Polyomino non-tiling problem:
Definition: A p-dimensional n-omino X is a union of integer lattice cubes (of size 1^p) such that
i) The interior of X is connected,
AND
ii) X is topologically equivalent to a closed p-dimensional disk.
[Note: I'm adding the nonstandard condition ii) in order to avoid trivial cases of non-tiling due to holes, such as the heptomino with its squares' centers arranged thus:
* *
* *
* * *
.]
In p-dimensional Euclidean space, find the least N for which there exists a p-dimensional N-omino, copies of which cannot tile p-space.
(E.g., it's known that F(2) = 7.)
Call this N by the notation F(p)
QUESTION I): Can F(p) be determined explicitly for all p ??? -------------
QUESTION II): In any case, can an asymptotic formulas be found ------------- for F(p), as p -> oo ???
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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