CIMPA School: Tilings and Tessellations
August 31, 2015
1
Grobner Basis and applications to tilings
Exo 1 What is the Grobner basis associated to the vertical and horizontal bars?
what is the link between these basis and the coloration of squares introduced
by F. Bassino?
Answer: The Grobner basis is B = {1 + X + ...X p−1 , 1 + ... + Y q−1 } for
the tilings by bars 1 × p and q × 1. Now, to the cell (i, j), is associated the
monomial X i Y j which can be reduced by B in
Exo 2 Grobner Basis can be used to give strong necessary conditions of
tilability. Here a example:
• Show that the Grobner basis on Z associated to the set of T tetraminos is
[X − 5, Y − 5, 8]. For this, you can prove that the polynomial associated
to a T tetramino belongs hX − 5, Y − 5, 8i. And reciprocally, that we can
reach the polyominoes associated to X −5,Y −5 and 8 from T tetraminoes.
• Explain without polynomial, directly on the polyomino, the meaning of
belonging to [X − 5, Y − 5, 8].
Exo 3 But sometime, we are disappointed:
• Show that the Grobner basis on Z associated to the set of L triminos is
[X − 1, Y − 1, 3]. What does it mean?
• Find how to Z-tile a unique square of weight 3 with L triminos.
Exo 4
• Consider that we are on the hexagonal lattice, what is the associated ideal
which corresponds to Z-tilability.
1
1/2
qb
1/2
0,1,L
start
1,1,R
qa
1/2
qc
1/2
qd
1/2
1/2
1/2
qe
2
The current candidate for the busy beaver for five states. It
is presumed that this Turing machine writes a maximum
number of 1’s before halting among all Turing machines
with five states and the tape alphabet {0, 1}. Proving this
conjecture is an open research problem.