1 research outputs found
Spatial chaos of Wang tiles with two symbols
This investigation completely classifies the spatial chaos problem in plane
edge coloring (Wang tiles) with two symbols. For a set of Wang tiles
, spatial chaos occurs when the spatial entropy
is positive. is called a minimal cycle generator if
and
whenever , where is the set of all periodic
patterns on generated by . Given a set of Wang
tiles , write , where , , are minimal cycle generators and
contains no minimal cycle generator except those contained in
. Then, the positivity of spatial
entropy is completely determined by .
Furthermore, there are 39 equivalent classes of marginal positive-entropy
(MPE) sets of Wang tiles and 18 equivalent classes of saturated zero-entropy
(SZE) sets of Wang tiles. For a set of Wang tiles ,
is positive if and only if contains an MPE set,
and is zero if and only if is a subset of an SZE
set