2 research outputs found
Sandpile toppling on Penrose tilings: identity and isotropic dynamics
We present experiments of sandpiles on grids (square, triangular, hexagonal)
and Penrose tilings. The challenging part is to program such simulator; and our
javacript code is available online, ready to play! We first present some
identity elements of the sandpile group on these aperiodic structures, and then
study the stabilization of the maximum stable configuration plus the identity,
which lets a surprising circular shape appear. Roundness measurements reveal
that the shapes are not approaching perfect circles, though they are close to
be. We compare numerically this almost isotropic dynamical phenomenon on
various tilings
On the Complexity of Asynchronous Freezing Cellular Automata
In this paper we study the family of freezing cellular automata (FCA) in the
context of asynchronous updating schemes. A cellular automaton is called
freezing if there exists an order of its states, and the transitions are only
allowed to go from a lower to a higher state. A cellular automaton is
asynchronous if at each time-step only one cell is updated. Given
configuration, we say that a cell is unstable if there exists a sequential
updating scheme that changes its state. In this context, we define the problem
AsyncUnstability, which consists in deciding if a cell is unstable or not. In
general AsyncUnstability is in NP, and we study in which cases we can solve the
problem by a more efficient algorithm.
We begin showing that AsyncUnstability is in NL for any one-dimensional FCA.
Then we focus on the family of life-like freezing CA (LFCA), which is a family
of two-dimensional two-state FCA that generalize the freezing version of the
game of life, known as life without death. We study the complexity of
AsyncUnstability for all LFCA in the triangular and square grids, showing that
almost all of them can be solved in NC, except for one rule for which the
problem is NP-complete