The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs, generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial conditions, at all sizes, there appears what we call an exact Sierpinski structure: the volume is filled with periodic patterns, glued together along straight lines, with the topology of a triangular Sierpinski gasket. Various lattices (square, hexagonal, kagome, …), initial conditions, and toppling rules show Sierpinski structures which are apparently unrelated and involve different mechanisms. As will be shown elsewhere, all these structures fall under one roof, and are in fact different projections of a unique mechanism pertinent to a family of deterministic surfaces in a four-dimensional lattice. This short note gives a description of this surface, and of the combinatorics associated to its construction.

Deterministic Abelian Sandpile and square-triangle tilings / S. Caracciolo, G. Paoletti, A. Sportiello (SPRINGER INDAM SERIES). - In: Combinatorial Methods in Topology and Algebra / [a cura di] B. Benedetti, E. Delucchi, L. Moci. - [s.l] : Springer, 2015 Aug 25. - ISBN 9783319201542.-127 (( convegno Combinatorial Methods in Topology and Algebra tenutosi a Cortona nel 2013 [10.1007/978-3-319-20155-9_23].

Deterministic Abelian Sandpile and square-triangle tilings

S. Caracciolo;
2015-08-25

Abstract

The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs, generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial conditions, at all sizes, there appears what we call an exact Sierpinski structure: the volume is filled with periodic patterns, glued together along straight lines, with the topology of a triangular Sierpinski gasket. Various lattices (square, hexagonal, kagome, …), initial conditions, and toppling rules show Sierpinski structures which are apparently unrelated and involve different mechanisms. As will be shown elsewhere, all these structures fall under one roof, and are in fact different projections of a unique mechanism pertinent to a family of deterministic surfaces in a four-dimensional lattice. This short note gives a description of this surface, and of the combinatorics associated to its construction.
Physics - Statistical Mechanics; Physics - Statistical Mechanics; High Energy Physics - Lattice; Mathematics - Combinatorics; Nonlinear Sciences - Cellular Automata and Lattice Gases
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
INdAM
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/334143
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