The three dimensional polyominoes of minimal area - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue The Electronic Journal of Combinatorics Année : 1996

The three dimensional polyominoes of minimal area

Laurent Alonso
  • Fonction : Auteur
Raphaël Cerf
  • Fonction : Auteur
  • PersonId : 1134082
  • IdRef : 085584789

Résumé

The set of the three dimensional polyominoes of minimal area and of volume $n$ contains a polyomino which is the union of a quasicube $j\times (j+\delta)\times (j+\theta)$, $\delta,\theta\in\{0,1\}$, a quasisquare $l\times (l+\epsilon)$, $\epsilon\in\{0,1\}$, and a bar $k$. This shape is naturally associated to the unique decomposition of $n=j(j+\delta)(j+\theta)+l(l+\epsilon)+k$ as the sum of a maximal quasicube, a maximal quasisquare and a bar. For $n$ a quasicube plus a quasisquare, or a quasicube minus one, the minimal polyominoes are reduced to these shapes. The minimal area is explicitly computed and yields a discrete isoperimetric inequality. These variational problems are the key for finding the path of escape from the metastable state for the three dimensional Ising model at very low temperatures. The results and proofs are illustrated by a lot of pictures.
Fichier principal
Vignette du fichier
minr.pdf (17.1 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03713061 , version 1 (04-07-2022)

Identifiants

Citer

Laurent Alonso, Raphaël Cerf. The three dimensional polyominoes of minimal area. The Electronic Journal of Combinatorics, 1996, 3 (1), ⟨10.37236/1251⟩. ⟨hal-03713061⟩
1 Consultations
3 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More