The three dimensional polyominoes of minimal area

  • Laurent Alonso
  • Raphaël Cerf


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.

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