a $T1$ on $s=4$, as we shall see.

A suitable $T1$ shifts a grain boundary
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In general, a grain boundary is shifted by a $T1$ at $s=p$, where $p$ labels the first heptagonal cell (Fig.\[fg4\]a). For example, the $T1$ (${\bf D} =...,.[-,.,.,.,.,+,.,.,+,.,.,.,.],-,.,...$) translates the sequence $...,6,[7,7,7,7,7,6,6,6,5,5,5,5,5],6,6,...$ into $...,6,6,[7,7,7,7,7,6,6,6,5,5,5,5,5],6,...$. Remember that this is easily done because neighboring cells are close to squares, meeting at critical vertices for $T1$ and ready to flip either way.

Thus, the sequence $(5,6,6),[(6,6,6,6,6),(6,6,6),(5,5,5,5,5)],6,6,\ldots$ (spheres $n=43-75$) is invariant under disappearance of $1$, together with a $T1$ on $s=4$ displacing the first grain boundary $[6,6,6,6,6,6,6,6,5,5,5,5,5]$ [^6]. Note that this thirteen-cell layer is too small to be a normal grain boundary (where the five innermost hexagons should be heptagons) but it has the correct topological charge of $+5$ that, with pentagon $s=1$, adds up to the charge $+6$ of an hemisphere. Let us call it a polar circle.

Other examples of sequences $c(s)$ are given in Appendix 1.

The sequence $c(s)$ and the local organization of cells into parastichies and grain boundaries are therefore invariant under disappearance of the first cell, or under growth, the inverse operation. This is a very important result that shows the malleability of 