If we allow_two_ sets of congruent pieces, then the regular pentagon can divided into various numbers of parts according to the Lucas numbers. Usually these are defined by the recursion $L_1=1,L_2=3, L_{n+1}=L_n+L_{n-1}$ for $n\ge2$. If we run the recursion backwards, we find $L_0=2$ and $l_{-n}=(-1)^nL_n$. We observe the (positive) numbers from this sequence in the duvisions below, where the two sets of congruent areas are represented by different colored dots. With an unbounded number of iterations the ratio of the numbers of lieces tends to $\phi=(1+\sqrt5)/2$, and the nodes fill in a tenfold symmetric quasilattice.
