We consider those initial segments of a nonstandard model $\frak M$ of Peano arithmetic (abbreviated by $P$) which can be obtained as unions or intersections of initial segments of $\frak M$ isomorphic to $\frak M$. For any consistent theory $T\supseteq P$ we find models of $T$ having collections of initial segments densely ordered by inclusion so that for any segment $I$ from such collection and any $k\in \omega$ the family $\c{A}_k^{\,\frak M}= \{\frak N|\,\frak N\subseteq_e \frak M. \frak N \prec_{\Sigma_k} \frak M, \frak N\cong \frak M\}$ can be partioned into two disjoint parts $\c{A}_1$, and $\c{A}_2$ satisfying $I= \bigcup\c{A}_1= \bigcap \c{A}_2$ i.e\. $I$ is a ``point of accumulation" for all families $\c{A}_k^{\,\frak M}$. We investigate. the order type of such collections of segments in the case of recursively saturated models of $P$.