at (330:0.65) [$a_4$]{}; (330:0.5) node\[fill=white,circle,inner sep=0.065cm\] circle (0.02cm);

-   Given $a \in L( \cZ )$, we say that ${\mathscr{X}}$ [*has a leapfrog converging to $a\in L({\mathscr{Z}})$*]{} if there is a sequence $\{x_i,y_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ of diagonals from ${\mathscr{X}}$ with $x_i \to a$ from below and $y_i \to a$ from above. (Convergence from below and above is explained in Definition \[def:convergence\_from\_below\_and\_above\].)

-   Given $a \in L({\mathscr{Z}})$, $z \in {\mathscr{Z}}$. We say that ${\mathscr{X}}$ [*has a right fountain at $z$ converging to $a$*]{} if there is a sequence $\{z, x_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ from ${\mathscr{X}}$ with $x_i \to a$ from below. We say that ${\mathscr{X}}$ [*has a left fountain at $z$ converging to $a$*]{} if there is a sequence $\{z, y_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ from ${\mathscr{X}}$ with $y_i \to a$ from above.

    We say that ${\mathscr{X}}$ [*has a fountain at $z$ converging to $a$*]{} if it has a right fountain and a left fountain at $z$ converging to $a$.

Here is our first main result. It is closely related to [@SvR thm. 7.17]. Given a set $\cX$ of diagonals of $\cZ$, we write $E({\mathscr{X}})$ for the corresponding set of indecomposable objects of ${\mathscr{C}}({\mathscr{Z}})$.

\[=Theorem \[T:cluster tilting\]\] \[thm:B\] Let ${\mathscr{X}}$ be a 