Consider the case that

$$\mathfrak{D}=\{(\mathbb{R}(1,0),(1+tx^{-1})^{\ell}),(\mathbb{R}(0,1),(1+ty^{-1})^{\ell})\}$$

for $\ell$ some positive integer. For $\ell=1$, it is easy to check that

$$\operatorname{{\mathsf{S}}}(\mathfrak{D})\setminus\mathfrak{D}=\{(\mathbb{R}_{\geq 0}(1,1),1+t^{2}x^{-1}y^{-1})\}.$$

Figure 9 shows explicitly what the automorphisms are as one traverses the depicted loop; the reader can easily check that the composition of the five automorphisms is the identity.

If $\ell=2$, then one finds

	$\displaystyle\operatorname{{\mathsf{S}}}(\mathfrak{D})\setminus\mathfrak{D}=$		$\displaystyle\{(\mathbb{R}(n+1,n),(1+t^{2n+1}x^{-(n+1)}y^{-n})^{2})|n\in\mathbb{Z},n\geq 1\}$
		$\displaystyle\cup$	$\displaystyle\{(\mathbb{R}(n,n+1),(1+t^{2n+1}x^{-n}y^{-(n+1)})^{2})|n\in\mathbb{Z},n\geq 1\}$
		$\displaystyle\cup$	$\displaystyle\{(\mathbb{R}(1,1),(1-t^{2}x^{-1}y^{-1})^{-4})\}.$

This was first found experimentally by myself and Siebert via a computer program, and the first verification of this was given in [14]. It also follows immediately from the results of [27] which will be explained in what follows.

If $\ell=3$, the situation becomes even more complicated. First, as noticed by Kontsevich, $\operatorname{{\mathsf{S}}}(\mathfrak{D})$ has a certain periodicity. Namely,

$$(\mathbb{R}_{\geq 0}(m_{1},m_{2}),f(x^{-m_{1}}y^{-m_{2}}))\in\operatorname{{\mathsf{S}}}(\mathfrak{D})$$

if and only if

$$(\mathbb{R}_{\geq 0}(3m_{1}-m_{2},m_{1}),f(x^{-(3m_{1}-m_{2})}y^{-m_{1}}))\in\operatorname{{\mathsf{S}}}(\mathfrak{D}),$$

provided that $m_{1},m_{2}$ and $3m_{1}-m_{2}$ are all positive. In addition, there are rays with support $\mathbb{R}_{\geq 0}(3,1)$ and $\mathbb{R}_{\geq 0}(1,3)$, hence by the periodicity, there are also rays with support

$$\mathbb{R}_{\geq 0}(8,3),\ \mathbb{R}_{\geq 0}(21,8),\ \ldots\ \ \ \text{and}\ \ \ \mathbb{R}_{\geq 0}(3,8),\ \mathbb{R}_{\geq 0}(8,21),\ \ldots$$

which converge to the rays of slope $(3\pm\sqrt{5})/2$, corresponding to the two distinct eigenspaces of the linear transformation $\begin{pmatrix}3&-1\\ 1&0\end{pmatrix}$. Each of these rays is of the form

$$\big(\mathbb{R}_{\geq 0}(m_{1},m_{2}),(1+t^{m_{1}+m_{2}}x^{-m_{1}}y^{-m_{2}})^{3}\big).$$

These are the only rays appearing outside of the cone generated by the rays of slope $(3\pm\sqrt{5})/2$. On the other hand, inside this cone, every rational slope occurs, and the attached functions are very complicated. For example, the function attached to the line of slope 1 is

$$\left(\sum_{n=0}^{\infty}{1\over 3n+1}\begin{pmatrix}4n\\ n\end{pmatrix}t^{2n}x^{-n}y^{-n}\right)^{9}.$$

Again, Siebert and I found this form via computer experiment, but it was verified by Reineke in [68]. Recently, Kontsevich has shown the functions attached to all these rays are algebraic. For example, if $g$ denotes the $9$-th root of the above function, it satisfies the equation

$$t^{2}x^{-1}y^{-1}g^{4}-g+1=0.$$

This series of examples also makes contact with a number of other interesting objects. On the one hand, Reineke in [68] gave an interpretation of the attached functions in terms of Euler characteristics of moduli spaces of representaions of the Kronecker $\ell$-quiver, the quiver with two vertices and $\ell$ arrows between them. On the other hand, these diagrams are also closely related to the cluster algebras defined by these quivers. This connection will be studied in more detail in forthcoming joint work with Keel, Kontsevich and others.