  We investigate the triangle singularity $f=x^a+y^b+z^c$, or $S=k[x,y,z]/(f)$,
attached to a weighted projective line $X$ given by the weight triple
$(a,b,c)$. We investigate the stable category of vector bundles on $X$ obtained
from the vector bundles by factoring out all line bundles. This category is
triangulated and has Serre duality. It is, moreover, naturally equivalent to
the stable category of graded maximal Cohen-Macaulay modules over $S$ (or
matrix factorizations of $f$), and then by results of Buchweitz and Orlov to
the graded singularity category of $f$. We show that this category is
fractional Calabi-Yau with a CY-dimension that is a function of the Euler
characteristic of $X$. We show the existence of a tilting object which has the
shape of an $(a-1)(b-1)(c-1)$-cuboid. Particular attention is given to the
weight types $(2,a,b)$, yielding an explanation of Happel-Seidel symmetry for a
class of important Nakayama algebras. In particular, the weight sequence
$(2,3,p)$ corresponds to an ADE-chain, the $E_n$-chain, extrapolating the
exceptional Dynkin cases $E_6$, $E_7$ and $E_8$ to a whole sequence of
triangulated categories.
