Consider the 2-Calabi--Yau triangulated category arising from the zigzag algebra of the An quiver. The braid group acts on this category by twists in spherical objects. Given a Bridgeland stability condition, we describe how to realise the spherical objects as a dense subset of a piecewise-linear manifold. This manifold is canonically associated to the category, and the braid group acts on it piecewise-linearly. We also describe how the manifold transforms under wall-crossings of the stability condition. The talk is based on joint work with Anand Deopurkar and Anthony M. Licata.