was only recently explored in [@litany2018deformable; @coma; @wang2018pixel2mesh; @kolotouros2019convolutional] for shape completion, non-linear facial morphable model construction and 3D reconstruction from single images, respectively. In this paper, we propose a novel representation learning and generative framework for fixed topology meshes. For this purpose, we formulate an ordering-based graph convolutional operator, contrary to the permutation invariant operators in the literature of Graph Neural Networks. In particular, similarly to image convolutions, for each vertex on the mesh, we enforce an explicit ordering of its neighbours, allowing a “1-1” mapping between the neighbours and the parameters of a learnable local filter. The order is obtained via a spiral scan, as proposed in [@spirals], hence the name of the operator, *Spiral Convolution*. This way we obtain anisotropic filters without sacrificing computational complexity, while simultaneously we explicitly encode the fixed graph connectivity. The operator can potentially be generalised to other domains that accept implicit local orderings, such as arbitrary mesh topologies and point clouds, while it is naturally equivalent to traditional grid convolutions. Via this equivalence, common CNN practices, such as dilated convolutions, can be easily formulated for meshes.

We use spiral convolution as a basic building block for hierarchical intrinsic mesh autoencoders, which we 