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Existing techniques to compress point cloud attributes leverage either geometric or video-based compression tools. We explore a radically different approach inspired by recent advances in point cloud representation learning. Point clouds can be interpreted as 2D manifolds in 3D space. Specifically, we fold a 2D grid onto a point cloud and we map attributes from the point cloud onto the folded 2D grid using a novel optimized mapping method. This mapping results in an image, which opens a way to apply existing image processing techniques on point cloud attributes. However, as this mapping process is lossy in nature, we propose several strategies to refine it so that attributes can be mapped to the 2D grid with minimal distortion. Moreover, this approach can be flexibly applied to point cloud patches in order to better adapt to local geometric complexity. In this work, we consider point cloud attribute compression; thus, we compress this image with a conventional 2D image codec. Our preliminary results show that the proposed folding-based coding scheme can already reach performance similar to the latest MPEG Geometry-based PCC (G-PCC) codec.