We introduce the polygon cloud, a compressible representation of three-dimensional geometry (including attributes, such as color), intermediate between polygonal meshes and point clouds. Dynamic polygon clouds, like dynamic polygonal meshes and dynamic point clouds, can take advantage of temporal redundancy for compression. In this paper, we propose methods for compressing both static and dynamic polygon clouds, specifically triangle clouds. We compare triangle clouds to both triangle meshes and point clouds in terms of compression, for live captured dynamic colored geometry. We find that triangle clouds can be compressed nearly as well as triangle meshes, while being more robust to noise and other structures typically found in live captures, which violate the assumption of a smooth surface manifold, such as lines, points, and ragged boundaries. We also find that triangle clouds can be used to compress point clouds with significantly better performance than previously demonstrated point cloud compression methods. For intra-frame coding of geometry, our method improves upon octree-based intra-frame coding by a factor of 5–10 in bit rate. Inter-frame coding improves this by another factor of 2–5. Overall, our proposed method improves over the previous state-of-the-art in dynamic point cloud compression by 33% or more.