There exists a need for unsupervised 3D segmentation on complex volumetric data, particularly when annotation ability is limited or discovery of new categories is desired. Using the observation that 3D data is innately hierarchical, we propose learning effective representations of 3D patches for unsupervised segmentation through a variational autoencoder with a hyperbolic latent space and a proposed gyroplane convolutional layer, which better models underlying hierarchical structure within a 3D image. We also introduce a hierarchical triplet loss and multi-scale patch sampling scheme to embed relationships across varying levels of granularity. We demonstrate the effectiveness of our hyperbolic representations for unsupervised 3D segmentation on a hierarchical toy dataset and the BraTS dataset.