We present Vermeer, a novel interactive 360° viewable 3D display. Like prior systems in this area, Vermeer provides viewpoint-corrected, stereoscopic 3D graphics to simultaneous users, 360° around the display, without the need for eyewear or other user instrumentation. Our goal is to over-come an issue inherent in these prior systems which - typically due to moving parts - restrict interactions to outside the display volume. Our system leverages a known optical illusion to demonstrate, for the first time, how users can reach into and directly touch 3D objects inside the display volume. Vermeer is intended to be a new enabling technology for interaction, and we therefore describe our hardware implementation in full, focusing on the challenges of combining this optical configuration with an existing approach for creating a 360° viewable 3D display. Initially we demonstrate direct involume interaction by sensing user input with a Kinect camera placed above the display. However, by exploiting the properties of the optical configuration, we also demonstrate novel prototypes for fully integrated input sensing alongside simultaneous display. We conclude by discussing limitations, implications for interaction, and ideas for future work.