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Volumetric displays provide interesting opportunities and challenges for 3D interaction and visualization, particularly when used in a highly interactive manner. We explore this area through the design and implementation of techniques for interactive direct manipulation of objects with a 3D volumetric display. Motion tracking of the user's fingers provides for direct gestural interaction with the virtual objects, through manipulations on and around the display's hemispheric enclosure. Our techniques leverage the unique features of volumetric displays, including a 360° viewing volume that enables manipulation from any viewpoint around the display, as well as natural and accurate perception of true depth information in the displayed 3D scene. We demonstrate our techniques within a prototype 3D geometric model building application.