Thinking Outside (and Inside) the Box

In many problems in commonsense reasoning and intelligent manufacturing, we need to reason about cutting, joining, and folding sheets of materials such as cardboard and metal. In this paper we introduce a first-order ontology of shape (called BoxWorld) that can support these applications. We reuse an existing ontology of shape for object recognition with 2D shapes (called CardWorld) and extend the axioms to three-dimensional shapes in the BoxWorld Ontology. A distinguishing characteristic of these ontologies is that they use only the notions of incidence and betweenness rather than Euclidean geometry as axiomatized by Hilbert and Tarski. Prelude When Alice went to the kitchen for her breakfast cereal, she opened the box by removing the tab from the slot at the top of the box. Discovering it empty, she reached for a new box, which she opened by detaching one side of the lid from the other. She closed the box by opening the slot on one slide of the lid and then inserted the tab into the slot. After breakfast, she disassembled the old box and dropped it into the recycling bin.