# Book: The Mother of All Tableaux

## Chapter: Geometry

### Blurb:

We conclude with a discussion of the remarkably well-behaved geometry of OT typologies, which gives a concrete sense to the notion of the Border Point Pair from which the MOAT derives. We show how a typology is represented on the permutohedron, a well-studied figure where each vertex corresponds to a linear order. We then show this representation reduces to the typohedron, a structure in which each vertex represents an entire grammar, leading to a visually accessible representation of the overall structure of a typology. We complete the discussion by offering proofs of several striking results announced by Jason Riggle, showing that when distance is defined between linear orders, grammars are convex regions analogous to disks and balls in the familiar Euclidean world.