1. Flat Faces in Block and Hole Polyhedra Walter Whiteley July 2015

2. Start with spherical block and hole polyhedra Block Hole Expanding Contracting (a) (b) (c) (d)

3. Recent Extension If triangulated sphere has one added cross-beam and resulting graph is 4 connected then redundantly rigid (Wendy Finbow-Singh, WW) Question is it generically globally rigid?

4. Flattening Extension Ask that ‘faces’ are kept as triangulated planes, with natural vertices, or vertices on natural edges. Specialized geometry – is this still ‘generically’ rigid? needs modification and extension of proof. Can be done (with Wendy Finbow-Singh).

5. Two Operations Add vertices as necessary along edges; “selected” an edge across face Split face along edge to create two faces. Need capacity to place faces on distinct planes without warping any of the other faces; Tool is the Steinitz sequence for (convex) spherical polyhedra. Analyzed in: How to design or describe a polyhedron, J. of Intelligent and Robotic Systems 11 (1994), 135-160

6. Face Split

7. Comments: This works for spherical polyhedra - 3 connected planar graph Does not change the selection of blocks and holes and connectivity criteria What about toroidal polyhedra? Is there a connectivity assumption that is sufficient? (e.g. 6 connected?)