We are interested in PLPAs that can open and close “all the way”. Let fold angle be the outer angle of the central fold in a PLPA or any PLS folded from a PLPA (e.g., the fold angle of a PLPA is zero). Note that, strictly speaking, a PLPA cannot close completely (e.g., at fold angle 180◦ ) by our definition of foldability, as the backdrop and ground are not allowed to overlap. We therefore formulate a foldable layout as: Definition 4 A PLPA T is said to be foldable if there is a fold transform f from T during which the fold angle increases monotonically from 0 to 180◦ − with arbitrarily small . 

This transform f is called the pop-up transform of T, and any f(t) for t ∈ [0, 1] is called popup-foldable. For example, the PLPA shown in Figure 3 (a) is foldable, and a popup-foldable surface at 90◦ fold angle is shown in (b). 3.3 Realizable layouts A foldable PLPA as defined above may not be able to “pop-up” as the user opens or closes the ground and backdrop. For example, the PLPA may contain pieces disconnected from the ground or backdrop that needs extra support to hold them in place.

 Also, the pop-up transform may require external forces along the fold lines in order to crease them in practice. Ideally, we would to create stable paper architecture where the pop-up transform does not require additional forces other than opening and closing the backdrop and ground by the user, and the paper at each stage during the transform is able to hold steady when the backdrop and ground are fixed. We can define realizability based on the same rigidity assumption we made above. Intuitively, a PLS that can hold steady when some patches are fixed is one that cannot be folded to another state while keeping the fixed patches unchanged. Also, if the PLPA at each time point of a continuous folding process is stable with respect to the backdrop and ground, then no external forces would be required to get to the next time from a previous time other than moving the backdrop and ground. More formally, we define: Definition 5 Given a PLS S and a subset of patches P ⊂ S, S is said to be stable with respect to P if there is no other PLS S 0 6= S that is foldable from S via a fold transform f, where P keeps stationary during f.