The Origami Forum

I have solved several Cp's namely some Hojyo's, Jason Ku's Elf, Miyajima's Knight with sword, that are kind of simple to do. You just find the axle of the model and then go mountain valley mountain valley mountain valley and so on the creases that touch the axle of the model and the base almost collapses. You just have to do a little extra work on the creases that do not touch the axle. Coincidentially all these bases are for human models.

If you do not get what I mean please check http://www.angelfire.com/co/cubo/exambase.html this is a hypothetical base that follows this principle.

My question is is there a name for this type of bases? What is that internal geometric condition that makes them easy to fold?

I think that's a generic principle not a base. If you have a crease that stays valley or mountain then when other creases hit they have to reverse as they cross it for the model to lie flat, except I guess in the case that they're being used together in a collapse (like the preliminary base).

I'll call them easy CPs.

Seriously, I'm not quite sure the bases can be classified like this. My gut feeling is that the the collapse mechanism of the CP depends on where you start - if you pick the "right" point to start, it becomes relatively easy to collapse.

Can you give a counterexample CP? It'll be easier to compare the differences between the two.

A counter example for an easy CP, one that you find the axle of it and just go mountain, valley, mountain, valley, ...?? I think I am not following you here dear wolf, for such a counter example (not counting box pleating models which are a different kind) would be almost any CP

My counter example would be Komatsu's lion [img]http://www.origami.gr.jp/~komatsu/gallery/lion-cp.gif[/img] but I think any of his models would do.

To explain better my point about the easy ones, two of them would be Hojyo's human bases at http://www.geocities.co.jp/HeartLand-Oa ... nbutu4.gif and http://www.geocities.co.jp/HeartLand-Oa ... iroku4.gif

I think this paper talks about what you're describing:
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When can you fold a map?
Arkin et al, Computational Geometry - Theory and Applications 29, 23-46 (2004)

We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by +/-180degrees. We first consider the analogous questions in one dimension lower-bending a segment into a flat object-which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds.

Next we explore simple foldability in two dimensions, and find a surprising contrast: "map" folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.
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So perhaps the CPs that are easy to fold, are those which can be flatfolded by the "simple foldings" described above. The ones that don't are flat-foldable but cannot be done with "simple foldings", or else they are not flat-foldable at all.