MIT折纸算法

vedio 1

Geomefric objects

1 linkage: rigid bars

2 paper: no stretch, no tears, no crossing

3 polyhedra: cut surface, one piece, no overlap

Question

Foldability: What structures fold at all/in particular way?

Design: What shapes (or particular property) can be folded & how?

Result

Universality: everything can be folded + algorithm

Decision: efficient algorithm to decide foldability

Hardness: computationally in fractable to decide foldability

1 linkages

Early motivation: convert
linear motion$\leftrightarrow$circular motion

Universality: linkage to sign your name (trace piecewise polynomial curve)

Rigidity: does a linkage fold at all?

linkages forbidding intersection

Reconfiguration: fold from config.A$\rightarrow$config.B

2 Paper

Foldability: which crease patterns fold flat?
-NP-hard
-easy for single vertex

Design: What shapes can be folded?

-any 2D polygon 3D polyhedra

3 Polyhedra

Unfolding:
-open: edge-unfolding convex polyhedra

Hinged dissections

any finite set of polygons of same area, they can be folded from one chain of polygons without collision.

vedio 2

piece of paper = 2D polygon with distinguished top/bottom
crease = line segment or curve drawn on paper
crease pattern = bunch of creases = planar graph drawn on paper
Folded state = finished origami
Flat folding = folded state lying in the plane
- call crease pattern flat foldable
Mountain crease = bottom sides touch
Valley crease = top sides touch
Mountain-valley assignment = which creases are mountains/valleys
Mountain-valley pattern = crease pattern +M/V assignment
Simple fold = fold along single line by $\pm 180^o$

Folding any shape:

every connected union of polygons in 3D, each with specified visible color (on each side), can be folded from sufficiently large piece of bicolor paper(双色纸)
Proof:
1 fold paper down to a long narrow strio(rectangle)
2 triangulate polygons (多边形划分为三角形)
3 cover each triangle at least one by using a zig-zag paralled to next edge starting opposite corner

Pseudo-efficiency: if you start with long strip, then can achieve

area (paper) = surface area + $\varepsilon$, for any $\varepsilon \gt 0$
Proof: Hamiltonian refinement

1D flat folding:
Piece of paper = (1D) line segment
Crease = point
Flat folding = lies on line

two operations:
1 end fold(末端折)
2 crimp（平折）

Characterization:
mountain-valley pattern is flat foldable
$\iff$ there is a sequence of crimps & end folds
$\iff$ mingling: for any maximal sequence of all V’s or M’s adjacent M or V or end on at least one side is nearer than adjacent V or M

Proof:
1 flat foldable = mingling
2 mingling $\implies$ end fold or crimp is possible
for each maximal sea of M’s or V’s, write
( if “left mingling”
[ if not
) if “right mingling”
] if not
3 crimp/end fold preserves flat foldability