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By Pogorelov, A.V.

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Example: Examine the crease pattern qd ) w(, $b1D, q1 ) w$b1, $D, qn ) w$, $b1D, q; ) wx$b;, b$D, qx ) w$b;, $D. 1 is the previously known analogous result for single-vertex flat folds. Let 7 ) 7Ld b 7 denote the angle between crease lines q7Ld and q7 . r-plane. 1 but also self-intersects. 1 becomes d L n L c c c L 1>d ) 1 L ; L c c c L 1> ) $ where the crease pattern has 1> ) z edges. ) This is sometimes stated as d b 1 L n bc c cb 1z ) (, and it is a necessary and sufficient condition for a single-vertex, fold to fold-flat.

LJXUH  A rabbit-ear molecule folds into a three-armed “starfish”.  0 %HUQ HW DO )LJXUH  We fold a quadrilateral into a four-armed starfish with a central valley. At this point, we regard the orientations of the valley folds as changeable: In the larger origami, some of them may be reversed from their initial assignment. 1 The arms of the starfish all point the same direction away from the spine in the flat origami, and the boundary of the original triangle is collinear. We shall fold the quadrilaterals as shown in Figure 3.

Akiyama, M. Kano, and M. , Tokyo, 1998. Lecture Notes in Computer Science, Vol. 1763, Springer, 104–117. [7] E. Demaine, M. Demaine, and A. Lubiw. Folding and one straight cut suffice. Proc. 10th ACM-SIAM Symp. Disc. Algorithms, 1999, 891–892.  0 %HUQ HW DO [8] H. -S. Tan. An upper bound for conforming Delaunay triangulations. Disc. Comput. Geom. 10 (1993) 197–213. [9] D. Eppstein. Faster circle packing with application to nonobtuse triangulation. Int. J. Comp. Geom. & Applications 7 (1997) 485–491.

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