Folding

The points accessible from $c$ by a single fold which leaves $a_1$, ..., $a_n$ fixed are exactly those points interior to or on the boundary of the intersection of the Circles through $c$ with centers at $a_i$, for $i=1$, ..., $n$. Given any three points in the plane $a$, $b$, and $c$, there is an Equilateral Triangle with Vertices $x$, $y$, and $z$ for which $a$, $b$, and $c$ are the images of $x$, $y$, and $z$ under a single fold. Given any four points in the plane $a$, $b$, $c$, and $d$, there is some Square with Vertices $x$, $y$, $z$, and $w$ for which $a$, $b$, $c$, and $d$ are the images of $x$, $y$, $z$, and $w$ under a sequence of at most three folds. Also, any four collinear points are the images of the Vertices of a suitable Square under at most two folds. Every five (six) points are the images of the Vertices of suitable regular Pentagon (Hexagon) under at most five (six) folds. The least number of folds required for $n\geq 4$ is not known, but some bounds are. In particular, every set of $n$ points is the image of a suitable Regular $n$-gon under at most $F(n)$ folds, where

$$F(n)\leq\cases{ {\textstyle{1\over 2}}(3n-2) & for $n$\ even\cr {\textstyle{1\over 2}}(3n-3) & for $n$\ odd.\cr}$$

The first few values are 0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, ... (Sloane's A007494).

See also Flexagon, Map Folding, Origami

References

Sabinin, P. and Stone, M. G. ``Transforming $n$-gons by Folding the Plane.'' Amer. Math. Monthly 102, 620-627, 1995.

Sloane, N. J. A. Sequence A007494 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.