The points accessible from by a single fold which leaves , ..., fixed are exactly those points
interior to or on the boundary of the intersection of the Circles through with centers at ,
for , ..., . Given any three points in the plane , , and , there is an Equilateral Triangle
with Vertices , , and for which , , and are the images of , ,
and under a single fold. Given any four points in the plane , , , and , there is some Square
with Vertices , , , and for which , , , and are the images of ,
, , and 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 is not known, but some bounds are. In
particular, every set of points is the image of a suitable Regular -gon under at
most folds, where

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

**References**

Sabinin, P. and Stone, M. G. ``Transforming -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.

© 1996-9

1999-05-26