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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
See also Flexagon, Map Folding, Origami
References
Sabinin, P. and Stone, M. G. ``Transforming
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.
-gons by Folding the Plane.'' Amer. Math. Monthly 102, 620-627, 1995.
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© 1996-9 Eric W. Weisstein