Folding
The points accessible from 
by a single fold which leaves 
, ..., 
fixed are exactly those pointsinterior 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 Trianglewith 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 Squarewith 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 theVertices of a suitable Square under at most two folds. Every five (six) points arethe images of the Vertices of suitable regular Pentagon (Hexagon) under atmost five (six) folds. The least number of folds required for 
is not known, but some bounds are. Inparticular, every set of points is the image of a suitable Regular -gon under atmost 
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).See also Flexagon, Map Folding, Origami
References
Sabinin, P. and Stone, M. G. ``Transforming Sloane, N. J. A. Sequence A007494in ``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.
- Phase Transition
- Phasor
- Phi Curve
- Phi Number System
- Phragmén-Lindêlöf Theorem
- Phyllotaxis
- Pi
- Piano Mover's Problem
- Picard's Existence Theorem
- Picard's Little Theorem
- Picard's Theorem
- Picard Variety
- Pick's Formula
- Pick's Theorem
- Picone's Theorem
- Piecewise Circular Curve
- Pie Cutting
- Pigeonhole Principle
- Pi Heptomino
- Pillai's Conjecture
- Pilot Vector
- Pinching Theorem
- Pinch Point
- Pine Cone Number
- Piriform