Paper Folding in Geometry
• Paper folding in Geometry was introduce 1983 by T. Sundar Row and R.C Yates who listed the axioms which plane Eclidean constructions were based on
Origin
• Started as historical backgrounds and fashionable wrappings
• As an aspect of Japanese history during the Heian period - Samurai warriors would exchange gifts, noshi, as a token of good luck which were paper folded with strips of dried fish or meat. -Origami butterflies were used to celebrate Shinto weddings by wrapping glasses of sake or rice wine folded like a butterfly to represent the bride an groom. origami-wikipedia, histry of origami)
• In spite what was said, one can fold paper along an arbitrary smooth curve. 1. portions of the paper on two sides of the curve meet at a zero angle 2. flattening out period in order to apply a crease in paper folding to geometry
Paper Folding
By: Eric Gold and Becky Mohl
Definition
• Origami (1603-1867) is the Japanese art of paper folding, and is the most commonly known (origami-wikipedia)
• “Kami is the Japanese word for paper (history of origami)
Definition
• The goal of origami is to create a representation of an object using geometric folds and patterns without cutting or gluing additional pieces of paper.
Axiom O3: Huzita
• Given two creases L1 and L2, there exists a unique fold that maps L1 onto L2.
• In 1191, The Italian-Japanese mathematician Humaiki Huzita added 6 more axioms of paper folding in terms of straight edges and compasses
Axiom O1: Yates
• Origami is a form of visual / sculptural representation that is defined primarily by the folding of the medium (usually paper). (origami history)
• “Ori” is the Japanese word for folding
• There exists a single fold connecting two distinct points.
Axiom O2: Yates
• Given two points P1 and P2, there exists a unique fold that maps P1 onto P2.
• Other countries such as China, Korea, Germany, and Spain have developed similar arts. (origami-wikipedia • Not certain whether started in Japan or China. • China 1st Century AD • Japan 6th Century AD
Paper Folding in Geometry
• A straight line becomes a crease or a fold
• A point is defined as the intersection between tow folds
• Folding paper is analogous to mirroring one half plane in a crease
• Folding means both drawing a crease and mapping one half of a plane onto another
Paper Folding In ry
• Folding is an isometry of the part of the plane on one side of the fold to another, the fold bearing the curve of fixed points of the isometry. The curve is straight because it has zero curvature.