Paper-Folding with Interactive Geometry Software
Exploring the folding of paper circles and triangles
using interactive geometry software
By Colin McAllister
August 2010
Abstract
The Huzita-Justin paper-folding Axiom 5, adapted for a circle, is explored using interactive geometry software. The axiom is generalised to other shapes, and applied to a triangle with rounded corners. An interesting configuration of folds is discovered when the triangle is equilateral. The properties of this configuration are explained by drawing a circle, of which the folds are diameters. An adjustable simulation of a folded paper triangle is used to demonstrate this explanation. A folding hypothesis is postulated for arbitrary shapes.
Huzita-Hatori Axiom 5
“Given two points p1 and p2 and a line l1we can make a fold that places p1 onto l1
and passes through the point p2.”
Axiom 5-C for a Circle
For two points p1 and p2 in a circle,Folds through p2 that place p1onto the boundary of the circle:
p1-p2>p2-circle: There are two such foldsp1-p2=p2-circle: One such fold.
p1-p2<p2-circle: The fold is impossible.
From: "Circle Origami Axioms", on MariaDroujkova's Math 2.0 Interest Group.
Geometric Model of Origami Circle Axiom 5-C
Circle Axiom on a Triangle with Rounded Corners
Folds of a Triangle with Rounded Corners
Folds of an Equilateral Triangle with Rounded Corners
Circle Intersecting an Equilateral Triangle
Simulated Fold of an Equilateral Triangle (1 of 2)
Simulated Fold of an Equilateral Triangle (2 of 2)
Summary
Circle origami Axiom 5-C was the trigger for this investigation.
Is the axiom valid for shapes other than circles?
Yes; Shapes that have a minimum radius of curvature of the boundary.
We choose triangles with rounded corners as an example of such shapes.
We can experiment with them using interactive geometry software.
We discover symmetric folds when the triangle is equilateral.
We can simulate folding of paper triangles, using interaction to control the folding, and hidden-line removal to represent two layers of paper.
The simulation helps us understand the geometry of the symmetric folds.
Reference
This slideshow is based on my geometry research paper: Paperfoldinggeometry.pdf by colinmca
on the geometry website: http://i2geo.netFurther references are given in that paper.
Circle Folding Axioms by Maria Droujkova et al, athttp://mathfuture.wikispaces.com/Circle+origami+axioms
Acknowledgement
I wish to thank Maria Droujkova, Linda Fahlberg-Stojanovska and my former school teacher Kenneth Blair for sharing their ideas and for their
enthusiasm in exploring and teaching mathematics.
Licence
This work is licenced under the Creative Commons Attribution 2.0 UK: England & Wales License. To view a copy of this licence, visit
http://creativecommons.org/licenses/by/2.0/uk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California
94105, USA.
To contact the author of this slideshow,email: [email protected]