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: colin.mcallister@ymail.com