Investigations of Paper Folding and Regular Polygons

Presented by:Ed Knote & Bhesh Mainali

University of Central Florida,                                                

                                      Phd. in Education, Mathematics Education 

Graduate Students

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• Mathematical Reflections In a Room with Many

Mirrors by Peter Hilton, Derek Holton, Jean Pedersen

• Chapter 4: Paper-Folding, Polyhedra-Building, and Number Theory

Introduction

Greeks were fascinated with the challenge of constructing regular convex polygons.

They wanted to construct them with Euclidean tools:• unmarked straightedge• compass

ObjectivesWe will perform, understand, &

explain:– Paper-folding procedure – Paper-folding construction of regular

convex octagons – Optimistically use Paper-folding to

construct regular convex heptagons

Key Terms• Folding and twisting (FAT-algorithm)• Optimistic strategy• Primary crease line• Secondary crease line• Dn Um-folding procedure

Prerequisite Skills• Angle relationships, Parallel Lines, and

Transversals

• Polygon Interior and Exterior Angle Sums

• Degree and Radian conversions

Parallel Lines & Transversals

Polygon Interior Angle Sums

Polygon Exterior Angle Sums

• Quadrilateral

• Pentagon

• Hexagon

Radian: A unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius.

Radian Degree

Degree Radian

Radian Degree

Regular Polygons & Radians

• How did we find the degree measure of each exterior angle of a regular polygon?

• What would that formula look like in radians?

FAT-Algorithm

FAT-Algorithm• Fold And Twist– Assume we have a nice strip of paper

with straight parallel edges–Mark your first vertex (near the left side)– Construct your angle (where b is the number of sides for your polygon)– Fold this angle in half and mark it– Then repeat process at equally spaced

vertices

FAT-Algorithm–What is the significance of the

angle ? –What are some angles in this form we

can easily construct?• What polygons do they relate to?

–What are some angles that we can not?

General Paper Folding• Each new crease line goes in the

forward (left to right) direction along the strip of paper

• Each new crease line always bisects the angle between the last crease line and the edge of the tape from which it originates.

Optimistic Strategy–What is a good estimate of on a

protractor?

– Lets take a look at our optimistic strategy.• Time to fold.

General Paper Folding• Each new crease line goes in the

forward (left to right) direction along the strip of paper

• Each new crease line always bisects the angle between the last crease line and the edge of the tape from which it originates.

General Paper Folding•  

Optimistic Strategy– Did your angle get closer to ?

–Why do you think this happens?

– Can we prove this mathematically?

– How can we show this in Excel?

Optimistic Strategy– Is this perfect or just a close estimate?

– Is this folding procedure the same for all polygons?

–What would it be for a pentagon.

General Paper Folding•  

Optimistic Strategy– Now you come up with the folding

procedure for a 13-gon.

Webpage

– Knote.pbworks.com• NCTM Paper Folding 2013