Teaching mathematics through paper folding Dissertation Thesis (author: Sándor Em˝oke) Dali Erzsébet, Orbán Lidia, Sándor Em˝oke, Tóth Tekla

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Dali Erzsébet, Orbán Lidia, Sándor Em˝oke, Tóth Tekla

Babe¸s-Bolyai University

Faculty of Mathematics and Computer Science

2020. july 16

BBU (CLuj Napoca) Teaching mathematics through paper folding 2020. july 16

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1 Introduction

History of paper folding Huzita–Hatori axioms

2 How to divide the paper inton equal parts?

How to divide a square inton equal parts?

Fujimoto’s approximation

3 Regular polygons in origami with knots 4 Exercises in paper folding

5 Bibliography

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use for padding and wrapping bronze mirror (Bronze mirrors preceded the glass mirrors of today.)

use for writing by 400 A.D.

use for toiletpaper by 700 A.D use for money by 960 A.D

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Origami butterflies appear in Japanese Shinto weddings around 1600s First origami book

”Sembazuru Orikata” (Secret Techniques of Thousand Crane Folding) published in 1797

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Tractatus de Sphera Mundi published in 1490

an Italian book by Matthias Geiger, published in 1629, for folding table napkins.

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kindergarten

Three basic types of folds are associated with him: theFolds of Life (basic folds that introduced kids to paper folding), theFolds of Truth (teaching basic principles of geometry), and theFolds of Beauty (more-advanced folds based on squares, hexagons, and octagons)

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Exercises in Paper Folding which used paper folding to demonstrate proofs of geometrical constructions.

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50,000 models, of which only a few hundred designs were presented as diagrams in his 18 books.

His most complex model is the cicada, it took him over 30 years to complete.

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shape

nowadays mathematicians are also interested in the relation of paper folding and mathematics

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theHuzita–Hatori axioms are a set of rules related to the mathematical principles of paper folding, describing the operations that can be made when folding a piece of paper.

the axioms were first discovered by Jacques Justinin 1986.

axioms 1 through 6 were rediscovered by Japanese-Italian mathematicianHumiaki Huzita

axiom 7 was rediscovered by Koshiro Hatoriin 2001; Robert J.

Lang also found axiom 7.

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You can make a fold through two points.

so we can find a line which overpass the points A and B

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You can fold one point onto another.

this is equivalent to finding the perpendicular bisector of the line segment [AB].

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You can fold one line (that is, the crease made by a previous fold, or the edge of the paper) onto another line.

this is equivalent to finding a bisector of the angle betweend₁ andd₂.

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You could make a fold through a point, so that your fold is perpendicular to a given line.

this is equivalent to finding a perpendicular todthat passes through A.

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You can make a fold through a point, so that it places another point onto a line.

we can construct this line with compass and straightedge.

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If you have two points and two lines, you can make a fold so that it places each point onto a line.

- this fold is called the Beloch fold after Margharita P. Beloch, who in 1936 showed using it that origami can be used to solve general cubic equations (we can’t construct this line with compass and straightedge).

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If you have a point and two lines, you can make a fold that is

perpendicular to one of the lines and places the point onto the other line.

we can construct this line with compass and straightedge.

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1 Introduction

5 Bibliography

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1 given an ABCD square paper.

2 make a fold that places AB ontoCD (to crease M N).

3 N is the midpoint of[BC].

4 make a fold along the crease passing through A andN.

5 fold along the diagonal BD.

6 {P}=BD∩AN

7 construct (with axiom 4) a line that perpendicular to BC and that passes through point P (to crease [RS]).

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3 N is the midpoint of[BC].

6 {P}=BD∩AN

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3 N is the midpoint of[BC]

6 {P}=BD∩AN

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6 {P}=BD∩AN

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6 {P}=BD∩AN

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6 {P}=BD∩AN

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6 {P}=BD∩AN

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6 {P}=BD∩AN

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6 {P}=BD∩AN

⇒ SB = BC 3

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ABCD square; (BC = 1) N ∈[BC],[CN]≡[N B]

{P}=BD∩AN P S ⊥BC,S∈[BC],

⇒ SB = BC 3 {P₁}=BD∩AS P1S1 ⊥BC,S1 ∈[BC],

⇒ S₁B = BC 4

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Let us take a square piece of paper: ABCD,BC= 1.

N ∈[BC],N B = 1 n {P}=BD∩AN

P S ⊥BC, andS ∈[BC], then SB = 1

n+ 1

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Step 1:

Let us take a rectangle piece of paper. The rectangle’s length is 1 unit.

Make a guess where 1 mark is. 5

The right side is roughly 4

5 of the paper.

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Step 2:

Fold and unfold the right side in half.

Then the position of the pinch 2 is near 3

5 and the right side is roughly 2

5 of the paper.

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Step 3:

Pinch the right side in half, where the lenght is near 2

5

The the position of the pinch 3 is near 4

5, so the left side of is roughly 4 of the paper and the 5 right side is roughly 1

5 of the paper.

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Step 4:

Fold and unfold the left side in half.

The pinch 4 is near 2 mark and the right side5 is roughly 3

5 of the paper.

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Step 5:

Fold and unfold the left side in half.

The position of the last pinch is very close to the 1

5 mark.

The last pinch is closer to 1

5, than the guess pinch.

We can repeat steps to get better

approximations.

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5

Position of pinch 2 is

1

5 +ε

+1 2

4

5 −ε

= 3 5+ ε

2 .

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5 2 2 5 2 5 4 - with each fold, the errorεis halved.

Position of pinch 4 is 2 5 +ε

8.

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The last pinch is at the position 1 5 + ε

16. Doing one round of Fujimoto approximation reduces the error by a factor of 16.

With each fold, the errorε is halved, so afternsteps the errorεwill be ε

2ⁿ.

Repeat steps, with last pinch to get better approximations

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1 Introduction

5 Bibliography

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1 Introduction

5 Bibliography

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Exercise 1.

Construct a square with exactly 1

4 the area of the original square.

Convince yourself that it is a square and has 1

4 of the area.

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Convince yourself that it is a square and has 1

4 of the area.

Solution:

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Exercise 2.

Construct a triangle with exactly 1

4 the area of the original square.

Convince yourself that it has 1

4 of the area.

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Convince yourself that it has 1

4 of the area.

Solution:

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Exercise 3.

Construct another triangle, also with 1

4 the area, that is not congruent to the first one you constructed. Convince yourself that it has 1

4 of the area.

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the first one you constructed. Convince yourself that it has 1

4 of the area.

Solution:

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Exercise 4.

Take a perfectly square piece of paper, and so fold it as to form an equilateral triangle in which the sides are the same length as those of the square.

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equilateral triangle in which the sides are the same length as those of the square.

Solution:

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1 Introduction

5 Bibliography

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Kazou Haga, ORIGAMICS Mathematical Explorations though Paper Folding Robert J. Lang, Origami and Geometric Constructions, 2010

Robert J. Lang, Origami (website)

Thomas Hull, Project Origami: Activities for Exploring Mathematics, Second Edition, Kindle Edition, 2006 Row, T. Sundara, Geometric Exercises in Paper Folding, 1958

Robert J. Lang, Origami4, Chapter 34, Tamara B. Veenstra, Fujimoto, number theory and a new folding technique?

Donovan A. Jonson, Paperfolding for the Mathematics Class, National Education Association, 1201 Sixteenth Street, N.W., Washington, D.C. 20036, 1957

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