origami and mathematics - heriot-watt universitysl37/slides/18_phdseminar.pdforigami comes from the...

Origami and mathematicswhy you are not just folding paperStefania Lisai

MACS PhD Seminar5th October 2018

Pretty pictures to get your attention

httpswwwquoracom

Why-dont-more-origami-inventors-follow-Yoshizawas-landmarkless-approach-to-design



httpsorigamiplusorigami-master-yoda


httpswwwquoracomWhy-is-origami-considered-art





httpviralpienet

the-art-of-paper-folding-just-got-taken-to-a-whole-new-level-with-3d-origami



httpwwwartfulmathscomblogcategoryorigami


httpwwwradionzconznationalprogrammesafternoonsaudio201835180

maths-and-crafts-using-crochet-and-origami-to-teach-mathematics

Meaning and historyOrigami comes from the Japanese words ori meaning folding and kamimeaning paper

g Paper folding was known in Europe Japan and China for long time In20th century different traditions mixed upg In 1986 Jacques Justin discovered axioms 1-6 but ignoredg In 1989 the first International Meeting of Origami Science and

Technology was held in Ferrara Italyg In 1991 Humiaki Huzita rediscovered axioms 1-6 and got all the gloryg In 2001 Koshiro Hatori discovered axiom 7

Meaning and historyOrigami comes from the Japanese words ori meaning folding and kamimeaning paperg Paper folding was known in Europe Japan and China for long time In20th century different traditions mixed up

g In 1986 Jacques Justin discovered axioms 1-6 but ignoredg In 1989 the first International Meeting of Origami Science and


Meaning and historyOrigami comes from the Japanese words ori meaning folding and kamimeaning paperg Paper folding was known in Europe Japan and China for long time In20th century different traditions mixed upg In 1986 Jacques Justin discovered axioms 1-6 but ignored

g In 1989 the first International Meeting of Origami Science andTechnology was held in Ferrara Italy

g In 1991 Humiaki Huzita rediscovered axioms 1-6 and got all the gloryg In 2001 Koshiro Hatori discovered axiom 7

Meaning and historyOrigami comes from the Japanese words ori meaning folding and kamimeaning paperg Paper folding was known in Europe Japan and China for long time In20th century different traditions mixed upg In 1986 Jacques Justin discovered axioms 1-6 but ignoredg In 1989 the first International Meeting of Origami Science and

Technology was held in Ferrara Italy



Technology was held in Ferrara Italyg In 1991 Humiaki Huzita rediscovered axioms 1-6 and got all the glory

g In 2001 Koshiro Hatori discovered axiom 7



Compass and Straightedge constructionBasic constructions with compass and straightedge

1 We can draw a line passing through 2 given points2 We can draw a circle passing through one point and centred in another3 We can find a point in the intersection of 2 non-parallel lines4 We can find one point in the intersection of a line and a circle (if 6= empty)5 We can find one point in the intersection of 2 given circle (if 6= empty)Using these constructions we can do other super cool things bisect anglesreflect points draw perpendicular lines find midpoint to segments draw theline tangent to a circle in a certain point etcWe cannot solve the three classical problems of ancient Greekgeometry using compass and straightedge

Compass and Straightedge constructionBasic constructions with compass and straightedge1 We can draw a line passing through 2 given points

2 We can draw a circle passing through one point and centred in another3 We can find a point in the intersection of 2 non-parallel lines4 We can find one point in the intersection of a line and a circle (if 6= empty)5 We can find one point in the intersection of 2 given circle (if 6= empty)Using these constructions we can do other super cool things bisect anglesreflect points draw perpendicular lines find midpoint to segments draw theline tangent to a circle in a certain point etcWe cannot solve the three classical problems of ancient Greekgeometry using compass and straightedge

Compass and Straightedge constructionBasic constructions with compass and straightedge1 We can draw a line passing through 2 given points2 We can draw a circle passing through one point and centred in another

3 We can find a point in the intersection of 2 non-parallel lines4 We can find one point in the intersection of a line and a circle (if 6= empty)5 We can find one point in the intersection of 2 given circle (if 6= empty)Using these constructions we can do other super cool things bisect anglesreflect points draw perpendicular lines find midpoint to segments draw theline tangent to a circle in a certain point etcWe cannot solve the three classical problems of ancient Greekgeometry using compass and straightedge

Compass and Straightedge constructionBasic constructions with compass and straightedge1 We can draw a line passing through 2 given points2 We can draw a circle passing through one point and centred in another3 We can find a point in the intersection of 2 non-parallel lines

4 We can find one point in the intersection of a line and a circle (if 6= empty)5 We can find one point in the intersection of 2 given circle (if 6= empty)Using these constructions we can do other super cool things bisect anglesreflect points draw perpendicular lines find midpoint to segments draw theline tangent to a circle in a certain point etcWe cannot solve the three classical problems of ancient Greekgeometry using compass and straightedge

Compass and Straightedge constructionBasic constructions with compass and straightedge1 We can draw a line passing through 2 given points2 We can draw a circle passing through one point and centred in another3 We can find a point in the intersection of 2 non-parallel lines4 We can find one point in the intersection of a line and a circle (if 6= empty)

5 We can find one point in the intersection of 2 given circle (if 6= empty)Using these constructions we can do other super cool things bisect anglesreflect points draw perpendicular lines find midpoint to segments draw theline tangent to a circle in a certain point etcWe cannot solve the three classical problems of ancient Greekgeometry using compass and straightedge

Compass and Straightedge constructionBasic constructions with compass and straightedge1 We can draw a line passing through 2 given points2 We can draw a circle passing through one point and centred in another3 We can find a point in the intersection of 2 non-parallel lines4 We can find one point in the intersection of a line and a circle (if 6= empty)5 We can find one point in the intersection of 2 given circle (if 6= empty)

Using these constructions we can do other super cool things bisect anglesreflect points draw perpendicular lines find midpoint to segments draw theline tangent to a circle in a certain point etcWe cannot solve the three classical problems of ancient Greekgeometry using compass and straightedge

Compass and Straightedge constructionBasic constructions with compass and straightedge1 We can draw a line passing through 2 given points2 We can draw a circle passing through one point and centred in another3 We can find a point in the intersection of 2 non-parallel lines4 We can find one point in the intersection of a line and a circle (if 6= empty)5 We can find one point in the intersection of 2 given circle (if 6= empty)Using these constructions we can do other super cool things bisect anglesreflect points draw perpendicular lines find midpoint to segments draw theline tangent to a circle in a certain point etc

We cannot solve the three classical problems of ancient Greekgeometry using compass and straightedge

Compass and Straightedge constructionBasic constructions with compass and straightedge1 We can draw a line passing through 2 given points2 We can draw a circle passing through one point and centred in another3 We can find a point in the intersection of 2 non-parallel lines4 We can find one point in the intersection of a line and a circle (if 6= empty)5 We can find one point in the intersection of 2 given circle (if 6= empty)Using these constructions we can do other super cool things bisect anglesreflect points draw perpendicular lines find midpoint to segments draw theline tangent to a circle in a certain point etcWe cannot solve the three classical problems of ancient Greekgeometry using compass and straightedge

Three geometric problems of antiquityDoubling the cube given a cube find the edge of another cube which hasdouble its volume ie solve

t3 = 2

Squaring the circle given a circle find the edge of a square which has thesame area as the circle ie solvet2 = π

Trisect the angles given an angle find another which is a third of it iesolvingt3 + 3at2 minus 3t minus a = 0with a = 1tan θ and t = tan ( θ3 minus π

2)

You can do 2 of these 3 things with origami guess which one isimpossible


t3 = 2Squaring the circle given a circle find the edge of a square which has thesame area as the circle ie solvet2 = π


2)





2)





2)


Justin-Huzita-Hatori Axioms

1 There is a fold passing through 2 given points2 There is a fold that places one point onto another3 There is a fold that places one line onto another4 There is a fold perpendicular to a given line and passing through agiven point5 There is a fold through a given point that places another point onto agiven line6 There is a fold that places a given point onto a given line and anotherpoint onto another line7 There is a fold perpendicular to a given line that places a given pointonto another line

Justin-Huzita-Hatori Axioms1 There is a fold passing through 2 given points

2 There is a fold that places one point onto another3 There is a fold that places one line onto another4 There is a fold perpendicular to a given line and passing through agiven point5 There is a fold through a given point that places another point onto agiven line6 There is a fold that places a given point onto a given line and anotherpoint onto another line7 There is a fold perpendicular to a given line that places a given pointonto another line

Justin-Huzita-Hatori Axioms1 There is a fold passing through 2 given points2 There is a fold that places one point onto another

3 There is a fold that places one line onto another4 There is a fold perpendicular to a given line and passing through agiven point5 There is a fold through a given point that places another point onto agiven line6 There is a fold that places a given point onto a given line and anotherpoint onto another line7 There is a fold perpendicular to a given line that places a given pointonto another line

Justin-Huzita-Hatori Axioms1 There is a fold passing through 2 given points2 There is a fold that places one point onto another3 There is a fold that places one line onto another

4 There is a fold perpendicular to a given line and passing through agiven point5 There is a fold through a given point that places another point onto agiven line6 There is a fold that places a given point onto a given line and anotherpoint onto another line7 There is a fold perpendicular to a given line that places a given pointonto another line

Justin-Huzita-Hatori Axioms1 There is a fold passing through 2 given points2 There is a fold that places one point onto another3 There is a fold that places one line onto another4 There is a fold perpendicular to a given line and passing through agiven point

5 There is a fold through a given point that places another point onto agiven line6 There is a fold that places a given point onto a given line and anotherpoint onto another line7 There is a fold perpendicular to a given line that places a given pointonto another line

Justin-Huzita-Hatori Axioms1 There is a fold passing through 2 given points2 There is a fold that places one point onto another3 There is a fold that places one line onto another4 There is a fold perpendicular to a given line and passing through agiven point5 There is a fold through a given point that places another point onto agiven line

6 There is a fold that places a given point onto a given line and anotherpoint onto another line7 There is a fold perpendicular to a given line that places a given pointonto another line

Justin-Huzita-Hatori Axioms1 There is a fold passing through 2 given points2 There is a fold that places one point onto another3 There is a fold that places one line onto another4 There is a fold perpendicular to a given line and passing through agiven point5 There is a fold through a given point that places another point onto agiven line6 There is a fold that places a given point onto a given line and anotherpoint onto another line

7 There is a fold perpendicular to a given line that places a given pointonto another line

Justin-Huzita-Hatori Axioms1 There is a fold passing through 2 given points2 There is a fold that places one point onto another3 There is a fold that places one line onto another4 There is a fold perpendicular to a given line and passing through agiven point5 There is a fold through a given point that places another point onto agiven line6 There is a fold that places a given point onto a given line and anotherpoint onto another line7 There is a fold perpendicular to a given line that places a given pointonto another line

Folding in thirds1minus x

x

y1minus x

x y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12

x2 + 14 = (x minus 1)2 Ntilde x = 3

8 Ntilde y2 = 1

214x = 1

3

Hagarsquos theorem

1minus xx

y1minus xx y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12

x2 + k2 = (1minus x )2 Ntilde x = 1minus k2

2 Ntilde y2 = k(1minus k)

1minus k2 = k1 + k

Hagarsquos theorem

If we choose k = 1N for some N isin N then

y2 = 1N

1 + 1N = 1N + 1

therefore starting from N = 2 we can obtain 1n for any n gt 2 hence anyrational m

n for 0 lt m lt n isin N

Geometric problems of antiquity double the cube

P Q

Geometric problems of antiquity double the cubeP

Q

x y

The wanted value is given by the ratio yx = 3radic2Doubling the cube is equivalent to solving the equation t3 minus 2 = 0

Geometric problems of antiquity trisect the angle

Q

θ

P


θ

P

A B

Q

The angle PAB is θ3 Trisecting the angle is equivalent to solving the equation

t3 + 3at2 minus 3t minus a = 0with a = 1tan θ and t = tan ( θ3 minus π2)

Geometric problems of antiquity square the circle

Unfortunately π is still transcendental even in the origami world Thisproblem is proved to be impossible in the folding paper theory

Solving the cubic equation t3 + at2 + bt + c = 0

K

L

M

Q

Qprime

P

P prime

slope(M)

P = (a 1) Q = (cb) L = x = minusc K = y = minus1The slope of M satisfies the equation


ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)

P = (a 1) Q = (cb) L = x = minusc K = y = minus1If a = 15 b = 15 c = 05 then t = slope(M) = minus15 satisfiest3 + at2 + bt + c = 0

Solving cubic equationsWant to solve t3 + at2 + bt + c = 0

φ = 4y = (x minus a)2ψ = 4cx = (y minus b)2

M = y = tx + uM is tangent to φ at R then u = minust2 minus at

M is tangent to ψ at S then u = b + ct

M is the crease that folds P onto K and Q onto L

Applications in real worldg Solar panels and mirrors for space

Applications in real world

g Solar panels and mirrors for spaceg Air bags

Applications in real worldg Solar panels and mirrors for spaceg Air bagsg Heart stents (Oxford)

Applications in real worldg Solar panels and mirrors for spaceg Air bagsg Heart stents (Oxford)g Self-folding robots (Harvard MIT)

Applications to real worldg Solar panels and mirrors for spaceg Air bagsg Heart stents (Oxford)g Self-folding robots (Harvard MIT)g Decorations for your flat

References[Lan08] Robert LangThe math and magic of origami

httpswwwyoutubecomwatchv=NYKcOFQCeno 2008[Lan10] Robert LangOrigami and geometric constructions

httpwhitemythcomsitesdefaultfilesdownloadsOrigamiOrigami

20TheoryRobert20J20Lang20-20Origami20Constructionspdf 2010[Mai14] Douglas MainFrom robots to retinas 9 amazing origami applications

httpswwwpopscicomarticlescience

robots-retinas-9-amazing-origami-applicationspage-4 2014[Tho15] Rachel ThomasFolding fractions

httpsplusmathsorgcontentfolding-numbers 2015[Wik17] WikipediaMathematics of paper folding mdash wikipedia the free encyclopedia

httpsenwikipediaorgwindexphptitle=Mathematics_of_paper_foldingamp

oldid=807827828 2017

Pictures

Introduction

Axioms of origami

Three classical problem in ancient geometry

Solving cubic equations

Applications


httpswwwquoracom

Why-dont-more-origami-inventors-follow-Yoshizawas-landmarkless-approach-to-design










httpviralpienet









































t3 = 2



2)





2)





2)





2)


















x

y1minus x

x y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12


8 Ntilde y2 = 1

214x = 1

3

Hagarsquos theorem

1minus xx

y1minus xx y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12



1minus k2 = k1 + k

Hagarsquos theorem


y2 = 1N

1 + 1N = 1N + 1




P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications










httpviralpienet









































t3 = 2



2)





2)





2)





2)


















x

y1minus x

x y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12


8 Ntilde y2 = 1

214x = 1

3

Hagarsquos theorem

1minus xx

y1minus xx y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12



1minus k2 = k1 + k

Hagarsquos theorem


y2 = 1N

1 + 1N = 1N + 1




P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications








































t3 = 2



2)





2)





2)





2)


















x

y1minus x

x y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12


8 Ntilde y2 = 1

214x = 1

3

Hagarsquos theorem

1minus xx

y1minus xx y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12



1minus k2 = k1 + k

Hagarsquos theorem


y2 = 1N

1 + 1N = 1N + 1




P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications




2)





2)





2)


















x

y1minus x

x y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12


8 Ntilde y2 = 1

214x = 1

3

Hagarsquos theorem

1minus xx

y1minus xx y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12



1minus k2 = k1 + k

Hagarsquos theorem


y2 = 1N

1 + 1N = 1N + 1




P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications




2)





2)


















x

y1minus x

x y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12


8 Ntilde y2 = 1

214x = 1

3

Hagarsquos theorem

1minus xx

y1minus xx y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12



1minus k2 = k1 + k

Hagarsquos theorem


y2 = 1N

1 + 1N = 1N + 1




P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications




2)


















x

y1minus x

x y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12


8 Ntilde y2 = 1

214x = 1

3

Hagarsquos theorem

1minus xx

y1minus xx y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12



1minus k2 = k1 + k

Hagarsquos theorem


y2 = 1N

1 + 1N = 1N + 1




P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications

















x

y1minus x

x y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12


8 Ntilde y2 = 1

214x = 1

3

Hagarsquos theorem

1minus xx

y1minus xx y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12



1minus k2 = k1 + k

Hagarsquos theorem


y2 = 1N

1 + 1N = 1N + 1




P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications

Hagarsquos theorem

1minus xx

y1minus xx y

α

β

β

α

α

β

α

β

β

α

β

k 1minus k

α

12 12



1minus k2 = k1 + k

Hagarsquos theorem


y2 = 1N

1 + 1N = 1N + 1




P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications

Hagarsquos theorem


y2 = 1N

1 + 1N = 1N + 1




P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications


P Q


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications


Q

x y



Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications


Q

θ

P


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications


θ

P

A B

Q






K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications




K

L

M

Q

Qprime

P

P prime

slope(M)



ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications


ψ

φ

K

L

M

Q

Qprime

P

P prime

R

S

1 slope(M)





















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications




















oldid=807827828 2017

Pictures

Introduction

Axioms of origami



Applications

origami and mathematics - heriot-watt universitysl37/slides/18_phdseminar.pdforigami comes from the...

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