pi.math.cornell.edu - Paper folding geometry: how origami beat … · 2013. 4. 1. · Postulates:...

Paper folding geometry:how origami beat Euclid

Sunday, March 3, 13

Sunday, March 3, 13

Double the altar

1

Sunday, March 3, 13

Double the cube

1

Sunday, March 3, 13

Double the cube

Sunday, March 3, 13

2

Double the cube

Sunday, March 3, 13

Double the cube?

Sunday, March 3, 13

22

2

Double the cube?

Sunday, March 3, 13

22

2

Double the cube?No, octuple the cube!

Sunday, March 3, 13

Double the cube

Volume of the cube = (edge)

3= 2

) edge = 3p2 ' 1.25992105...

Sunday, March 3, 13

1

Double the cube

Vol=1


3= 2

) edge = 3p2 ' 1.25992105...

Sunday, March 3, 13

Vol=2

Double the cube

3p2


3= 2

) edge = 3p2 ' 1.25992105...

Sunday, March 3, 13

Double the square

Sunday, March 3, 13

Double the square

1

Sunday, March 3, 13

Sunday, March 3, 13

Postulates:1. Let it be granted that a straight line may be drawn from any one point to any other point.

Sunday, March 3, 13

Postulates:1. Let it be granted that a straight line may be drawn from any one point to any other point.2. Let it be granted that a finite straight line may be produced to any length in a straight line.

Sunday, March 3, 13

Postulates:1. Let it be granted that a straight line may be drawn from any one point to any other point.2. Let it be granted that a finite straight line may be produced to any length in a straight line.3. Let it be granted that a circle may be described with any center at any distance from that center.

Sunday, March 3, 13


4. All right angles are equal.

Sunday, March 3, 13


4. All right angles are equal.5. If two straight lines meet a third straight line so as to make the two interior angles on the same side less than two right angles, then those two lines will meet if extended indefinitely on the side on which the angles are less than two right angles.

Sunday, March 3, 13


4. All right angles are equal.5. Given a straight line and a point not on that line, there exists exactly one line through that point that is parallel to the given line.

Sunday, March 3, 13

Sunday, March 3, 13

Proposition 1: To construct an equilateral triangle on a given finite straight line.

Sunday, March 3, 13

Proposition 9: To bisect a given rectilinear angle.

Sunday, March 3, 13

Proposition 10: To bisect a given finite straight line.

Sunday, March 3, 13

Sunday, March 3, 13

Vol=2

3p2

Double the cube


3= 2

) edge = 3p2 ' 1.25992105...

Sunday, March 3, 13

What points can we construct?

Sunday, March 3, 13

What numbers can we construct?

Sunday, March 3, 13


0

Sunday, March 3, 13


0 1

Sunday, March 3, 13

2


0 1

Sunday, March 3, 13

2


0 112

Sunday, March 3, 13

32


0 112

Sunday, March 3, 13

32


0 112

14

Sunday, March 3, 13

32


0 112

14

34

Sunday, March 3, 13

32


0 112

14

34

32

Sunday, March 3, 13

32


0 112

14

34

32

18

Sunday, March 3, 13

2k.All integers and all fractions with denominator

32


0 112

14

34

32

18

Sunday, March 3, 13

2k.All integers and all fractions with denominator

32


0 112

14

34

32

18

And what else?Sunday, March 3, 13

Thales’ theorem (intercept theorem):

Sunday, March 3, 13

ab

c d


Sunday, March 3, 13

ab

c d

b

a=

d

c


Sunday, March 3, 13

Sunday, March 3, 13

A = B = height of Thales

Sunday, March 3, 13

C = 32m+ 230m2 = 147m


Sunday, March 3, 13

C = 32m+ 230m2 = 147m

D

A=

C

B


Sunday, March 3, 13

C = 32m+ 230m2 = 147m

D

A=

C

B


D = 147m

Sunday, March 3, 13

ab

c d

b

a=

d

c


Sunday, March 3, 13

ab

1 ?

b

a=

?

1

Sunday, March 3, 13

ab

1 ba

b

a=

?

1

Sunday, March 3, 13

ab

1 ba

b

a=

?

1

We can construct all fractions!

Sunday, March 3, 13

ab

1 ba

b

a=

?

1

We can construct all fractions!

÷a

Sunday, March 3, 13

a

1

?

d

?a =

d1

Sunday, March 3, 13

a

1 d

?a =

d1

a⇥ d

Sunday, March 3, 13

a

1 d

⇥a ?a =

d1

a⇥ d

Sunday, March 3, 13

a

1 d

⇥a ?a =

d1

a⇥ d

We can multiply, divide, add and subtract.

Sunday, March 3, 13

a

1 d

⇥a ?a =

d1

a⇥ d

We can multiply, divide, add and subtract.

And what else?

Sunday, March 3, 13

1

Sunday, March 3, 13

1

p2

Sunday, March 3, 13

All square roots!

Sunday, March 3, 13

All square roots!

p2

1

1

Sunday, March 3, 13

All square roots! 1

p2

1

1

Sunday, March 3, 13

All square roots! 1

12 +p22=?2

p2

1

1

Sunday, March 3, 13

All square roots! 1

12 +p22=?2

p3

p2

1

1

Sunday, March 3, 13

All square roots! 1

12 +p22=?2

1

p3

p2

1

1

Sunday, March 3, 13

All square roots! 1

12 +p22=?2

1

12 +p32=?2

p3

p2

1

1

Sunday, March 3, 13

All square roots! 1

12 +p22=?2

1

12 +p32=?2

p4p

3p2

1

1

Sunday, March 3, 13

All square roots!

Sunday, March 3, 13

All square roots! Ver no quadro que produz todas as \sqrt{k}Espiral de Teodoro (até \sqrt{17}. Em 1958, E. Teuffel provou que duas hipotenusas da espiral nunca colidem.

Sunday, March 3, 13

We can construct

Sunday, March 3, 13

We can constructp2

Sunday, March 3, 13

We can constructp2p3

Sunday, March 3, 13


p53

Sunday, March 3, 13


p53

6p7

Sunday, March 3, 13


p53

6p7

1 + 6p7

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

56

p3� 2

q107

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

56

p3� 2

q107

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

56

p3� 2

q107

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

56

p3� 2

q107

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

56

p3� 2

q107

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

etc.

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

And what about ?p

1 + 3p2

etc.

Sunday, March 3, 13

1

a

?

?

a? =

?1

All square roots?

⇥?

Sunday, March 3, 13

1

a

?

?

a? =

?1

? =pa

All square roots?

⇥?

Sunday, March 3, 13

All square roots!

1 a

Sunday, March 3, 13

All square roots!

1 a

pa

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

etc.

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

p1 + 3

p2

etc.

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

p1 + 3

p2

2 +q

67� 3 59p12

etc.

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

p1 + 3

p2

2 +q

67� 3 59p12 q

5+p7

7+p5

etc.

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

p1 + 3

p2

2 +q

67� 3 59p12

rp3457 � 2

q91�

pp5 + 33p

2+p3

q5+

p7

7+p5

etc.

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

p1 + 3

p2

2 +q

67� 3 59p12

rp3457 � 2

q91�

pp5 + 33p

2+p3

q5+

p7

7+p5

etc.

8p47

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

p1 + 3

p2

2 +q

67� 3 59p12

rp3457 � 2

q91�

pp5 + 33p

2+p3

q5+

p7

7+p5

etc.

etc.

8p47

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

p1 + 3

p2

2 +q

67� 3 59p12

rp3457 � 2

q91�

pp5 + 33p

2+p3

q5+

p7

7+p5

And what else?

etc.

etc.

8p47

Sunday, March 3, 13


p53

6p7

1 + 6p7

23 + 5

p11

4�2p15

22

� 913 +137

p10

8p5

p2p3

q2341

�4 + 12q

1992

3�p22

�22+ 45p67

56

p3� 2

q107

p21100

20 �525

p26 + 43p

7+ 3883

p99

p1 + 3

p2

2 +q

67� 3 59p12

rp3457 � 2

q91�

pp5 + 33p

2+p3

q5+

p7

7+p5

And what else? Nothing else!

etc.

etc.

8p47

Sunday, March 3, 13

Sunday, March 3, 13

In general, it’s impossible to trisect an angle with ruler and compass.✓

Sunday, March 3, 13

In general, it’s impossible to trisect an angle with ruler and compass.

In general, the number cos is not constructible with ruler and compass.

✓

✓3

Sunday, March 3, 13



✓

1

✓3

Sunday, March 3, 13



✓

1

✓3

↵cos ↵

Sunday, March 3, 13



✓

1cos ✓ = 4(cos ✓3 )

3 � 3(cos ✓3 )

✓3

↵cos ↵

Sunday, March 3, 13



✓

1cos ✓ = 4(cos ✓3 )

3 � 3(cos ✓3 )12 = 4x

3 � 3x=)

✓3

↵cos ↵

Sunday, March 3, 13

Sunday, March 3, 13

Given a circle, build a square with the same area.

Sunday, March 3, 13


r

area of the circle = ⇡r2

Sunday, March 3, 13


r


area of the square = (edge)

2

Sunday, March 3, 13


r



2

edge =p⇡r

Sunday, March 3, 13


r

Is constructible with ruler and compass?

p⇡



2

edge =p⇡r

Sunday, March 3, 13


r

Is constructible with ruler and compass?

⇡



2

edge =p⇡r

Sunday, March 3, 13


r

Is constructible with ruler and compass? No.

⇡



2

edge =p⇡r

Sunday, March 3, 13

Sunday, March 3, 13

n = 2k ⇥ product of distinct Fermat primes

Sunday, March 3, 13

Fn = 22n + 1


Sunday, March 3, 13

Fn = 22n + 1


F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537, F5 is not prime, . . .

Sunday, March 3, 13

Sunday, March 3, 13

Dividing into thirdsDarren Scotthttp://[email protected]

2/3

1/3

2

3

1

3

2

3

Sunday, March 3, 13


2/3

1/3

2

3

1

3

2

3

12

x

1� x

Sunday, March 3, 13


2/3

1/3

2

3

1

3

2

3

12

x

1� x38

12

Sunday, March 3, 13


2/3

1/3

2

3

1

3

2

3

12

x

1� x38

12

12

Sunday, March 3, 13

Axiom 1:

Sunday, March 3, 13

Axiom 1:

Axiom 2:

Sunday, March 3, 13

Axiom 1: Axiom 2:

Sunday, March 3, 13

Axiom 1: Axiom 2:

Axiom 3:

Sunday, March 3, 13

Axiom 1: Axiom 2: Axiom 3:

Sunday, March 3, 13


Axiom 4:

Sunday, March 3, 13


Axiom 4:

Axiom 5:

Sunday, March 3, 13


Axiom 4: Axiom 5:

Sunday, March 3, 13


Axiom 4: Axiom 5:

Axiom 6:

Sunday, March 3, 13



Sunday, March 3, 13



Axiom 7:

Sunday, March 3, 13

Peter Messer’s construction of :3p2

A

B

Sunday, March 3, 13


P

A

B

A

B

Sunday, March 3, 13


P

A

B

A

B

BP = 3p2 AP

Sunday, March 3, 13

P

A

B

r

3p2 r

Sunday, March 3, 13

P

A

B

P

A

B

r

3p2 r

r

3p2 r

Sunday, March 3, 13

P

A

B

P

A

B3p2

1

r

3p2 r

r

3p2 r

Sunday, March 3, 13

P

A

B

P

A

B3p2

1

r

3p2 r

r

3p2 r

3p2 r

Sunday, March 3, 13

A B

C

Angle trisection:

Sunday, March 3, 13

A B

C

A B

C

D

A

Angle trisection:

Sunday, March 3, 13

A B

C

A B

C

D

A

Angle trisection:

BÂD = 13 BÂC

Sunday, March 3, 13

And...?

Sunday, March 3, 13

No.

And...?

Sunday, March 3, 13

Fn = 22n + 1


F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537, F5 is not prime, . . .Sunday, March 3, 13

Fn = 22n + 1


n = 2k3l ⇥ product of distinct Pierpont primes

Sunday, March 3, 13

2u3v + 1



Sunday, March 3, 13

2u3v + 1

2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, . . .


Sunday, March 3, 13

Sunday, March 3, 13

= {points at the same distance from the point O and the line l}

Sunday, March 3, 13

Axiom 6:

Sunday, March 3, 13

y

2=� 4x

focus: (�1, 0)directrix: x = 1

directrix: y = �1/2focus: (0, 1/2)

x

2=2y

Sunday, March 3, 13

y

2 = �4x 2y dydx

= �4 y = �2m

(x, y) = (�1m

2,

�2m

)

x

2 = 2y 2x = 2dydx

x = m (x, y) = (m, m2

2)

Sunday, March 3, 13

y

2 = �4x 2y dydx

= �4 y = �2m

(x, y) = (�1m

2,

�2m

)

x

2 = 2y 2x = 2dydx

x = m (x, y) = (m, m2

2)

y = mx+ b

Sunday, March 3, 13

y

2 = �4x 2y dydx

= �4 y = �2m

(x, y) = (�1m

2,

�2m

)

x

2 = 2y 2x = 2dydx

x = m (x, y) = (m, m2

2)

y = mx+ b m

2

2= mm+ b

�2m

= m�1m2

+ b

Sunday, March 3, 13

y

2 = �4x 2y dydx

= �4 y = �2m

(x, y) = (�1m

2,

�2m

)

x

2 = 2y 2x = 2dydx

x = m (x, y) = (m, m2

2)

y = mx+ b m

2

2= mm+ b

�2m

= m�1m2

+ b b = �m

2

2= � 1

m

Sunday, March 3, 13

y

2 = �4x 2y dydx

= �4 y = �2m

(x, y) = (�1m

2,

�2m

)

x

2 = 2y 2x = 2dydx

x = m (x, y) = (m, m2

2)

y = mx+ b m

2

2= mm+ b

�2m

= m�1m2

+ b b = �m

2

2= � 1

m

m3 = 2

Sunday, March 3, 13

y

2 = �4x 2y dydx

= �4 y = �2m

(x, y) = (�1m

2,

�2m

)

x

2 = 2y 2x = 2dydx

x = m (x, y) = (m, m2

2)

y = mx+ b m

2

2= mm+ b

�2m

= m�1m2

+ b b = �m

2

2= � 1

m

m3 = 2 m = 3p2

Sunday, March 3, 13

Sunday, March 3, 13

References:

•Euclid’s elements: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html (with explanations of the proofs and a geometry applet) and http://www.math.ubc.ca/~cass/Euclid/byrne.html (“in which coloured diagrams and symbols are used instead of letters for the greater ease of learners”)

• Wikipedia page about the origami axioms: http://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms

• Wikipedia page about origami math results: http://en.wikipedia.org/wiki/Mathematics_of_paper_folding

• Robert Lang’s webpage has loads of materials (including many of the photos of fancy origami): http://www.langorigami.com/ . In particular, this paper has a lot a information: http://www.langorigami.com/science/math/hja/origami_constructions.pdf

• A TED talk about origami math things, but not quite the same content as this talk: http://www.ted.com/talks/robert_lang_folds_way_new_origami.html

• Folding a regular heptagon! http://www.math.sjsu.edu/~alperin/TotallyRealHeptagon.pdf

Sunday, March 3, 13
http://aleph0.clarku.edu/~djoyce/java/elements/elements.htmlhttp://aleph0.clarku.edu/~djoyce/java/elements/elements.htmlhttp://www.math.ubc.ca/~cass/Euclid/byrne.htmlhttp://www.math.ubc.ca/~cass/Euclid/byrne.htmlhttp://www.math.ubc.ca/~cass/Euclid/byrne.htmlhttp://www.math.ubc.ca/~cass/Euclid/byrne.htmlhttp://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axiomshttp://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axiomshttp://en.wikipedia.org/wiki/Mathematics_of_paper_foldinghttp://en.wikipedia.org/wiki/Mathematics_of_paper_foldinghttp://www.langorigami.com/http://www.langorigami.com/http://www.langorigami.com/science/math/hja/origami_constructions.pdfhttp://www.langorigami.com/science/math/hja/origami_constructions.pdfhttp://www.ted.com/talks/robert_lang_folds_way_new_origami.htmlhttp://www.ted.com/talks/robert_lang_folds_way_new_origami.htmlhttp://www.math.sjsu.edu/~alperin/TotallyRealHeptagon.pdfhttp://www.math.sjsu.edu/~alperin/TotallyRealHeptagon.pdf

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