Paper folding : how origami beat

Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Double the altar

1

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 Double the cube

2

Sunday, March 3, 13 Double the cube?

Sunday, March 3, 13 Double the cube?

2

2 2

Sunday, March 3, 13 Double the cube? No, octuple the cube!

2

2 2

Sunday, March 3, 13 Double the cube

Volume of the cube = (edge)3 =2

edge = p3 2 1.25992105... ) '

Sunday, March 3, 13 Double the cube

Volume of the cube = (edge)3 =2

edge = p3 2 1.25992105... ) '

Vol=1

1

Sunday, March 3, 13 Double the cube

Volume of the cube = (edge)3 =2

edge = p3 2 1.25992105... ) '

Vol=2

p3 2

Sunday, March 3, 13 Double the square

Sunday, March 3, 13 Double the square

1

Sunday, March 3, 13 Double the square

1

Sunday, March 3, 13 Double the square

1

Sunday, March 3, 13 Double the square

1

Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Postulates: 1. Let it be granted that a straight 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.

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 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.

4. All right are equal.

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.

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 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.

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 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.

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 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.

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 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.

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 to the given 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.

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 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.

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 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.

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

Sunday, March 3, 13 Proposition 1: To construct an equilateral triangle on a given finite straight line.

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 .

Sunday, March 3, 13 Proposition 9: To bisect a given rectilinear angle.

Sunday, March 3, 13 Proposition 9: To bisect a given rectilinear angle.

Sunday, March 3, 13 Proposition 9: To bisect a given rectilinear angle.

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 Proposition 10: To bisect a given finite straight line.

Sunday, March 3, 13 Proposition 10: To bisect a given finite straight line.

Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Double the cube

Volume of the cube = (edge)3 =2

edge = p3 2 1.25992105... ) '

Vol=2

p3 2

Sunday, March 3, 13 What points can we construct?

Sunday, March 3, 13 What points can we construct?

Sunday, March 3, 13 What points can we construct?

Sunday, March 3, 13 What points can we construct?

Sunday, March 3, 13 What points can we construct?

Sunday, March 3, 13 What points can we construct?

Sunday, March 3, 13 What points can we construct?

Sunday, March 3, 13 What numbers can we construct?

Sunday, March 3, 13 What numbers can we construct?

0

Sunday, March 3, 13 What numbers can we construct?

0 1

Sunday, March 3, 13 What numbers can we construct?

0 1 2

Sunday, March 3, 13 What numbers can we construct?

0 1 1 2 2

Sunday, March 3, 13 What numbers can we construct?

0 1 1 2 2 3

Sunday, March 3, 13 What numbers can we construct?

1 4 0 1 1 2 2 3

Sunday, March 3, 13 What numbers can we construct?

1 3 4 4 0 1 1 2 2 3

Sunday, March 3, 13 What numbers can we construct?

1 3 3 4 4 2 0 1 1 2 2 3

Sunday, March 3, 13 What numbers can we construct?

1 3 3 4 4 2 0 1 1 1 2 8 2 3

Sunday, March 3, 13 What numbers can we construct?

1 3 3 4 4 2 0 1 1 1 2 8 2 3

All integers and all fractions with denominator 2 k .

Sunday, March 3, 13 What numbers can we construct?

1 3 3 4 4 2 0 1 1 1 2 8 2 3

All integers and all fractions with denominator 2 k . And what else?

Sunday, March 3, 13 Thales’ theorem (intercept theorem):

Sunday, March 3, 13 Thales’ theorem (intercept theorem):

Sunday, March 3, 13 Thales’ theorem (intercept theorem):

Sunday, March 3, 13 Thales’ theorem (intercept theorem):

b a

c d

Sunday, March 3, 13 Thales’ theorem (intercept theorem):

b d b = a c a

c d

Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 A = B = height of Thales

Sunday, March 3, 13 A = B = height of Thales 230m C = 32m + 2 = 147m

Sunday, March 3, 13 A = B = height of Thales 230m C = 32m + 2 = 147m D C = A B

Sunday, March 3, 13 A = B = height of Thales 230m C = 32m + 2 = 147m D C = A B D = 147m

Sunday, March 3, 13 Thales’ theorem (intercept theorem):

b d b = a c a

c d

Sunday, March 3, 13 b ? b = a 1 a

1 ?

Sunday, March 3, 13 b ? b = a 1 a

1 b a

Sunday, March 3, 13 We can construct all fractions!

b ? b = a 1 a

1 b a

Sunday, March 3, 13 We can construct all fractions!

b ? b a = ÷ a 1 a

1 b a

Sunday, March 3, 13 ? ? d a = 1 a

1 d

Sunday, March 3, 13 a d ? d ⇥ a = 1 a

1 d

Sunday, March 3, 13 a d a ? d ⇥ ⇥ a = 1 a

1 d

Sunday, March 3, 13 We can multiply, divide, add and subtract.

a d a ? d ⇥ ⇥ a = 1 a

1 d

Sunday, March 3, 13 We can multiply, divide, add and subtract.

a d a ? d ⇥ ⇥ a = 1 a

1 d

And what else?

Sunday, March 3, 13 1

Sunday, March 3, 13 p2

1

Sunday, March 3, 13 All square roots!

Sunday, March 3, 13 All square roots!

1 p2 1

Sunday, March 3, 13 All square roots! 1

1 p2 1

Sunday, March 3, 13 All square roots! 1

1 p2 1 2 12 + p2 =?2

Sunday, March 3, 13 All square roots! 1

p3

1 p2 1 2 12 + p2 =?2

Sunday, March 3, 13 1 All square roots! 1

p3

1 p2 1 2 12 + p2 =?2

Sunday, March 3, 13 1 All square roots! 1

p3

1 p2 1 2 12 + p2 =?2 2 12 + p3 =?2

Sunday, March 3, 13 1 All square roots! 1 p4 p3

1 p2 1 2 12 + p2 =?2 2 12 + p3 =?2

Sunday, March 3, 13 All square roots!

Sunday, March 3, 13 Ver no quadro que produz All square roots! 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 Ver no quadro que produz All square roots! 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 Ver no quadro que produz All square roots! 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 construct p2

Sunday, March 3, 13 We can construct p2 p3

Sunday, March 3, 13 We can construct p2 p3

p53

Sunday, March 3, 13 We can construct p2 p3

6p7

p53

Sunday, March 3, 13 We can construct p2 p3

6p7 1+6p7

p53

Sunday, March 3, 13 We can construct p2 p3

6p7 1+6p7

p53 2 p 3 +5 11

Sunday, March 3, 13 We can construct p2 p3

4 2p15 p 22 6 7 1+6p7

p53 2 p 3 +5 11

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 1+6p7

p53 2 p 3 +5 11

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 1+6p7

5 p3 2 10 6 7 q p53 2 p 3 +5 11

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 1+6p7

8 5 10 p3 2 p5 6 7 q p53 2 p 3 +5 11

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 1+6p7

8 5 10 p3 2 p5 6 7 q p53 p2 p3 2 p 3 +5 11

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 1+6p7

8 5 p 10 p 6 3 2 7 5 23 q 41 p53 q p2 p3 2 p 3 +5 11

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 1+6p7

8 5 p 10 p 6 3 2 7 5 23 q 41 p53 q p2 p3 2 p 3 +5 11 4 + 12 19 92 q

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 3 p22 1+6p7 22+ 4 p67 5 8 5 p 10 p 6 3 2 7 5 23 q 41 p53 q p2 p3 2 p 3 +5 11 4 + 12 19 92 q

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 3 p22 1+6p7 22+ 4 p67 5 8 5 p 10 p 6 3 2 7 5 23 q 41 p53 q p2 p3 2 p 3 +5 11 19 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 3 p22 1+6p7 22+ 4 p67 5 8 5 p 10 etc. p 6 3 2 7 5 23 q 41 p53 q p2 p3 2 p 3 +5 11 19 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 We can construct p2 And what about 1+3 p 2 ? p 3 p 9 + 1 p10 13 37 4 2p15 p 22 6 7 3 p22 1+6p7 22+ 4 p67 5 8 5 p 10 etc. p 6 3 2 7 5 23 q 41 p53 q p2 p3 2 p 3 +5 11 19 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 All square roots?

a ? a ? = 1

? ? ⇥

1 ?

Sunday, March 3, 13 All square roots?

a ? a ? = 1

?=pa ? ? ⇥

1 ?

Sunday, March 3, 13 All square roots!

1 a

Sunday, March 3, 13 All square roots!

1 a

Sunday, March 3, 13 All square roots!

1 a

Sunday, March 3, 13 All square roots!

1 a

Sunday, March 3, 13 All square roots!

pa

1 a

Sunday, March 3, 13 We can construct p2 p3 9 + 1 p10 13 37 4 2p15 p 22 6 7 3 p22 1+6p7 22+ 4 p67 5 8 5 10 p3 2 etc. p5 6 7 23 q 41 p53 q p2 p3 2 +5p11 19 3 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 We can construct p2 1+3p2 p 3 p 9 + 1 p10 13 37 4 2p15 p 22 6 7 3 p22 1+6p7 22+ 4 p67 5 8 5 10 p3 2 etc. p5 6 7 23 q 41 p53 q p2 p3 2 +5p11 19 3 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 We can construct p2 1+3p2 p 3 p 9 1 p 13 + 37 10 59 2+ 67 3 p 4 2p15 12 6p7 22 q 3 p22 1+6p7 22+ 4 p67 5 8 5 10 p3 2 etc. p5 6 7 23 q 41 p53 q p2 p3 2 +5p11 19 3 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 We can construct p2 1+3p2 p 3 p 9 1 p 13 + 37 10 59 2+ 67 3 p 4 2p15 12 6p7 22 q 5+p7 7+p5 3 p22 1+6p7 22+ 4 p67 q 5 8 5 10 p3 2 etc. p5 6 7 23 q 41 p53 q p2 p3 2 +5p11 19 3 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 We can construct p2 1+3p2 p 3 p 9 1 p 13 + 37 10 59 2+ 67 3 p 4 2p15 12 6p7 22 q 5+p7 7+p5 3 p22 1+6p7 22+ 4 p67 q 5 8 5 10 p3 2 etc. p5 6 7 23 q 41 p q 53 p345 33 p2 p 7 2 91 5+ p p p 2+ 3 3 r q 2p+5p11 19 3 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 We can construct p2 1+3p2 p 3 p p8 9 1 p 47 13 + 37 10 59 2+ 67 3 p 4 2p15 12 6p7 22 q 5+p7 7+p5 3 p22 1+6p7 22+ 4 p67 q 5 8 5 10 p3 2 etc. p5 6 7 23 q 41 p q 53 p345 33 p2 p 7 2 91 5+ p p p 2+ 3 3 r q 2p+5p11 19 3 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 We can construct p2 1+3p2 p 3 p p8 9 1 p 47 13 + 37 10 59 2+ 67 3 p 4 2p15 12 6p7 22 q 5+p7 7+p5 3 p22 1+6p7 22+ 4 p67 q 5 8 5 10 p3 2 etc. p5 6 7 23 q 41 p q 53 p345 33 p2 etc. p 7 2 91 5+ p p p 2+ 3 3 r q 2p+5p11 19 3 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 20 5 p7 83

Sunday, March 3, 13 We can construct p2 1+3p2 p 3 p p8 9 1 p 47 13 + 37 10 59 2+ 67 3 p 4 2p15 12 6p7 22 q 5+p7 7+p5 3 p22 1+6p7 22+ 4 p67 q 5 8 5 10 p3 2 etc. p5 6 7 23 q 41 p q 53 p345 33 p2 etc. p 7 2 91 5+ p p p 2+ 3 3 r q 2p+5p11 19 3 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 And what else? 20 5 p7 83

Sunday, March 3, 13 We can construct p2 1+3p2 p 3 p p8 9 1 p 47 13 + 37 10 59 2+ 67 3 p 4 2p15 12 6p7 22 q 5+p7 7+p5 3 p22 1+6p7 22+ 4 p67 q 5 8 5 10 p3 2 etc. p5 6 7 23 q 41 p q 53 p345 33 p2 etc. p 7 2 91 5+ p p p 2+ 3 3 r q 2p+5p11 19 3 4 + 12 92 p 21 q 100 52 p26 + 43 + 38 p99 And what else? Nothing else! 20 5 p7 83

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 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

1

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

↵ 1 cos ↵

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

↵ 1 cos ↵ cos ✓ = 4(cos ✓ )3 3(cos ✓ ) 3 3

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

↵ 1 cos ↵

cos ✓ = 4(cos ✓ )3 3(cos ✓ ) = 1 =4x3 3 x 3 3 ) 2

Sunday, March 3, 13 Sunday, March 3, 13 Given a circle, build a square with the same area.

Sunday, March 3, 13 Given a circle, build a square with the same area.

Sunday, March 3, 13 Given a circle, build a square with the same area.

r

area of the circle = ⇡r2

Sunday, March 3, 13 Given a circle, build a square with the same area.

r

area of the circle = ⇡r2 area of the square = (edge)2

Sunday, March 3, 13 Given a circle, build a square with the same area.

r

area of the circle = ⇡r2 area of the square = (edge)2 edge = p⇡r

Sunday, March 3, 13 Given a circle, build a square with the same area.

r

area of the circle = ⇡r2 area of the square = (edge)2 edge = p⇡r

Is p ⇡ constructible with ruler and compass?

Sunday, March 3, 13 Given a circle, build a square with the same area.

r

area of the circle = ⇡r2 area of the square = (edge)2 edge = p⇡r

Is ⇡ constructible with ruler and compass?

Sunday, March 3, 13 Given a circle, build a square with the same area.

r

area of the circle = ⇡r2 area of the square = (edge)2 edge = p⇡r

Is ⇡ constructible with ruler and compass? No.

Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 n =2k product of distinct Fermat primes ⇥

Sunday, March 3, 13 n =2k product of distinct Fermat primes ⇥

2n Fn =2 +1

Sunday, March 3, 13 n =2k product of distinct Fermat primes ⇥

2n Fn =2 +1 F0 =3,F1 =5,F2 = 17,F3 = 257,F4 = 65537,F5 is not prime,...

Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Dividing into thirds Darren Scott http://www.floppyfish.net [email protected]

2 2

3 2/3 3 1

3 1/3

Sunday, March 3, 13 Dividing into thirds Darren Scott http://www.floppyfish.net [email protected]

2 2

3 2/3 3 1

3 1/3

Sunday, March 3, 13 Dividing into thirds Darren Scott http://www.floppyfish.net [email protected]

2 2

3 2/3 3 1

3 1/3

Sunday, March 3, 13 Dividing into thirds Darren Scott http://www.floppyfish.net [email protected]

2 2

3 2/3 3 1

3 1/3

Sunday, March 3, 13 Dividing into thirds Darren Scott http://www.floppyfish.net [email protected]

1 2 x 2 1 x 2 3 2/3 3 1

3 1/3

Sunday, March 3, 13 Dividing into thirds Darren Scott http://www.floppyfish.net [email protected]

1 1 2 2 x 3 2 1 x 2 8 3 2/3 3 1

3 1/3

Sunday, March 3, 13 Dividing into thirds Darren Scott http://www.floppyfish.net [email protected]

1 1 1 2 2 2 x 3 2 1 x 2 8 3 2/3 3 1

3 1/3

Sunday, March 3, 13 1:

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 1: Axiom 2: Axiom 3:

Axiom 4:

Sunday, March 3, 13 Axiom 1: Axiom 2: Axiom 3:

Axiom 4:

Sunday, March 3, 13 Axiom 1: Axiom 2: Axiom 3:

Axiom 5:

Axiom 4:

Sunday, March 3, 13 Axiom 1: Axiom 2: Axiom 3:

Axiom 4: Axiom 5:

Sunday, March 3, 13 Axiom 1: Axiom 2: Axiom 3:

Axiom 6:

Axiom 4: Axiom 5:

Sunday, March 3, 13 Axiom 1: Axiom 2: Axiom 3:

Axiom 4: Axiom 5: Axiom 6:

Sunday, March 3, 13 Axiom 1: Axiom 2: Axiom 3:

Axiom 7:

Axiom 4: Axiom 5: Axiom 6:

Sunday, March 3, 13 Axiom 1: Axiom 2: Axiom 3:

Axiom 7:

Axiom 4: Axiom 5: Axiom 6:

Sunday, March 3, 13 Peter Messer’s construction of p 3 2 :

B

A

Sunday, March 3, 13 Peter Messer’s construction of p 3 2 :

B B

P

A A

Sunday, March 3, 13 Peter Messer’s construction of p 3 2 :

B B

P

A A

BP = p3 2 AP

Sunday, March 3, 13 B

p3 2 r

P

r

A

Sunday, March 3, 13 B B

p3 2 r p3 2 r

P P

r r

A A

Sunday, March 3, 13 p3 2 B B

p3 2 r p3 2 r

P P

r r

A A 1

Sunday, March 3, 13 p3 2 B B

p3 2 r p3 2 r

P P p3 2 r

r r

A A 1

Sunday, March 3, 13 :

C

A B

Sunday, March 3, 13 Angle trisection:

C C

D

A B AA B

Sunday, March 3, 13 Angle trisection:

C C

D

A B AA B

ˆ 1 ˆ BAD = 3 BAC

Sunday, March 3, 13 Angle trisection:

C C

D

A B AA B

ˆ 1 ˆ BAD = 3 BAC

Sunday, March 3, 13 Angle trisection:

C C

D

A B AA B

ˆ 1 ˆ BAD = 3 BAC

Sunday, March 3, 13 Angle trisection:

C C

D

A B AA B

ˆ 1 ˆ BAD = 3 BAC

Sunday, March 3, 13 And...?

Sunday, March 3, 13 And...?

Sunday, March 3, 13 And...?

No.

Sunday, March 3, 13 n =2k product of distinct Fermat primes ⇥

2n Fn =2 +1

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

Sunday, March 3, 13 n =2k product of distinct Fermat primes ⇥

2n Fn =2 +1

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

Sunday, March 3, 13 n =2k product of distinct Fermat primes ⇥

2n Fn =2 +1

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

Sunday, March 3, 13 n =2k3l product of distinct Pierpont primes ⇥

2n Fn =2 +1

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

Sunday, March 3, 13 n =2k3l product of distinct Pierpont primes ⇥

2u3v +1

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

Sunday, March 3, 13 n =2k3l product of distinct Pierpont primes ⇥

2u3v +1 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257,...

Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 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 directrix: y = 1/2 focus: (0, 1/2) x2 =2y

y2 = 4x focus: ( 1, 0) directrix: x =1 Sunday, March 3, 13 directrix: y = 1/2 focus: (0, 1/2) x2 =2y

y2 = 4x focus: ( 1, 0) directrix: x =1 Sunday, March 3, 13 dy m2 x2 =2y 2x =2 x = m (x, y)=(m, ) dx 2 dy 2 1 2 y2 = 4x 2y = 4 y = (x, y)=( , ) dx m m2 m

Sunday, March 3, 13 dy m2 x2 =2y 2x =2 x = m (x, y)=(m, ) dx 2 dy 2 1 2 y2 = 4x 2y = 4 y = (x, y)=( , ) dx m m2 m

Sunday, March 3, 13 dy m2 x2 =2y 2x =2 x = m (x, y)=(m, ) dx 2 dy 2 1 2 y2 = 4x 2y = 4 y = (x, y)=( , ) dx m m2 m

Sunday, March 3, 13 dy m2 x2 =2y 2x =2 x = m (x, y)=(m, ) dx 2 dy 2 1 2 y2 = 4x 2y = 4 y = (x, y)=( , ) dx m m2 m

y = mx + b

Sunday, March 3, 13 dy m2 x2 =2y 2x =2 x = m (x, y)=(m, ) dx 2 dy 2 1 2 y2 = 4x 2y = 4 y = (x, y)=( , ) dx m m2 m m2 = mm + b y = mx + b 2 2 1 = m + b m m2

Sunday, March 3, 13 dy m2 x2 =2y 2x =2 x = m (x, y)=(m, ) dx 2 dy 2 1 2 y2 = 4x 2y = 4 y = (x, y)=( , ) dx m m2 m m2 = mm + b 2 m2 1 y = mx + b b = = 2 1 2 m = m + b m m2

Sunday, March 3, 13 dy m2 x2 =2y 2x =2 x = m (x, y)=(m, ) dx 2 dy 2 1 2 y2 = 4x 2y = 4 y = (x, y)=( , ) dx m m2 m m2 = mm + b 2 m2 1 y = mx + b b = = 2 1 2 m = m + b m m2

3 m =2

Sunday, March 3, 13 dy m2 x2 =2y 2x =2 x = m (x, y)=(m, ) dx 2 dy 2 1 2 y2 = 4x 2y = 4 y = (x, y)=( , ) dx m m2 m m2 = mm + b 2 m2 1 y = mx + b b = = 2 1 2 m = m + b m m2

3 3 m =2 m = p2

Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 Sunday, March 3, 13 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 : 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