Origami and mathematics: why you are not just folding paper - - PowerPoint PPT Presentation

Origami and mathematics:

why you are not just folding paper Stefania Lisai PG Colloquium 24th November 2017

Pretty pictures to get your attention

https://www.quora.com/ Why-dont-more-origami-inventors-follow-Yoshizawas-landmarkless-approach-to-design/

Pretty pictures to get your attention

Pretty pictures to get your attention

https://origami.plus/origami-master-yoda

Pretty pictures to get your attention

https://www.quora.com/Why-is-origami-considered-art

Pretty pictures to get your attention

Pretty pictures to get your attention

Pretty pictures to get your attention

Pretty pictures to get your attention

http://viralpie.net/ the-art-of-paper-folding-just-got-taken-to-a-whole-new-level-with-3d-origami/

Pretty pictures to get your attention

Pretty pictures to get your attention

http://www.artfulmaths.com/blog/category/origami

Pretty pictures to get your attention

http://www.radionz.co.nz/national/programmes/afternoons/audio/201835180/ maths-and-crafts-using-crochet-and-origami-to-teach-mathematics

Meaning and history

Origami comes from ori meaning ”folding”, and kami meaning ”paper”. g Paper folding was known in Europe, Japan and China for long

• time. In 20th century, different traditions mixed up.

g In 1986, Jacques Justin discovered axioms 1-6, but ignored. g In 1989, the first International Meeting of Origami Science and Technology was held in Ferrara, Italy. g In 1991, Humiaki Huzita rediscovered axioms 1-6 and got all the glory. g In 2001, Koshiro Hatori discovered axiom 7.

Compass and Straightedge construction

Basic constructions with compass and straightedge:

• 1. We can draw a line passing through 2 given points;
• 2. 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 lines;
• 4. We can find one point in the intersection of a line and a circle (if

= ∅);

• 5. We can find one point in the intersection of 2 given circle (if = ∅).

Compass and Straightedge construction

Basic constructions with compass and straightedge:

• 1. We can draw a line passing through 2 given points;
• 2. 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 lines;
• 4. We can find one point in the intersection of a line and a circle (if

= ∅);

• 5. We can find one point in the intersection of 2 given circle (if = ∅).

Using these constructions, we can do other super cool things: bisect angles, reflect points, draw perpendicular lines, find midpoint to segments, draw the line tangent to a circle in a certain point, etc...

Compass and Straightedge construction

Basic constructions with compass and straightedge:

• 1. We can draw a line passing through 2 given points;
• 2. 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 lines;
• 4. We can find one point in the intersection of a line and a circle (if

= ∅);

• 5. We can find one point in the intersection of 2 given circle (if = ∅).

Using these constructions, we can do other super cool things: bisect angles, reflect points, draw perpendicular lines, find midpoint to segments, draw the line tangent to a circle in a certain point, etc... We cannot solve the three classical problems of ancient Greek geometry using compass and straightedge!

Three geometric problems of antiquity

Doubling the cube: given a cube, find the edge of another cube which has double its volume, i.e. solve t3 = 2.

Three geometric problems of antiquity

Doubling the cube: given a cube, find the edge of another cube which has double its volume, i.e. solve t3 = 2. Squaring the circle: given a circle, find the edge of a square which has the same area as the circle, i.e. solve t2 = π.

Three geometric problems of antiquity

Doubling the cube: given a cube, find the edge of another cube which has double its volume, i.e. solve t3 = 2. Squaring the circle: given a circle, find the edge of a square which has the same area as the circle, i.e. solve t2 = π. Trisect the angles: given an angle, find another which is a third of it, i.e. solving t3 + 3at2 − 3t − a = 0 with a =

1 tan θ and t = tan

θ

3 − π 2

• .

Three geometric problems of antiquity

Doubling the cube: given a cube, find the edge of another cube which has double its volume, i.e. solve t3 = 2. Squaring the circle: given a circle, find the edge of a square which has the same area as the circle, i.e. solve t2 = π. Trisect the angles: given an angle, find another which is a third of it, i.e. solving t3 + 3at2 − 3t − a = 0 with a =

1 tan θ and t = tan

θ

3 − π 2

• .

You can do 2 of these 3 things with origami: guess which one is impossible?

Justin-Huzita-Hatori Axioms

• 1. There is a fold passing through 2 given points;
• 2. There is a fold that places one point onto another;
• 3. There is a fold that places one line onto another;
• 4. There is a fold perpendicular to a given line and passing through a

given point;

• 5. There is a fold through a given point that places another point onto

a given line;

• 6. There is a fold that places a given point onto a given line and

another point onto another line;

• 7. There is a fold perpendicular to a given line that places a given

point onto another line.

Folding in thirds

1 − x x y α β β α β α 1/2 1/2

x2 + 1 4 = (x − 1)2 ⇒ x = 3 8 ⇒ y 2 = 1 2

1 4

x = 1 3

Haga’s theorem

1 − x x y α β β α α β k 1 − k

x2 + k2 = (1 − x)2 ⇒ x = 1 − k2 2 ⇒ y 2 = k(1 − k) 1 − k2 = k 1 + k

Haga’s theorem

If we choose k = 1

N , for some N ∈ N, then

y 2 = 1/N 1 + 1/N = 1 N + 1, therefore starting from N = 2 we can obtain 1

n for any n > 2, hence any

rational m

n for 0 < m < n ∈ N.

Geometric problems of antiquity: double the cube

P Q

Geometric problems of antiquity: double the cube

P Q x y

The wanted value is given by the ratio y

x =

3

√ 2. Doubling the cube is equivalent to solving the equation t3 − 2 = 0.

Geometric problems of antiquity: trisect the angle

Q θ P

Geometric problems of antiquity: trisect the angle

θ P A B Q

The angle PAB is θ

3.

Trisecting the angle is equivalent to solving the equation t3 + 3at2 − 3t − a = 0 with a =

1 tan θ and t = tan

θ

3 − π 2

• .

Geometric problems of antiquity: square the circle

Unfortunately, π is still transcendental, even in the origami world. This problem is proved to be impossible in the folding paper theory.

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

K L M

Q Q′ P P′ slope(M)

P = (a, 1), Q = (c, b), L = {x = −c}, K = {y = −1}. The slope of M satisfies the equation.

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

ψ φ K L M

Q Q′ P P′ R S 1 slope(M)

P = (a, 1), Q = (c, b), L = {x = −c}, K = {y = −1}. If a = 1.5, b = 1.5, c = 0.5, then t = slope(M) = −1.5 satisfies t3 + at2 + bt + c = 0.

Solving cubic equations

Want to solve t3 + at2 + bt + c = 0. φ = {4y = (x − a)2}, ψ = {4cx = (y − b)2}, M = {y = tx + u}. M is tangent to φ at R, then u = −t2 − at, M is tangent to ψ at S, then u = b + c

t .

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

Applications in real world

g Solar panels and mirrors for space;

Applications in real world

g Solar panels and mirrors for space; g Air bags;

Applications in real world

g Solar panels and mirrors for space; g Air bags; g Heart stents (Oxford);

Applications in real world

g Solar panels and mirrors for space; g Air bags; g Heart stents (Oxford); g Self-folding robots (Harvard, MIT);

Applications to real world

g Solar panels and mirrors for space; g Air bags; g Heart stents (Oxford); g Self-folding robots (Harvard, MIT); g Decorations for my bedroom.

References

[lan08] The math and magic of origami — robert lang. https://www.youtube.com/watch?v=NYKcOFQCeno, 2008. [Lan10] Robert Lang. Origami and geometric constructions. http://whitemyth.com/sites/default/files/downloads/Origami/Origami% 20Theory/Robert%20J.%20Lang%20-%20Origami%20Constructions.pdf, 2010. [Mai14] Douglas Main. From robots to retinas: 9 amazing origami applications. https://www.popsci.com/article/science/ robots-retinas-9-amazing-origami-applications#page-4, 2014. [Tho15] Rachel Thomas. Folding fractions. https://plus.maths.org/content/folding-numbers, 2015. [Wik17] Wikipedia. Mathematics of paper folding — wikipedia, the free encyclopedia. "https://en.wikipedia.org/w/index.php?title=Mathematics_of_paper_folding&

• ldid=807827828", 2017.