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 t 3 = 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 t 3 = 2. Squaring the circle: given a circle, find the edge of a square which has the same area as the circle, i.e. solve t 2 = 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 t 3 = 2. Squaring the circle: given a circle, find the edge of a square which has the same area as the circle, i.e. solve t 2 = π . Trisect the angles: given an angle, find another which is a third of it, i.e. � θ � solving t 3 + 3 at 2 − 3 t − a = 0 with a = 1 3 − π tan θ and t = tan . 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 t 3 = 2. Squaring the circle: given a circle, find the edge of a square which has the same area as the circle, i.e. solve t 2 = π . Trisect the angles: given an angle, find another which is a third of it, i.e. � θ � solving t 3 + 3 at 2 − 3 t − a = 0 with a = 1 3 − π tan θ and t = tan . 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 / 2 1 / 2 β α x 1 − x α y β β α 1 x 2 + 1 x = 3 y 2 = 1 x = 1 4 = ( x − 1 ) 2 4 ⇒ ⇒ 8 2 3

Haga’s theorem k 1 − k α β x y 1 − x α β β α x = 1 − k 2 2 = k ( 1 − k ) y k x 2 + k 2 = ( 1 − x ) 2 ⇒ ⇒ = 2 1 − k 2 1 + k

Haga’s theorem If we choose k = 1 N , for some N ∈ N , then y 1 / N 1 2 = 1 + 1 / N = 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 y x Q √ The wanted value is given by the ratio y 3 x = 2. Doubling the cube is equivalent to solving the equation t 3 − 2 = 0.

Geometric problems of antiquity: trisect the angle Q θ P

Geometric problems of antiquity: trisect the angle Q P θ A B The angle � PAB is θ 3 . Trisecting the angle is equivalent to solving the equation t 3 + 3 at 2 − 3 t − a = 0 � θ � 1 3 − π with a = tan θ and t = tan . 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 t 3 + at 2 + bt + c = 0 L Q P M slope( M ) Q ′ P ′ K P = ( a , 1 ) , Q = ( c , b ) , L = { x = − c } , K = { y = − 1 } . The slope of M satisfies the equation.

Solving the cubic equation t 3 + at 2 + bt + c = 0 L ψ Q P φ M R 1 slope( M ) Q ′ S P ′ K 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 t 3 + at 2 + bt + c = 0 .

Solving cubic equations Want to solve t 3 + at 2 + bt + c = 0. φ = { 4 y = ( x − a ) 2 } , ψ = { 4 cx = ( y − b ) 2 } , M = { y = tx + u } . M is tangent to φ at R , then u = − t 2 − 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& oldid=807827828" , 2017.