Origami and mathematics: why you are not just folding paper Stefania Lisai MACS PhD Seminar 5th October 2018
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Meaning and history Origami comes from the Japanese words ori meaning "folding", and kami meaning "paper".
Meaning and history Origami comes from the Japanese words 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.
Meaning and history Origami comes from the Japanese words 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.
Meaning and history Origami comes from the Japanese words 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.
Meaning and history Origami comes from the Japanese words 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.
Meaning and history Origami comes from the Japanese words 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:
Compass and Straightedge construction Basic constructions with compass and straightedge: 1. We can draw a line passing through 2 given points;
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;
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;
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 � = ∅ );
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. solving t 3 + 3 at 2 − 3 t − a = 0 � θ � 1 3 − π with a = 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. solving t 3 + 3 at 2 − 3 t − a = 0 � θ � 1 3 − π with a = 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
Justin-Huzita-Hatori Axioms 1. There is a fold passing through 2 given points;
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;
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;
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;
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;
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