Paper folding geometry: how origami beat Euclid 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 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. 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 angles 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 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 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 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 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 .