Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models Visualizing Hyperbolic Geometry Evelyn Lamb University of Utah May 8, 2014 Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models 1 Euclidean Geometry 2 A Spherical Interlude 3 Hyperbolic geometry 4 Models Visualizing Hyperbolic Geometry 1st example of axiomatic approach to mathematics 23 definitions, 5 axioms, 5 postulates (This is part of the Pythagorean theorem in Oliver Byrne's 1847 edition of the text.) Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models Euclid's Elements c. 300 BCE Visualizing Hyperbolic Geometry 23 definitions, 5 axioms, 5 postulates (This is part of the Pythagorean theorem in Oliver Byrne's 1847 edition of the text.) Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models Euclid's Elements c. 300 BCE 1st example of axiomatic approach to mathematics Visualizing Hyperbolic Geometry (This is part of the Pythagorean theorem in Oliver Byrne's 1847 edition of the text.) Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models Euclid's Elements c. 300 BCE 1st example of axiomatic approach to mathematics 23 definitions, 5 axioms, 5 postulates Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models Euclid's Elements c. 300 BCE 1st example of axiomatic approach to mathematics 23 definitions, 5 axioms, 5 postulates (This is part of the Pythagorean theorem in Oliver Byrne's 1847 edition of the text.) Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models One of these postulates is not like the others A straight line segment can be drawn joining any two points. A straight line segment can be extended indefinitely in a straight line. Given any straight line segment, a circle can be drawn having the segment as a radius and one endpoint as center. All right angles are congruent. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A few statements that are equivalent to the 5th postulate Playfair's axiom: Between a line L and a point P not on L, there is exactly one line through P that does not intersect L. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A few statements that are equivalent to the 5th postulate The Pythagorean Theorem Visualizing Hyperbolic Geometry There exists a triangle whose angles add up to 180◦. All triangles have the same sum of angles. There is no upper limit to the area of a triangle. There exists a pair of triangles that are similar but not congruent. There exists a quadrilateral in which all angles are right angles. Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A few statements that are equivalent to the 5th postulate The sum of interior angles in a triangle is 180◦. Visualizing Hyperbolic Geometry All triangles have the same sum of angles. There is no upper limit to the area of a triangle. There exists a pair of triangles that are similar but not congruent. There exists a quadrilateral in which all angles are right angles. Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A few statements that are equivalent to the 5th postulate The sum of interior angles in a triangle is 180◦. There exists a triangle whose angles add up to 180◦. Visualizing Hyperbolic Geometry There is no upper limit to the area of a triangle. There exists a pair of triangles that are similar but not congruent. There exists a quadrilateral in which all angles are right angles. Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A few statements that are equivalent to the 5th postulate The sum of interior angles in a triangle is 180◦. There exists a triangle whose angles add up to 180◦. All triangles have the same sum of angles. Visualizing Hyperbolic Geometry There exists a pair of triangles that are similar but not congruent. There exists a quadrilateral in which all angles are right angles. Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A few statements that are equivalent to the 5th postulate The sum of interior angles in a triangle is 180◦. There exists a triangle whose angles add up to 180◦. All triangles have the same sum of angles. There is no upper limit to the area of a triangle. Visualizing Hyperbolic Geometry There exists a quadrilateral in which all angles are right angles. Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A few statements that are equivalent to the 5th postulate The sum of interior angles in a triangle is 180◦. There exists a triangle whose angles add up to 180◦. All triangles have the same sum of angles. There is no upper limit to the area of a triangle. There exists a pair of triangles that are similar but not congruent. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A few statements that are equivalent to the 5th postulate The sum of interior angles in a triangle is 180◦. There exists a triangle whose angles add up to 180◦. All triangles have the same sum of angles. There is no upper limit to the area of a triangle. There exists a pair of triangles that are similar but not congruent. There exists a quadrilateral in which all angles are right angles. Visualizing Hyperbolic Geometry Spoiler alert: they were wrong. Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models For 2000 years, mathematicians tried to \prove" the fifth postulate using the other four. In other words, they wanted to show that in order for the first four postulates to hold, the fifth had to as well. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models For 2000 years, mathematicians tried to \prove" the fifth postulate using the other four. In other words, they wanted to show that in order for the first four postulates to hold, the fifth had to as well. Spoiler alert: they were wrong. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models Alternative Versions of the Fifth Postulate If the fifth postulate didn't hold, what could happen? Playfair's Axiom: Between a line L and a point P not on L, there is exactly one line through P that does not intersect L. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models How Could We Change the Fifth Postulate? If the fifth postulate doesn't hold, what does that mean? Playfair's Axiom, elliptic style: Between a line L and a point P not on L, there are no lines through P that do not intersect L. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models How Could We Change the Fifth Postulate? If the fifth postulate doesn't hold, what does that mean? Playfair's Axiom, hyperbolic style: Between a line L and a point P not on L, there are infinitely many lines through P that do not intersect L. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A spherical interlude These alternatives to Playfair's Axiom might seem bizarre, but if you've ever flown on a plane, you're familiar with one of them already. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models A spherical interlude Yes, we live on a sphere, so the shortest distance between two points doesn't \look like" a straight line on a map. Visualizing Hyperbolic Geometry None of them work for the Earth. Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models Remember these statements that are equivalent to the 5th postulate? The sum of interior angles in a triangle is 180◦. All triangles have the same sum of angles. There is no upper limit to the area of a triangle. Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models Remember these statements that are equivalent to the 5th postulate? The sum of interior angles in a triangle is 180◦. All triangles have the same sum of angles. There is no upper limit to the area of a triangle. None of them work for the Earth. Visualizing Hyperbolic Geometry Spherical geometry is sitting right under our feet, but it was harder to discover hyperbolic geometry. Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models What about the other version? Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models What about the other version? Spherical geometry is sitting right under our feet, but it was harder to discover hyperbolic geometry. Visualizing Hyperbolic Geometry (\For God's sake, I beseech you, give it up. Fear it no less than sensual passions because it too may take all your time and