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Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Visualizing Hyperbolic Geometry Evelyn Lamb University of Utah May 8, 2014 Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry 1 Euclidean Geometry 2 A Spherical Interlude 3 Hyperbolic geometry 4 Models Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Euclid’s Elements c. 300 BCE Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude Euclid’s Elements c. 300 BCE 1st example of axiomatic approach to mathematics Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude Euclid’s Elements c. 300 BCE 1st example of axiomatic approach to mathematics 23 definitions, 5 axioms, 5 postulates Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude 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 Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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 Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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 Models 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 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 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 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 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 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 Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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 Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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 Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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 Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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 Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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 Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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 Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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 Models 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 Outline Euclidean Geometry A Spherical Interlude What about the other version? Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry What about the other version? Spherical geometry is sitting right under our feet, but it was harder to discover hyperbolic geometry. Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Bolyai and Lobachevsky “When the time is ripe for certain things, they appear at different places in the manner of violets coming to light in early spring.”-Farkas Bolyai, urging his son János to publish his work on non-Euclidean geometry Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Bolyai and Lobachevsky “When the time is ripe for certain things, they appear at different places in the manner of violets coming to light in early spring.”-Farkas Bolyai, urging his son János to publish his work on non-Euclidean 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 deprive you of your health, peace of mind and happiness in life”-Farkas Bolyai, a few years earlier, urging his son János to stop studying non-Euclidean geometry) Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Bolyai and Lobachevsky “When the time is ripe for certain things, they appear at different places in the manner of violets coming to light in early spring.”-Farkas Bolyai, urging his son János to publish his work on non-Euclidean 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 deprive you of your health, peace of mind and happiness in life”-Farkas Bolyai, a few years earlier, urging his son János to stop studying non-Euclidean geometry) János Bolyai and Nikolai Lobachevsky independently developed (discovered?) hyperbolic geometry in the 1820s. Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry How hyperbolic geometry is different from Euclidean geometry Given 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 Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry How hyperbolic geometry is different from Euclidean geometry Given a line L and a point P not on L, there are infinitely many lines through P that do not intersect L. The sum of angles in a triangle is strictly less that 180◦ . Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry How hyperbolic geometry is different from Euclidean geometry Given a line L and a point P not on L, there are infinitely many lines through P that do not intersect L. The sum of angles in a triangle is strictly less that 180◦ . The angles of a triangle determine its side lengths. (No such thing as similar but not congruent.) Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry A Spherical Interlude So what does it look like?! Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude So what does it look like?! Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude So what does it look like?! Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude So what does it look like?! Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude So what does it look like?! Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude Poincaré Disk All the lines shown are “straight” from the point of view of hyperbolic geometry because distance is defined differently here than it is in the Euclidean plane. Like a map of the Earth, the Poincaré Disk tells truths and lies. It tells the truth about angles and lies about area. Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry Circle Limit I M.C. Escher Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Circle Limit I Doug Dunham Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Circle Limit I Doug Dunham Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Circle Limit I Doug Dunham Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Circle Limit I Escher Visualizing Hyperbolic Geometry Dunham Models Outline Euclidean Geometry Circle Limit I Doug Dunham Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Circle Limit III M.C. Escher Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Circle Limit III Doug Dunham Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Circle Limit III Doug Dunham Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Physical Models In hyperbolic geometry, area increases more quickly than in Euclidean geometry. A circle of radius 1 has area larger than π. Many physical models of the hyperbolic plane put “too much area” around vertices or edges of flat shapes. Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry Soccer Balls Euclidean “soccer ball” Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Soccer Balls Spherical soccer ball Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Soccer Balls Hyperbolic “soccer ball” Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry Hyperbolic Fish Katie Mann Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic crochet Gabriele Meyer Visualizing Hyperbolic Geometry Hyperbolic geometry Models Outline Euclidean Geometry Hyperbolic Crochet Daina Taimina Visualizing Hyperbolic Geometry A Spherical Interlude Hyperbolic geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Hyperbolic Blanket Design by Helaman Ferguson, construction by Jeff Weeks Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 3D Printing Designed by Henry Segerman and printed by Shapeways Visualizing Hyperbolic Geometry Models Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry How to play along at home More on Escher and math: Doug Dunham, Bill Casselman, Doris Schattschneider Hyperbolic geometry maze (either Poincaré disk or Beltrami-Klein model): David Madore Instructions to build your own hyperbolic soccer ball: Keith Henderson, Cabinet Magazine Template for hyperbolic fish: Katie Mann (Teaching) How to crochet hyperbolic shapes: Daina Taimina, Institute for Figuring Instructions for other models of hyperbolic space: David Henderson (Experiencing Geometry), Jeff Weeks (Geometry Games) 3D printed model: Henry Segerman, Shapeways Or you could grow some kale Visualizing Hyperbolic Geometry Models
