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Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Visualizing Hyperbolic Geometry Evelyn Lamb August 22, 2016 Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry A spherical interlude 1 Euclidean Geometry 2 A spherical interlude 3 Hyperbolic geometry 4 Mathematical Models 5 Physical Models Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Euclid’s Elements Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Euclid’s Elements Built on 23 definitions, 5 axioms, 5 postulates Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Equivalently, 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Equivalently, 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models A few other 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 interior angle sum. 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. The Pythagorean Theorem: a2 + b 2 = c 2 . Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models A few other 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 interior angle sum. 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. The Pythagorean Theorem: a2 + b 2 = c 2 . Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models A few other 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 interior angle sum. 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. The Pythagorean Theorem: a2 + b 2 = c 2 . Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models A few other 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 interior angle sum. 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. The Pythagorean Theorem: a2 + b 2 = c 2 . Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models A few other 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 interior angle sum. 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. The Pythagorean Theorem: a2 + b 2 = c 2 . Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models A few other 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 interior angle sum. 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. The Pythagorean Theorem: a2 + b 2 = c 2 . Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models A few other 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 interior angle sum. 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. The Pythagorean Theorem: a2 + b 2 = c 2 . Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Pythagorean Theorem Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models 2000 years of... Mathematicians (paraphrased): Ugh, I hate that postulate! It is not as self-evident as the other postulates! Can I prove it from the other postulates? Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry Evelyn Lamb Visualizing Hyperbolic Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Alternatives to the Parallel Postulate Playfair’s Axiom: Between a line L and a point P not on L, there is exactly one line through P that do not intersect L. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Alternatives to the Parallel Postulate 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Alternatives to the Parallel Postulate 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry A spherical interlude Evelyn Lamb Visualizing Hyperbolic Geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models 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. Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models 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. Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Statements equivalent to the Euclidean parallel postulate The sum of interior angles in a triangle is 180◦ . All triangles have the same interior angle sum. There is no upper limit to the area of a triangle. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Statements equivalent to the Euclidean parallel postulate The sum of interior angles in a triangle is 180◦ . All triangles have the same interior angle sum. There is no upper limit to the area of a triangle. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Statements equivalent to the elliptic parallel postulate The sum of interior angles in a triangle is 180◦ . The sum of interior angles in a triangle is greater than 180◦ . All triangles have the same interior angle sum. The interior angle sum of a triangle depends on its area. There is no upper limit to the area of a triangle. There is a largest triangle. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Here ends the spherical interlude Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Playfair’s axiom, hyperbolic style Evelyn Lamb Visualizing Hyperbolic Geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Statements equivalent to the hyperbolic parallel postulate The sum of interior angles in a triangle is 180◦ . The sum of interior angles in a triangle is less than 180◦ . All triangles have the same interior angle sum. The interior angle sum of a triangle depends on its area. There is no upper limit to the area of a triangle. There is a largest triangle. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude So what does it look like?! Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude So what does it look like?! Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude So what does it look like?! Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude So what does it look like?! Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude So what does it look like?! Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Poincaré Disk {z = x + iy ∈ C : |z| < 1} Inner product: If (a, b), (c, d) are tangent vectors at z, then 4(ac + bd) (a, b) · (c, d) = (1 − |z|2 )2 Norm: 1 2(a2 + b 2 ) 2 ||(a, b)|| = 1 − |z|2 Metric: 2|z2 − z1 | tanh d(z1 , z2 ) = |1 − z1 z2 | Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical 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. Truth: angles Lies: area, straightness, distance Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Circle Limit I Image: M.C. Escher Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Circle Limit I Image: Doug Dunham Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Circle Limit I Image: Doug Dunham Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Circle Limit I Image: Doug Dunham Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Circle Limit I Image: Doug Dunham Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Circle Limit III Image: M.C. Escher Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Circle Limit III Image: Doug Dunham Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Circle Limit III Image: Doug Dunham Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Tile Yourself Tool: Malin Christersson Background: Firth of Forth Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Upper half-plane model {z = x + iy ∈ C : y > 0} Inner product: If (a, b) and (c, d) are tangent vectors at ac + bd z = x + iy , then (a, b) · (c, d) = . y2 Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Upper half-plane model and Poincaré disk Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry A spherical interlude Now with more lizards Image: M.C. Escher Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Revisiting the hyperbolic parallel postulate equivalents Between 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 interior angles in a triangle is less than 180◦ . The interior angle sum of a triangle depends on its area. There is a largest triangle. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models 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. Evelyn Lamb Visualizing Hyperbolic Geometry Outline Euclidean Geometry A spherical interlude Soccer Balls Euclidean soccer ball Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Soccer Balls Spherical soccer ball Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Soccer Balls Hyperbolic soccer ball Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Soccer Balls Truths: area, distance Lie: angles Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic Fish Image by Katie Mann Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Hyperbolic crochet Images by Gabriele Meyer Evelyn Lamb Visualizing Hyperbolic Geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic Crochet Image by Daina Taimina Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Hyperbolic Blanket Design by Helaman Ferguson, construction by Jeff Weeks Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic Plane Skirt Andrea Hawksley Evelyn Lamb Visualizing Hyperbolic Geometry Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models 3D Printing Designed by Henry Segerman and printed by Shape ways Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models 3D Printing Designed by Henry Segerman and printed by Shapeways Evelyn Lamb Visualizing Hyperbolic Geometry Physical Models Outline Euclidean Geometry Edibles Evelyn Lamb Visualizing Hyperbolic Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models Outline Euclidean Geometry A spherical interlude Hyperbolic geometry Mathematical Models Physical Models How to play along at home Escher and math: Doug Dunham, Bill Casselman, Doris Schattschneider Picture tiling tool: Malin Christersson Instructions for building your own hyperbolic soccer ball: Keith Henderson, Cabinet Magazine How to crochet hyperbolic things: Daina Taimina, Institute for Figuring 3D printed models: Henry Segerman, Shapeways Instructions for other models of hyperbolic space: David Henderson (Experiencing Geometry), Jeff Weeks (Geometry Games) How to grow kale: almanac.com Hyperbolic geometry maze: David Madore HyperRogue game Evelyn Lamb Visualizing Hyperbolic Geometry