May 11, 2015

- 1.History of Non-EuclideanGeometry http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non- Euclidean_geometry.html http://en.wikipedia.org/wiki/Non-Euclidean_geometry

2. Euclids Postulatesfrom Elements, 300BCTo draw a straight line from any point to any 1.other.To produce a finite straight line continuously in a 2.straight line.To describe a circle with any centre and distance. 3.That all right angles are equal to each other. 4.That, if a straight line falling on two straight lines 5.make the interior angles on the same side lessthan two right angles, if produced indefinitely,meet on that side on which are the angles lessthan the two right angles. 3. What is up with #5? 5. That, if a straight line falling on two straight lines make theinterior angles on the same side less than two right angles, ifproduced indefinitely, meet on that side on which are theangles less than the two right angles. Equivalently, Playfairs Axiom: Given a line and a point not on the line, it ispossible to draw exactly one line through the given point parallelto the line. To each triangle, there exists a similar triangle of arbitrarymagnitude. The sum of the angles of a triangle is equal to two right angles. Through any point in the interior of an angle it is always possibleto draw a line which meets both sides of the angle. 4. Can the 5th Postulate be provenfrom the other 4? Ptolemy tried (~150 BC) Proclus tried (~450BC) Wallis tried (1663) Saccheri tried (1697) This attempt was important, he tried proof by contradiction Legendre tried for 40 years (1800s) Others tried, making the 5th postulate the hot problem in elementary geometryDAmbert called it the scandal of elementary geometry 5. Gauss and his breakthrough Started working on it at age 15 (1792) Still nothing by age 36 Decided the 5th postulate was independent of the other 4. Wondered, what if we allowed 2 linesthrough a single point to BOTH be parallel toa given line The Birth ofnon-Euclidean Geometry!!! Never published his work, he wanted to avoid controversy. 6. Bolyais Strange New World Gauss talked with Farkas Bolyai about the 5th postulate. Farkas told his son Janos, but said dont waste one hour's time on that problem. Janos wrote daddy in 1823 sayingI have discovered things so wonderful that I was astounded ... out of nothing I have created a strange new world. 7. Bolyais Strange New World Bolyai took 2 years to write a 24 page appendix about it. After reading it, Gauss told a friend, I regard this young geometer Bolyai as a genius of the first order Then wrecked Bolyai by telling him that he discovered this all earlier. 8. Lobachevsky Lobachevsky also published a work about replacing the 5th postulate in 1829. Published in Russian in a local university publication, no one knew about it. Wrote a book, Geometrical investigations on the theory of parallels in 1840. Lobachevsky's Parallel Postulate. There exist two lines parallel to a given line through a given point not on the line. 9. 5th postulate controversy Bolyais appendix Lobachevskys book the endorsement of Gauss but the mathematical community wasnt accepting it.WHY? 10. 5th postulate controversy Many had spent years trying to prove the 5th postulate from the other 4. They still clung to the belief that they could do it. Euclid was a god. To replace one of his postulates was blasphemy. It still wasnt clear that this new system was consistent. 11. Riemann Riemann wrote his doctoral dissertation under Gauss (1851) he reformulated the whole concept of geometry, now called Riemannian geometry. Instead of axioms involving just points and lines, helooked at differentiable manifolds (spaces which arelocally similar enough to Euclidean space so that onecan do calculus) whose tangent spaces are innerproduct spaces, where the inner products vary smoothlyfrom point to point. This allows us to define a metric (from the innerproduct), curves, volumes, curvature 12. Consistent by Beltrami Beltrami wrote Essay on the interpretation of non-Euclidean geometry In it, he created a model of 2D non-Euclidean geometry within Consistentby Beltrami3D Euclidean geometry. This provided a model for showing the consistency on non-Euclidean geometry. 13. Eternity by Klein Klein finished the work started by Beltrami Showed there were 3 types of (non-)Euclidean geometry:Hyperbolic Geometry (Bolyai-Lobachevsky-Gauss). 1.Elliptic Geometry (Riemann type of 2.spherical geometry)Euclidean geometry. 3. 14. The Geometries Comparison of Major Two-Dimensional Geometries. Smith, The Nature of Mathematics, p. 501 15. Hyperbolic geometry There are infinitely many lines througha single point which are parallel to agiven line The Klein Model The Poincare Model 16. Hyperbolic geometry Used in Einstein's theory of general relativity If a triangle is constructed out of three rays of light, then in generalthe interior angles do not add up to 180 degrees due to gravity. Arelatively weak gravitational field, such as the Earth's or the sun's, isrepresented by a metric that is approximately, but not exactly,Euclidean. 17. Hyperbolic geometry Used in Einstein's theory of general relativity If a triangle is constructed out of three rays of light, then in generalthe interior angles do not add up to 180 degrees due to gravity. Arelatively weak gravitational field, such as the Earth's or the sun's, isrepresented by a metric that is approximately, but not exactly,Euclidean. 18. Theory of Relativity General relativity is a theory of gravitation Some of the consequences of general relativity are: Time speeds up at higher gravitational potentials.Even rays of light (which are weightless) bend in the presence of a gravitational field.Orbits change in the direction of the axis of a rotating object in a way unexpected in Newton'stheory of gravity. (This has been observed in the orbit of Mercury and in binary pulsars).The Universe is expanding, and the far parts of it are moving away from us faster than the speed oflight. This does not contradict the theory of special relativity, since it is space itself that isexpanding.Frame-dragging, in which a rotating mass quot;drags alongquot; the space time around it. http://en.wikipedia.org/wiki/Theory_of_relativity 19. Theory of Relativity Special relativity is a theory of the structure of spacetime. Special relativity is based on two postulates which are contradictory in classical mechanics:1. The laws of physics are the same for all observers in uniform motion relative to one another(Galileo's principle of relativity), 2. The speed of light in a vacuum is the same for all observers, regardless of their relative motionor of the motion of the source of the light. The resultant theory has many surprising consequences. Some of these are: Time dilation: Moving clocks are measured to tick more slowly than an observer's quot;stationaryquot;clock.Length contraction: Objects are measured to be shortened in the direction that they are movingwith respect to the observer.Relativity of simultaneity: two events that appear simultaneous to an observer A will not besimultaneous to an observer B if B is moving with respect to A.Mass-energy equivalence: E = mc, energy and mass are equivalent and transmutable.http://en.wikipedia.org/wiki/Theory_of_relativity 20. Einstein and GPS GPS can give position, speed, and heading in real-time, accurate to without 5-10 meters. To be this accurate, the atomic clocks must be accurate to within 20-30 nanoseconds. Special Relativity predicts that the on- board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their http://www.ctre.iastate.edu/educweb/ce352/lec24/gps.htm relative motion. 21. Einstein and GPS Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away. As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocksin each GPS satellite should get ahead of ground-based clocksby 45 microseconds per day. 22. Einstein and GPS If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day! 23. Elliptic geometry There are no parallel lineshttp://www.joelduffin.com/opensource/globe/ http://gc.kls2.com/ 24. Elliptic geometry Captain Cook was a mathematician.An Observation of an Eclipse of the Sun at the Island ofNewfoundland, August 5, 1766, with the Longitude of the placeof Observation deduced from it. Cook made an observation of the eclipse in latitude 47 36 19, in Newfoundland. He compared it with an observation at Oxford on the same eclipse, then computed the difference of longitude of the places of observation, taking into account the effect of parallax, and the the shape of the earth. Parallax: the apparent shift of an object against the background that is caused by a change in the observer's position. 25. Projective geometry Projective Geometry developedindependent of non-Euclideangeometry. In the beginning, mathematicians usedEuclidean geometry for their calculations. Riemann showed it was consistent withoutthe 5th postulate.

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