history of non euclidean geometry
TRANSCRIPT
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
Euclid’s Postulatesfrom Elements, 300BC
1. To draw a straight line from any point to anyother.
2. To produce a finite straight line continuously in astraight line.
3. To describe a circle with any centre and distance.4. That all right angles are equal to each other.5. That, if a straight line falling on two straight lines
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.
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, Playfair’s Axiom: Given a line and a point not on the line, it is
possible 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 possible
to draw a line which meets both sides of the angle.
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 geometry
D’Ambert called it“the scandal of elementary geometry”
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 of non-Euclidean Geometry!!!
Never published his work, he wanted to avoid controversy.
Bolyai’s Strange New World
Gauss talked with Farkas Bolyai about the 5thpostulate.
Farkas told his son Janos, but said don’t “waste onehour's time on that problem”.
Janos wrote daddy in 1823 saying
“I have discovered things so wonderfulthat I was astounded ... out of nothingI have created a strange new world.”
Bolyai’s Strange New World
Bolyai took 2 years to write a 24 page appendixabout it.
After reading it, Gauss told a friend,
“I regard this young geometer Bolyai as agenius of the first order”
Then wrecked Bolyai by telling him that hediscovered this all earlier.
Lobachevsky
Lobachevsky also published a work about replacing the 5thpostulate in 1829.
Published in Russian in a local university publication, no oneknew about it.
Wrote a book, Geometrical investigations on the theory ofparallels in 1840.
Lobachevsky's Parallel Postulate. There exist two linesparallel to a given line through a given point not onthe line.
5th postulate controversy
Bolyai’s appendix Lobachevsky’s book the endorsement of Gauss…
but the mathematical community wasn’t accepting it.
WHY?
5th postulate controversy
Many had spent years trying to prove the 5thpostulate from the other 4. They still clung to thebelief that they could do it.
Euclid was a god. To replace one of his postulateswas blasphemy.
It still wasn’t clear that this new system wasconsistent.
Riemann
Riemann wrote his doctoral dissertationunder Gauss (1851)he reformulated the whole concept of geometry,
now called Riemannian geometry. Instead of axioms involving just points and lines, he
looked 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…
Consistent by Beltrami
Consistent by Beltrami
Beltrami wrote Essay on the interpretationof non-Euclidean geometry
In it, he created a model of 2Dnon-Euclidean geometry within3D Euclidean geometry.
This provided a model for showing theconsistency on non-Euclidean geometry.
Eternity by Klein
Klein finished the work started byBeltrami
Showed there were 3 types of(non-)Euclidean geometry:
1. Hyperbolic Geometry (Bolyai-Lobachevsky-Gauss).2. Elliptic Geometry (Riemann type of
spherical geometry)3. Euclidean geometry.
The Geometries
Comparison of Major Two-Dimensional Geometries. Smith, The Nature of Mathematics, p. 501
Hyperbolic geometry
There are infinitely many lines througha single point which are parallel to agiven line
The Klein Model The Poincare Model
Hyperbolic geometry
Used in Einstein's theory of general relativity If a triangle is constructed out of three rays of light, then in general
the 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.
Hyperbolic geometry
Used in Einstein's theory of general relativity If a triangle is constructed out of three rays of light, then in general
the 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.
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's
theory 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 of
light. This does not contradict the theory of special relativity, since it is space itself that isexpanding.
Frame-dragging, in which a rotating mass "drags along" the space time around it.
http://en.wikipedia.org/wiki/Theory_of_relativity
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 motion
or 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 "stationary"
clock. Length contraction: Objects are measured to be shortened in the direction that they are moving
with respect to the observer. Relativity of simultaneity: two events that appear simultaneous to an observer A will not be
simultaneous to an observer B if B is moving with respect to A. Mass-energy equivalence: E = mc², energy and mass are equivalent and transmutable.
http://en.wikipedia.org/wiki/Theory_of_relativity
Einstein and GPS
GPS can give position, speed, andheading in real-time, accurate towithout 5-10 meters.
To be this accurate, the atomicclocks must be accurate to within20-30 nanoseconds.
Special Relativity predicts that the on-board atomic clocks on the satellitesshould fall behind clocks on theground by about 7 microsecondsper day because of the slower tickingrate due to the time dilation effect of theirrelative motion. http://www.ctre.iastate.edu/educweb/ce352/lec24/gps.htm
Einstein and GPS
Further, the satellites are in orbits high above the Earth, where thecurvature of spacetime due to the Earth's mass is less than it is at theEarth's surface.
A prediction of General Relativity is that clocks closer to a massiveobject will seem to tick more slowly than those located further away.As such, when viewed from the surface of the Earth, the clocks on thesatellites appear to be ticking faster than identical clocks on theground.
A calculation using General Relativity predicts that the clocksin each GPS satellite should get ahead of ground-based clocksby 45 microseconds per day.
Einstein and GPS
If these effects were not properly taken into account, anavigational fix based on the GPS constellationwould be false after only 2 minutes, and errors inglobal positions would continue to accumulate at arate of about 10 kilometers each day!
Elliptic geometry
There are no parallel lines
http://www.joelduffin.com/opensource/globe/ http://gc.kls2.com/
Elliptic geometry
Captain Cook was a mathematician. ‘An Observation of an Eclipse of the Sun at the Island of
Newfoundland, 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 computedthe difference of longitude of the places of observation, taking into account theeffect of parallax, and the the shape of the earth.
Parallax: the apparent shift of an object against the background that is caused by achange in the observer's position.
Projective geometry
Projective Geometry developedindependent of non-Euclideangeometry.In the beginning, mathematicians used
Euclidean geometry for their calculations.Riemann showed it was consistent without
the 5th postulate.