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Euclidean geometry: the only and the first in the past

NON-EUCLIDEAN GEOMETRIES. Euclidean geometry: the only and the first in the past. THE 5 th EUCLID’S AXIOM.

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Euclidean geometry: the only and the first in the past

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  1. NON-EUCLIDEAN GEOMETRIES Euclidean geometry: the only and the first in the past

  2. THE 5th EUCLID’S AXIOM “If two lines m and l meet a third line n, so as to make the sum of angles 1 and 2 less than 180°, then the lines m and l meet on that side of the line n on which the angles 1 and 2 lie. If the sum is 180° then m and l are parallel”

  3. Playfair’s axiom:”given a line g and a point P not on that line, there is one and only one line g’ on the plane of P and g which passes through P and does not meet g’” Playfair’s axiom didn’t satisfy mathematicians 18th century Mathematicians : a new tack Saccheri: demonstration by contradiction

  4. GAUSS A genius child Many scientific interests Challenge to Euclid’s axiom: “ given a point P outside a line l there are more than one parallel line through P ” …a new kind of geometry! Fear of publishing studies After his death  his work discovered

  5. FROM GAUSS TO LOBACHEVSKY & BOLYAI • Gauss: the first to discover the non Euclidean geometry but unknown • Fame to Lobachevsky & Bolyai : first to publish works about non Euclidean geometry

  6. GAUSS’S NON-EUCLIDEAN GEOMETRY New Axiom: Given a line l and a point P. There are infinite non secant and two parallel lines to l through P Gauss’sparallel axiom Gauss’s non Euclidean geometry: based on contradiction of 5th Euclidean axiom Creating new theorems • Sum of internal angles in triangle <180° • Triangle area depends on sum of its angles • If two triangles have equal angles respectively, they are congruent • Angle A depends on distance l-P

  7. RIEMANN’S NON-EUCLIDEAN GEOMETRY 19th mathematician's interest in second axiom Riemann: endlessness and infinite length of straight lines Alternative to Euclide’s parallel axiom Saccheri and Gauss : similarities and differences with Riemann Georg Friedrich Bernhard Riemann (1826-1866)

  8. NEW INTERPRETATION OF LINES Cylindrical surface  Euclidean theorems continue to hold. Model of Riemann’s non Euclidean geometry: spherical surface.

  9. THE APPLICABILITY OF NON-EUCLIDEAN GEOMETRIES Non-Euclidean geometries: • Applicable? • More functional? • More effective? Impossible answers New geometries • Rejected • Just mathematical speculation • Man’s experience Euclidean geometry : taken for granted

  10. INITIAL CONCEPTS • Point • Line • Plane • Cannot be directly defined • Properties defined by axioms

  11. NON-EUCLIDEAN GEOMETRIES • All perpendiculars to a line meet in a point • Triangles: sum of angles more than 180° • Why Greeks didn’t hit upon non Euclidean geometries

  12. NON-EUCLIDEAN GEOMETRIES Application of non Euclidean geometry: surveyors’ example Relativity theory: path of light in space-time system

  13. Maths doesn’t offer truths Maths can evolve Maths needs axioms Maths works with numbers Maths uses deductive method MATHS Vs SCIENCE Science uses experimental method Science works with energies, masses, velocities and forces

  14. IMPLICATIONS FOR OUR CULTURE Non Euclidean geometry revolutionized science Mathematical laws are merely nature’s approximate descriptions Experiences confirm Euclidean geometry Philosophers cannot prove truths Human mind’s limits

  15. CREDITS: Written by all 5aC students: M. Alberghini, L. Barbieri, R. Bellini, L. Bortolamasi, L. Bovini, M. Briamo, S. A. Brundisini, V. Ceccarelli, G. Cervellati,M. Ignesti, S. Milani, E. Nicotera, L. Porcarelli, S. Quadretti, S. Romano, M. Sturniolo, G. Tarozzi, G. S. Virgallito, M. Zanotti, F. Zoni Slideshow by: Lorenzo Bovini, Marco Sturniolo, Giulia Tarozzi A special thank to the teachers that have this project made possible Mrs Maria Luisa Pozzi Lolli and Mrs Angela Rambaldi

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