Euclidean geometry: the only and the first in the past

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 THE 5th EUCLIDS AXIOM

Playfairs axiom didnt satisfy mathematicians18th century Mathematicians : a new tackSaccheri: demonstration by contradictionPlayfairs 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

GAUSSA genius childMany scientific interestsChallenge to Euclids 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

FROM GAUSS TO LOBACHEVSKY & BOLYAIGauss: the first to discover the non Euclidean geometry but unknown Fame to Lobachevsky & Bolyai : first to publish works about non Euclidean geometry

GAUSSS NON-EUCLIDEAN GEOMETRYGausss non Euclidean geometry: based on contradiction of 5th Euclidean axiomNew Axiom:Given a line l and a point P. There are infinite non secant and two parallel lines to l through PCreating new theoremsSum of internal angles in triangle

RIEMANNS NON-EUCLIDEAN GEOMETRYGeorg Friedrich Bernhard Riemann(1826-1866)

19th mathematician's interest in second axiomRiemann: endlessness and infinite length of straight linesAlternative to Euclides parallel axiomSaccheri and Gauss : similarities and differences with Riemann

NEW INTERPRETATION OF LINESCylindrical surface Euclidean theorems continue to hold.Model of Riemanns non Euclidean geometry: spherical surface.

THE APPLICABILITY OF NON-EUCLIDEAN GEOMETRIESNew geometriesRejected Just mathematical speculationMans experienceEuclidean geometry : taken for grantedNon-Euclidean geometries: Applicable?More functional?More effective?Impossible answers

INITIAL CONCEPTSPointLinePlane Cannot be directly defined Properties defined by axioms

All perpendiculars to a line meet in a pointTriangles: sum of angles more than 180Why Greeks didnt hit upon non Euclidean geometriesNON-EUCLIDEAN GEOMETRIES

Application of non Euclidean geometry: surveyors exampleRelativity theory: path of light in space-time systemNON-EUCLIDEAN GEOMETRIES

MATHS Vs SCIENCEMaths doesnt offer truthsMaths can evolveMaths needs axiomsMaths works with numbersMaths uses deductive methodScience uses experimental methodScience works with energies, masses, velocities and forces

IMPLICATIONS FOR OUR CULTURENon Euclidean geometry revolutionized scienceMathematical laws are merely natures approximate descriptionsExperiences confirm Euclidean geometryPhilosophers cannot prove truthsHuman minds limits

A special thank to the teachers that have this project made possible Mrs Maria Luisa Pozzi Lolli and Mrs Angela Rambaldi 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