1. Euclidean geometry: the only and the first in the past

2. “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 5 th EUCLID’S AXIOM

3. Playfair’s axiom didn’t satisfy mathematicians 18th century Mathematicians : a new tack Saccheri: demonstration by contradiction 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’”

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 Gauss’s non Euclidean geometry: based on contradiction of 5 th Euclidean axiom New Axiom: Given a line l and a point P. There are infinite non secant and two parallel lines to l through P 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 Gauss’s parallel axiom

7. RIEMANN’S NON-EUCLIDEAN GEOMETRY Georg Friedrich Bernhard Riemann (1826-1866) 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

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 New geometries Rejected Just mathematical speculation Man’s experience Euclidean geometry : taken for granted Non-Euclidean geometries: Applicable? More functional? More effective? Impossible answers

10. INITIAL CONCEPTS 1.Point 2.Line 3.Plane Cannot be directly defined Properties defined by axioms

11. 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 NON-EUCLIDEAN GEOMETRIES

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

13. MATHS Vs SCIENCE Maths doesn’t offer truths Maths can evolve Maths needs axioms Maths works with numbers Maths uses deductive method 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. A special thank to the teachers that have this project made possible Mrs Maria Luisa Pozzi Lolli and Mrs Angela Rambaldi Written by all 5 a C 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