Download Non-Euclidean Geometry

Euclidean geometry: the only and the first in the past
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”
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
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
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
GAUSS’S NON-EUCLIDEAN
GEOMETRY
New Axiom:
Gauss’s parallel axiom
Gauss’s non
Euclidean
geometry:
based on
contradiction of
5th Euclidean
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
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)
NEW INTERPRETATION OF
LINES
•Cylindrical surface  Euclidean theorems
continue to hold.
•Model of Riemann’s non Euclidean
geometry: spherical surface.
THE APPLICABILITY OF NONEUCLIDEAN GEOMETRIES
Non-Euclidean geometries:
• Applicable?
• More functional?
• More effective?
Impossible answers
Euclidean geometry :
taken for granted
New geometries
• Rejected
• Just mathematical
speculation
• Man’s experience
INITIAL CONCEPTS
1. Point
2. Line
3. Plane
• Cannot be directly defined
• Properties defined by axioms
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
NON-EUCLIDEAN
GEOMETRIES
• Application of non Euclidean geometry:
surveyors’ example
• Relativity theory: path of light in
space-time system
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
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
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