# Matematica | GEOMETRY IV

## Matematica GEOMETRY IV

 0512300013 DIPARTIMENTO DI MATEMATICA MATHEMATICS 2013/2014

 OBBLIGATORIO YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2010 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/03 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 ANNA DI CONCILIO T
Objectives
TRAINING PURPOSES
-KNOWLEDGE AND UNDERSTANDING
THE COURSE OF GEOMETRY IV DEVELOPS IN CONTINUITY WITH THE COURSE OF GEOMETRY III. THE KNOWLEDGE AND UNDERSTANDING OF THREE DIMENSIONAL EUCLIDEAN SPACE IS COMPLETED BY STUDYING EULER’S THEOREMS OF DECOMPOSITION OF ROTATIONS AND CLASSIFICATION OF ISOMETRIES. THE FIVE PLATONIC SOLIDS ARE INTRODUCED AND THEIR GROUPS OF SYMMETRIES ARE DETERMINED BY USING CONCRETE MODELS TOO. REAL PROJECTIVE SPACES OF ANY DIMENSION ARE CONSIDERED, THEIR GRAPHICAL STRUCTURE IS INVESTIGATED AND SEVERAL MODELS ARE CONSTRUCTED. A PARTICULAR ATTENTION IS DEVOTED TO THE REAL PROJECTIVE SPACE OF DIMENSION THREE THOUGHT OF AS THE REALM IN WHICH QUADRICS LIVE. PROJECTIVE, AFFINE, METRIC CLASSIFICATIONS FOR QUADRICS ARE GIVEN. IN THEIR GEOMETRICAL INVESTIGATION TRANSLATION OF SYNTHETIC PROPERTIES IN ANALYTIC PROBLEMS AND VICE VERSA ARE PROPOSED HAVING AS TRAINING PURPOSE THAT OF IMPROVING APPLYING ABILITY. METRIC SPACES ARE THE NATURAL ENVIRONMENT OF GEOMETRIES. THE ULTRAMETRIC SPACES WITH THEIR ASTONISHING PROPERTIES GENERATE A STRONG INTEREST AND EXTRAORDINARY CURIOSITY. FINALLY, THE METRIC GEOMETRY, BASED ON A PURELY METRIC DEFINITION OF STRAIGHT LINE, IS INTRODUCED AND THAT ASSOCIATED WITH THE MINKOWSKI METRIC IS INVESTIGATED BY UNDERLINING ANALOGIES AND DIFFERENCES WITH EUCLIDEAN PLANE GEOMETRY AND ITS BETTER WORKING IN AN URBAN CONTEXT. TOPICS AND METHODS ARE ORGANIZED TO ORIGINATE SCIENTIFIC CURIOSITY AND TO INTEGRATE EXPERIENCE, KNOWLEDGE AND UNDERSTANDING. PROBLEMS ARE SUGGESTED AND TYPICAL MATHEMATICAL APPROACHES ARE CONSIDERED TO IMPROVE THE APPLYING ABLILITY AND CAPACITY OF INVENTION IN DEMONSTRATION. THE TOOLS AND METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. ASSIGNMENTS, HINTS, SUGGESTIONS ARE GIVEN TO IMPROVE APPLYING ABILITY AND INVENTION IN DEMONSTRATION.
AT THE END OF THE COURSE THE STUDENTS SHOULD HAVE
• ACQUIRED THE BASIC INTRODUCED CONCEPTS, AS , FOR EXAMPLE, THE PROPERTIES OF EUCLIDEAN ISOMETRIES, THE GROUPS OF SYMMETRIES OF PLATONIC SOLIDS, THREE-DIMENSIONAL PROJECTIVE SPACE AND QUADRICS IMMERSED IN IT.
• UNDERSTOOD THE RELATIONS AMONG THEM JOINTLY WITH THE USED TECHNIQUES TO STATE THEM, AS THE TRANSLATION OF GEOMETRIC PROBLEMS IN ALGEBRAIC PROBLEMS
• ACQUIRED A DEEP KNOWLEDGE AND UNDERSTANDING OF THE EUCLIDEAN WORLD AND OF OTHER DIFFERENT NON-EUCLIDEAN , NON –ARCHIMEDEAN CONTEXTS AS WELL.
-APPLYING KNOWLEDGE AND UNDERSTANDING
PROBLEMS ARE SUGGESTED AND TYPICAL MATHEMATICAL APPROACHES ARE CONSIDERED TO IMPROVE THE APPLYING ABILITY AND CAPACITY OF INVENTION IN DEMONSTRATION. AT THE END OF THE COURSE THE STUDENTS MUST BE ABLE :
• TO USE EFFICIENTLY THE PROPOSED TECHNIQUES BY THEIR APPLICATION IN CONSTRUCTING SIGNIFICANT EXAMPLES AND IN SOLVING PROBLEMS AND EXERCISES
• TO ANALYZE PROBLEMS, EXPLAIN CONCEPTS AND MAKE PROOFS.
-MAKING JUDGEMENTS
THE STUDENTS SHOULD ENRICH THEIR MAKING JUDGEMENTS BY DEEPENING SUGGESTED MATHEMATICAL ARGUMENTS AND TOOLS. BY THEMSELVES, THEY MUST JOIN UP THE DISCIPLINE WITH ALREADY ACQUIRED KNOWLEDGE, IN PARTICULAR WITH THE COMPUTER-SCIENCE INSTRUMENTS.
-COMMUNICATION SKILLS
ONE OF THE PURPOSE OF THE COURSE IS TO MAKE THE STUDENTS ABLE TO EXPRESS THEIR MINDING IN A CLEAR AND RIGOROUS WAY.
-LEARNING SKILLS
BY THEMSELVES, THE STUDENTS SHOULD BE ABLE TO UNDERSTAND FURTHER AND MORE DIFFICULT GEOMETRICAL CONCEPTS, TO RECOGNIZE TO DIFFERENT GEOMETRIES THE SAME DIGNITY, TO DISCRIMINATE THEIR RELATIVE INVARIANTS AND, FINALLY, TO INTEGRATE EXPERIENCE, KNOWLEDGE AND LEARNING RELATED TO THE COURSE OF GEOMETRY IV.
Prerequisites
FINITE-DIMENSIONAL VECTOR SPACES. LINEAR EQUATIONS. ALGEBRAIC EQUATIONS. BILINEAR AND QUADRATIC FORMS. DIAGONALIZATION. CONICS. AFFINITIES. EUCLIDEAN ISOMETRIES. METRIC SPACES.
Contents
THREE-DIMENSIONAL EUCLIDEAN SPACE. EULER'S THEOREMS. THE FIVE PLATONIC SOLIDS.DUALITY AND EULER-POINCARÉ CHARACTERISTIC. THE GROUPS OF SYMMETRIES OF PLATONIC SOLIDS. REAL PROJECTIVE SPACES AND THEIR GRAPHICAL STRUCTURE. CONSTRUCTION OF MODELS. PROJECTIVE GEOMETRY. DETAILED STUDY OF THE THREE-DIMENSIONAL REAL PROJECTIVE SPACE. QUADRICS AND THEIR PROJECTIVE, AFFINE, METRIC CLASSIFICATIONS. METRIC GEOMETRY. METRIC SPACES. ULTRAMETRIC SPACES, THEIR PROPERTIES AND EXAMPLES. MANHATTAN METRIC AND MINKOWSKI PLANE GEOMETRY.
Teaching Methods
THE CHOISE OF THE FIVE PLATONIC SOLIDS, WHICH ARE CRYSTALS, AND THEIR GROUPS OF SYMMETRIES, INTERESTING FINITE GROUPS, INTRODUCES THE EULER-POINCARÉ CHARACTERISTIC, ONE OF TWO TOPOLOGICAL CHARACTERS CLASSIFYING SURFACES AND AT THE SAME TIME GIVES A BETTER APPROACH TO THE UNDERSTANDING OF THE
NATURE‘S STRUCTURE. THE MAIN PURPOSE OF STUDYING QUADRICS IS THE ANALYTIC RECASTING, BY THE MEANS OF ALREADY ACQUIRED MATHEMATICAL INSTRUMENTS, OF SYNTHETIC PROPERTIES, WHICH FURTHER IMPROVES APPLYING ABILITY. THE METRIC SPACES, WHICH ARE THE NATURAL ENVIRONMENT OF GEOMETRIES, SERVE TO EXPLAIN ONCE AGAIN THAT THERE IS NO ABSOLUTE GEOMETRY, BUT ONLY GEOMETRIES CLOSER THAN SOME OTHER ONES TO THE WORKING CONTEXT AS FOR INSTANCE THE MINKOWSKI GEOMETRY ASSOCIATED WITH THE MANHATTAN OR TAXI-METRIC, MORE APPROPRIATE THAN THE EUCLIDEAN DISTANCE TO DESCRIBE AN URBAN CONTEXT WITH A CARTESIAN STRUCTURE. THE ULTRAMETRIC SPACES WITH THEIR ASTONISHING PROPERTIES, SO FAR FROM THE USUAL VISUAL PERCEPTION, GENERATE A STRONG INTEREST AND EXTRAORDINARY CURIOSITY. THE TOOLS AND METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. ASSIGNMENTS IMPROVING APPLYING ABILITY AND INVENTION IN DEMONSTRATION ARE GIVEN.
Verification of learning
A FINAL ORAL EXAMINATION TO VALUE THE KNOWLEDGE OF THE ARGUMENTS TREATED IN THE COURSE, THE LEVEL OF UNDERSTANDING PERFORMED MATHEMATICAL APPROACHES AND THE COMMUNICATION SKILLS, THE OPENING TO DISCUSSION, THE ORIGINALITY IN DEMONSTRATION.
Texts

[1] G. CASTELNUOVO LEZIONI DI GEOMETRIA ANALITICA MILANO 1931
[2] E. F. KRAUSE TAXICAB GEOMETRY : AN ADVENTURE IN NON-EUCLIDEAN GEOMETRY DOVER
PUBLICATIONS 1986
[3] A. ROBINSON NON-STANDARD ANALYSIS PRINCETON UNIVERSITY PRESS 1974
[4] E. SERNESI GEOMETRIA 1 BOLLATI BORINGHIERI 1989
[5] E. SERNESI GEOMETRIA 2 BOLLATI BORINGHIERI 1994