1. Conceptual bases of mathematics: primitive concepts in arithmetic, geometry, probability; the idea of demonstration; mathematics, philosophy and scientific knowledge.
2. The discrete and the continuous. Euclidean geometry, natural numbers, the real line. Conceptual, epistemological, linguistic and didactic nodes of teaching and learning mathematics.
3. Mathematics in culture: the social and economic role of mathematics, mathematics in education, the community
international mathematics.
2. The discrete and the continuous. Euclidean geometry, natural numbers, the real line. Conceptual, epistemological, linguistic and didactic nodes of teaching and learning mathematics.
3. Mathematics in culture: the social and economic role of mathematics, mathematics in education, the community
international mathematics.
Curriculum
teacher profile teaching materials
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.
V. Nikulin, I. Shafarevich "Geometries and groups" Springer ed.
Fruizione: 20410343 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica L-35 BRUNO ANDREA, SUPINO PAOLA
Programme
1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.
Core Documentation
R. Trudeau: "La rivoluzione non Euclidea" Bollati BoringhieriV. Nikulin, I. Shafarevich "Geometries and groups" Springer ed.
Type of delivery of the course
Standing lecturesType of evaluation
Written test and oral exam teacher profile teaching materials
notes.
Fruizione: 20410343 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica L-35 BRUNO ANDREA, SUPINO PAOLA
Programme
The subject integrates with exercises and insights the activities planned by the colleague owner of MC310, with whom the lessons will be coordinated.Core Documentation
John Stillwell, The Four Pillars of Geometry (Undergraduate Texts in Mathematics) 2005th Edition;notes.
Type of delivery of the course
Traditional in the classroom, unless otherwise indicated due to the COVID emergency.Attendance
The subject integrates with exercises and insights the activities planned by the colleague owner of MC310, with whom the lessons will be coordinated.Type of evaluation
joint test with the course of the co-teacher of MC310. teacher profile teaching materials
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.
V. Nikulin, I. Shafarevich "Geometries and groups" Springer ed.
Fruizione: 20410343 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica L-35 BRUNO ANDREA, SUPINO PAOLA
Programme
1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.
Core Documentation
R. Trudeau: "La rivoluzione non Euclidea" Bollati BoringhieriV. Nikulin, I. Shafarevich "Geometries and groups" Springer ed.
Type of delivery of the course
Standing lecturesType of evaluation
Written test and oral exam teacher profile teaching materials
notes.
Fruizione: 20410343 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica L-35 BRUNO ANDREA, SUPINO PAOLA
Programme
The subject integrates with exercises and insights the activities planned by the colleague owner of MC310, with whom the lessons will be coordinated.Core Documentation
John Stillwell, The Four Pillars of Geometry (Undergraduate Texts in Mathematics) 2005th Edition;notes.
Type of delivery of the course
Traditional in the classroom, unless otherwise indicated due to the COVID emergency.Attendance
The subject integrates with exercises and insights the activities planned by the colleague owner of MC310, with whom the lessons will be coordinated.Type of evaluation
joint test with the course of the co-teacher of MC310. teacher profile teaching materials
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.
V. Nikulin, I. Shafarevich "Geometries and groups" Springer ed.
Fruizione: 20410343 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica L-35 BRUNO ANDREA, SUPINO PAOLA
Programme
1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.
Core Documentation
R. Trudeau: "La rivoluzione non Euclidea" Bollati BoringhieriV. Nikulin, I. Shafarevich "Geometries and groups" Springer ed.
Type of delivery of the course
Standing lecturesType of evaluation
Written test and oral exam teacher profile teaching materials
notes.
Fruizione: 20410343 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica L-35 BRUNO ANDREA, SUPINO PAOLA
Programme
The subject integrates with exercises and insights the activities planned by the colleague owner of MC310, with whom the lessons will be coordinated.Core Documentation
John Stillwell, The Four Pillars of Geometry (Undergraduate Texts in Mathematics) 2005th Edition;notes.
Type of delivery of the course
Traditional in the classroom, unless otherwise indicated due to the COVID emergency.Attendance
The subject integrates with exercises and insights the activities planned by the colleague owner of MC310, with whom the lessons will be coordinated.Type of evaluation
joint test with the course of the co-teacher of MC310. teacher profile teaching materials
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.
V. Nikulin, I. Shafarevich "Geometries and groups" Springer ed.
Fruizione: 20410343 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica L-35 BRUNO ANDREA, SUPINO PAOLA
Programme
1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.
Core Documentation
R. Trudeau: "La rivoluzione non Euclidea" Bollati BoringhieriV. Nikulin, I. Shafarevich "Geometries and groups" Springer ed.
Type of delivery of the course
Standing lecturesType of evaluation
Written test and oral exam teacher profile teaching materials
notes.
Fruizione: 20410343 MC310 - ISTITUZIONI DI MATEMATICHE COMPLEMENTARI in Matematica L-35 BRUNO ANDREA, SUPINO PAOLA
Programme
The subject integrates with exercises and insights the activities planned by the colleague owner of MC310, with whom the lessons will be coordinated.Core Documentation
John Stillwell, The Four Pillars of Geometry (Undergraduate Texts in Mathematics) 2005th Edition;notes.
Type of delivery of the course
Traditional in the classroom, unless otherwise indicated due to the COVID emergency.Attendance
The subject integrates with exercises and insights the activities planned by the colleague owner of MC310, with whom the lessons will be coordinated.Type of evaluation
joint test with the course of the co-teacher of MC310.