Compass-and-straightedge geometric constructions. Constructible numbers. Geometric transformations of the Euclidean plane. Isometries: reflections, translations, rotations, glide reflections. Similarities and their properties. Homotheties. Fixed points. Circle inversions. Homographies of the Riemann sphere and their properties. Euclid’s fifth postulate. Construction of a model of hyperbolic geometry. Notes on elliptic geometry of the sphere.
Maria Dedo’ – Trasformazioni geometriche –1999 – Decibel Zanichelli
Michael Artin– Algebra – 1997 Bollati Boringhieri
R. Courant, H. Robbins - Che cos'e’ la matematica? -2000 Boringhieri
Maria Dedo’ – Trasformazioni geometriche – 1999 – Decibel Zanichelli
Michael Artin– Algebra – 1997 Bollati Boringhieri
R. Courant, H. Robbins - Che cos'e’ la matematica? -2000 Boringhieri
Learning Objectives
Knowledge acquired: to learn about issues related to the historical and logical foundations of geometry.
Competence acquired: development of a flexible scientific attitude useful to critically analyze mathematical problems.
Skills acquired: To use the acquired knowledge to solve theoretical problems; to prepare educational activities for secondary school; to set mathematics in a wider cultural context.
Prerequisites
The knowledge of elementary algebra and geometry acquired in high school and during the first and second year of the bachelor's degree in mathematics are sufficient.
Teaching Methods
CFU: 9
Total hours of the course: 225
Hours reserved to private study and other indivual formative activities: 153
Contact hours for: Lectures (hours): 72 (possibly including 2-4 laboratory hours to introduce educational softwares)
Frequency of lectures, practice and lab: not compulsory, but recommended.
Teaching Tools:
http://donatopertici.wordpress.com/
Office hours: by appointment
Dipartimento di Matematica e Informatica "Ulisse Dini"
Viale Morgagni, 67/a
50134 FIRENZE
Tel: 055 4237125
Email: donato.pertici@unifi.it
Type of Assessment
Oral exam.
Course program
Compass-and-straightedge geometric constructions. Fundamental constructions. Constructible numbers. The three construction problems of antiquity. Regular Polygons. Geometric transformations of the Euclidean plane. Isometries: reflections, translations, rotations, glide reflections. Structure theorem. Fagnano’s Problem. Classical triangle centers. Similarities and their properties. Homotheties. Fixed points. Circle inversions. Homographies of the Riemann sphere and their properties. Euclid’s fifth postulate. Construction of a model of hyperbolic geometry: Poincare’ half-plane (or Poincare’ hyperbolic disk). Intersecting, parallel and ultraparallel lines. Hyperbolic isometries. Notes on elliptic geometry of the sphere and on spherical triangles.