Complementary Mathematics

Credits

Prerequisites

Geometry 1.

Scientific-disciplinary sector (SSD)

MAT/04 Complementary Mathematics.

Examination method

Oral exam.

Learning
objectives

Acquisition of a historical and critical awareness of the theories and methods of mathematics through a synergistic comparison between Hilbert’s axiomatic approach to Euclidean geometry and projective geometry.

Syllabus

Elements of projective geometry. The notion of cross-ratio. Projective frames. Projectivities between forms of the first kind and between forms of the second kind. Affinities. Similarities. Isometries. Circle inversion. Foundational aspects of geometry: the axiomatic approach from Euclid to Hilbert. Axiomatic foundation of Euclidean plane geometry. The problem of completeness/continuity/categoricity. The Euclidean line and the real numbers. The parallel postulate and its history. Non-Euclidean geometries. Hyperbolic plane geometry. The Klein and Poincaré models. Elliptic geometry and spherical geometry.

Expected learning
outcomes

On completion of the course, students must demonstrate that they

understand and know the differences between Hilbert’s axiomatic approach to Euclidean geometry and projective geometry. They must also know the historical and epistemological aspects of the birth of non-Euclidean geometries;
can apply the knowledge acquired to solving problems and exercises of varying complexity
can communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
can identify the most appropriate methods to analyse and solve a problem relating to the course topics and interpret the results correctly.

Learning outcomes
to be assessed

Clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Complementary Mathematics