Geometries

Description:

For centuries people were convinced that the only "correct geometry" - geometry related to the real word - is the Euclidean geometry. However in XIX century it was discovered that other geometries may be as "true" as the Euclidean one and, moreover, that they may be related to the real word as well.

The main examples of non-Euclidean geometry are spherical geometry (in particular, geometry of the globe) and hyperbolic geometry (used for example as Minkowski time-space in general relativity). All three geometries (hyperbolic, Euclidean and spherical) share a number of properties, but in the other aspects they are very different. For example, a non-Euclidean triangle is determined (up to isometry) by three angles, i.e. AAA congruence of triangles holds for non-Euclidean triangles. One can consider also other geometries such as Möbius geometry, affine geometry, projective geometry, finite geometry...

In this project we will follow Klein's famous idea that geometry is actually a set considered together with a group of symmetries acting on it. We will think about different groups acting on various spaces and mesuring how symmetric is the space. We will also think how these geometries are related to each other. Various topics may include:

• axiomatic approach to geometry;
• neutral geometry;
• spherical geometry;
• Möbius geometry;
• various models of hyperbolic geometry;
• comparison of spherical, Euclidean and hyperbolic geometries;
• affine and projective geometries;
• tesselations and discrete group actions in different geometries;
• regular polytopes;
• geometric structures on surfaces and manifolds.

Resources:

• A.B.Sossinsky , "Geometries" ----
• Caroline Series, "Hyperbolic geometry" ----

email: Anna Felikson