Key Concepts in Non-Euclidean Geometries

Non-Euclidean geometries challenge traditional Euclidean concepts, exploring unique spaces like hyperbolic and elliptic geometries. These geometries reveal fascinating properties of shapes and angles, impacting fields from art to physics, and deepen our understanding of the universe's structure.

1. Hyperbolic geometry

   • Considers a space where the parallel postulate of Euclidean geometry does not hold; through a point not on a line, there are infinitely many lines that do not intersect the original line.
   • Utilizes models such as the Poincaré disk and the hyperboloid model to visualize hyperbolic planes.
   • Features triangles with angle sums less than 180 degrees, leading to unique properties in triangle geometry.
   • Has applications in various fields, including art, architecture, and theoretical physics, particularly in the study of spacetime.

2. Elliptic geometry

   • Based on the idea that there are no parallel lines; all lines eventually intersect, similar to the surface of a sphere.
   • The sum of angles in a triangle exceeds 180 degrees, which leads to different geometric properties compared to Euclidean and hyperbolic geometries.
   • Can be represented on the surface of a sphere, where great circles serve as "lines."
   • Important in the study of global properties of surfaces and has implications in cosmology and the theory of relativity.

3. Spherical geometry

   • A specific case of elliptic geometry where the surface is a sphere, and the geometry is defined by great circles.
   • Involves triangles where the sum of angles is greater than 180 degrees, and the concept of distance is based on arcs of great circles.
   • Useful in navigation, astronomy, and understanding the geometry of the Earth and other celestial bodies.
   • Provides insights into the behavior of shapes and figures in a curved space, contrasting with flat Euclidean geometry.

4. Projective geometry

   • Focuses on properties that remain invariant under projection, emphasizing the relationships between points, lines, and planes.
   • Does not distinguish between parallel and intersecting lines, as all lines are considered to meet at a point at infinity.
   • Useful in computer graphics, art, and perspective drawing, as it models how objects appear from different viewpoints.
   • Forms the foundation for understanding duality in geometry, where points and lines can be interchanged in certain contexts.

5. Riemannian geometry

   • A branch of differential geometry that studies curved surfaces and manifolds, incorporating concepts of distance and angles in non-flat spaces.
   • Utilizes the Riemannian metric to define lengths and angles, allowing for the exploration of geometric properties in higher dimensions.
   • Fundamental in the study of general relativity, where the curvature of spacetime is described using Riemannian geometry.
   • Provides tools for understanding complex structures in mathematics and physics, including the behavior of geodesics and curvature.