Geometry isn't just flat shapes on paper. There are different types, each with unique rules. Euclidean geometry deals with flat surfaces, while non-Euclidean geometries explore curved spaces like spheres and saddle shapes.

These different geometries change how lines, angles, and shapes behave. Understanding them helps us solve real-world problems, from navigation to Einstein's theories. It's like seeing the world through different mathematical lenses.

Euclidean and Non-Euclidean Geometries

Euclidean vs non-Euclidean geometries

• Euclidean geometry
  • Describes planar geometry on a flat surface where parallel lines never intersect
  • Sum of angles in a triangle always equals 180° (equilateral, isosceles, scalene)
  • Squares and other regular polygons (pentagons, hexagons) exist
• Spherical geometry
  • Studies geometry on the surface of a sphere where lines are great circles
  • No parallel lines exist since all lines eventually intersect
  • Sum of angles in a triangle always exceeds 180° (spherical excess)
• Hyperbolic geometry
  • Examines geometry on a saddle-shaped surface or hyperbolic plane (pseudosphere)
  • Infinite number of parallel lines can be drawn through a point not on a given line
  • Sum of angles in a triangle is always less than 180° (angular defect)
  • Regular polygons with 90° angles (squares, regular pentagons) do not exist

Postulates of non-Euclidean geometries

• Euclidean geometry: Euclid's fifth postulate (parallel postulate)
  • For a line and a point not on the line, there is exactly one line through the point parallel to the given line
• Spherical geometry: Parallel postulate does not hold
  • No parallel lines exist on a sphere since all great circles intersect
• Hyperbolic geometry: Parallel postulate replaced by the hyperbolic parallel postulate
  • For a line and a point not on the line, there are at least two distinct lines through the point parallel to the given line
  • Hyperbolic parallel postulate allows for multiple parallels and non-intersecting lines

Historical impact of non-Euclidean geometries

• Challenged long-held belief that Euclidean geometry was the only consistent geometric system
• Opened new avenues for mathematical research and applications (topology, differential geometry)
• Demonstrated independence of the parallel postulate from other Euclidean postulates
• Contributed to development of abstract algebra and concept of mathematical structures (groups, rings, fields)
• Influenced development of Einstein's theory of general relativity (curved spacetime)

Applications of geometric principles

• Use Euclidean geometry principles for problems involving planar surfaces
  1. Apply distance formulas (Pythagorean theorem)
  2. Calculate angle measurements and area (triangles, quadrilaterals, circles)
• Use spherical geometry principles for problems involving spherical surfaces
  1. Calculate great circle distances (navigation, geodesics)
  2. Apply spherical triangle properties (angles, area, spherical excess)
• Use hyperbolic geometry principles for problems involving hyperbolic surfaces
  1. Apply hyperbolic distance formulas (Poincaré disk model, Klein model)
  2. Utilize hyperbolic triangle properties (angles, area, angular defect)
• Recognize appropriate geometry based on problem context
  • Planar surfaces (architectural designs, surveying): Euclidean geometry
  • Spherical surfaces (Earth, celestial navigation): Spherical geometry
  • Hyperbolic surfaces (crochet models, special relativity): Hyperbolic geometry