Geometry // Domain_02
NON-EUCLIDEAN
Curved Space. Breaking the Fifth Postulate.
The Fifth Postulate Problem
Euclid said: "If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough."
Translation: Parallel lines never meet.
Non-Euclidean Geometry asks: "What if they do?"
Violation Detected: Axiom 5
Elliptic
Curvature
Positive (K > 0)
Parallel Lines
There are NO parallel lines. All lines eventually meet (like lines of longitude meeting at the poles).
[Image of spherical geometry parallel lines]Triangles
Angles sum to > 180°. Triangles look "fat" or bloated.
Hyperbolic
Curvature
Negative (K < 0)
Parallel Lines
There are INFINITE parallel lines through a single point. Space expands too fast for them to touch.
Triangles
Angles sum to < 180°. Triangles look "thin" or pinched.
Escher's Infinite Limit