Non-Euclidean Geometry
Euclid's 2,000-Year Old Mystery
For over 2,000 years, mathematicians treated Euclid’s Elements as the absolute truth. However, one specific rule, 'the Fifth Postulate' (or The Parallel Postulate) felt different. Many tried to prove it was a 'theorem' derived from the first four postulates, but they all failed.
In 1820-1830s, mathematicians like Janos Bolyai, Carl Friedrich Gauss and Nikolai Lobachevsky asked:
"What if we just assume the Fifth Postulate is wrong? We have already tested Euclid’s five postulates on flat paper. Let's leave the flat world."
Test Euclid's Postulates on New Surfaces
Imagine your 'plane' is no longer a flat sheet of paper. Use a ball (sphere) or observe the sphere model below.
- Use a piece of string. Can you form a straight line segment joining any two points on the sphere? (or try it on the model below)
- If you keep forming that straight line, does it go on forever without touching itself, or does it intersect?
- Does Euclid’s 1st Postulate (joining two points with a line) still work the same way?
Let's test Euclid's Fifth Postulate:
"Given a line and a point not on that line, there is exactly one parallel line through that point."
- Imagine the Great Circle (Equator/ longitude) is your line. Pick a point (i.e., North Pole). Can you form any line through that point that never crosses the Equator? (or try it on the model above)
- Repeat it on a hemisphere shape (or try it on the model below: 3D hemisphere/ Poincaré disk model)
- Through a point not on a line, how many lines can you draw that never touch the original line?
Because these surfaces do not satisfy Euclid’s Parallel Postulate, we call this Non-Euclidean Geometry.