I never got beyond calculus in school, and I’ve heard this term thrown around by smart math and science people bit have no clue what it means or why it’s special.
In: Mathematics
Euclid had 5 postulates, things we assume to be true because they are obvious
1. A line can be drawn between two points
2. A line segment can be extended infinitely
3. A circle can be drawn with any radius
4. All right angles are equal
5. Any two lines with a 3rd line crossing them both such that the sum of the interior angles on one side is less than 180°, the lines will eventually intersect on that side.
The first 4 are all pretty straight forward, we can see why those are true because it’s obvious. The problem is the 5th postulate.
Euclidean geometry is when we assume the 5th postulate to be true. This just happens to be when the space we are working in is flat.
If we have space that is curved in some way, the other 4 postulates hold true, but the 5th doesn’t. This is noneuclidean space.
An example of noneuclidean space would be the surface of a sphere. This would be what we call positive curvature. Take the Earth, for example, at the equator, the lines of longitude are parallel, but at the poles, they intersect. In Euclidean space, parallel lines never intersect (which is provable with the use of the 5th postulate).
Mathematicians, including Euclid, hated the 5th postulate because it was so much more complicated than the other 4, so many mathematicians spent years of their lives trying to either prove it or resolve it in some other way.
Latest Answers