What is Non-Euclidean Geometry?
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Euclid’s work generally assumes you are drawing on graph paper or an equivalent, or that you’ve made a shape out of graph paper like an origami cube.
Non-Euclidian geometry looks at things with odd an a-symetriccal shapes like a lampshade, an apple, a mug (with a handle), a globe, and so on.
As an example, think of that BIG screen you see in Mission Control if you’ve ever seen a rocket launch or a movie about rockets/spaceships. You know how the screen is flat and has the rocket doing a sort of wavy sideways “S” shape over the surface?
That “S” shape is actually a straight line…if you draw the route on a globe. But in order to convert it to Euclidian geometry it becomes an S shape and looks kinked up.
In the same way, you could (in theory) take the BIGMAP(TM) down and fold it in such a way that the line would be straight — but then the entire rest of the map would be all warped out of shape.
Two lines of latitude (flatways) around the Earth are parallel, but if you were to rotate them to be at angles with the Equator they would turn into ovals and/or cease to be parallel. The surface of the sphere has a continuous curvature that Euclidian Geometery does not fully acount for; it can figure surface area and radius and all that, but the fact that it is one curved surface rather than the intersection of two flat surfaces requires additional maths if you want to calculate properties of the surface of the sphere (as compared to properties of a cube).
And that’s about as simple as I can make it unfortunately – it (Euclidian) is a simplification of 3 dimensions into the intersection of one or more 2D surfaces. Non-Euclidian Geometry attempts to acccount for the full 3D but it makes things a bit more involved, and for simple problems like following a map from your home to your job, or for building a house those extra (3D) considerations are not necessary.
(In analogy, Newton’s Laws of Motion allow us to send spacecraft to other worlds, no problem; but if we want to build a GPS system we have to account for relativity and that means factoring in Einstein as an extra step).
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