What is Non-Euclidean Geometry?
In: 408
Imagine a really large sheet of paper and we’re like ants, standing on the same spot. You go North 3 miles, I go East 4 miles, so now we’re 5 miles apart. That’s Euclidean.
Or we’re standing 1 mile apart (doesn’t matter which way). You go to the North 100 miles, I go to the North 100 miles, we’re still 1 mile apart.
Also there’s no boundary to the paper, it can extend as far North / South / West / East as you need it to be. Those are properties of Euclidean geometry.
But you can imagine in a different world, if you’re standing here, and I start from your position but keep going North, I’ll suddenly come back from the South and go back to your same spot. That’s not Euclidean. In fact, the Earth is like that (because Earth is round).
Or in a yet stranger world, I might come back to the same spot again but also from the North, and suddenly the whole world is mirror-imaged. So if you had a pimple in your left cheek, now I’ll see your pimple to be in your right cheek. This is definitely not Euclidean.
Non-Euclidean geometry is a type of geometry that is different from the familiar geometry we learn in school, called Euclidean geometry. In Euclidean geometry, the rules and theorems are based on the idea that lines are straight and parallel lines never meet. In non-Euclidean geometry, the rules and theorems are different because the lines are not straight and parallel lines can meet.
It’s geometry on surfaces that are not flat. Like spheres, toroids (donut-shaped), and others. Like another person said on a sphere you can have a triangle with all corners being 90°.
There are three types of curvature: Zero curvature, a flat surface like a sheet of paper; Positive curvature, like a ball or sphere; and negative curvature, like a saddle or Pringle chip.
Non-Euclidean geometry describes the second and third kinds I listed, which affects what things like polygons and parallel lines look and act like.
Euclidean geometry is geometry where the shortest distance between two points is a straight line.
Non Euclidean geometry is all geometry where that isn’t true. Now, considering that the definition of a straight line is also “the shortest path between two points” then, in non Euclidean geometry, there is no such thing as a straight line
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).
A long time ago a guy named Euclid basically defined a set of “rules” for geometry, most of which seem fairly obvious at first, like, “there is exactly one way to draw a straight line between any two points,” or “straight lines go on forever”.
However, it turns out these rules are only true because we’re used to doing geometry on flat planes or 3d space. If you get a little creative with how you define space, these rules don’t always hold anymore, at which point we call your type of geometry “non-euclidean”.
One really easy example is the surface of the Earth! The Earth is a sphere, and so we can do things on it that we can’t do in a flat plane. For example, if you started walking in a perfectly straight line, you’d eventually circle the whole planet and arrive back where you started, which violates the rule that straight lines must go on forever. Also, there is no longer always exactly one way to draw a straight line between two points; if you start at the north pole, for example, you could walk in a straight line in any direction you want and you would still end up at the south pole eventually.
I’m not sure how to explain even Euclidean geometry to a 5 year old. Triangles do different things in different geometric spaces. Those triangles are consistent within those spaces. You can solve some problems more easily in those spaces.
Consider this: You’re at the North Pole. You go 10 miles South. You turn 90 degrees to the left, go 1 mile. You turn 90 degrees left, go 10 miles, and you end up back at the North Pole.
You can’t do that in Euclidian geometry. wrapping lines around the outside of a sphere like that (the sphere being the earth) breaks some of the things that Euclidian geometry is based on. For example, it’s possible to draw multiple lines on a globe connecting any two points. You can’t do that in Euclidian geometry.
Instead of geometry on a flat surface, it’s geometry on a curved surface. Things get weird.
Latest Answers