Question Corner and Discussion Area
I have recently been studying Euclid (the "father" of geometry), and was amazed to find out about the existence of a non-Euclidean geometry. Being as curious as I am, I would like to
Question Corner and Discussion Area
I have recently been studying Euclid (the "father" of geometry), and was amazed to find out about the existence of a non-Euclidean geometry. Being as curious as I am, I would like to know about non-Euclidean geometry. Thanks!!!All of Euclidean geometry can be deduced from just a few properties (called "axioms") of points and lines. With one exception (which I will describe below), these properties are all very basic and self-evident things like "for every pair of distinct points, there is exactly one line containing both of them".
This approach doesn't require you to get into a philosophical definition of what a "point" or a "line" actually is. You could attach those labels to any concepts you like, and as long as those concepts satisfy the axioms, then all of the theorems of geometry are guaranteed to be true (because the theorems are deducible purely from the axioms without requiring any further knowledge of what "point" or "line" means).
Although most of the axioms are extremely basic and self-evident, one is less so. It says (roughly) that if you draw two lines each at ninety degrees to a third line, then those two lines are parallel and never intersect. This statement, called Euclid's Parallel Postulate, seems more like a theorem than an obvious and self-evident property, and for centuries people tried and failed to prove it from the other axioms.
Eventually it was discovered that it is independent of the other axioms, in the sense that it is logically self-consistent to have some things called "lines" and other things called "points" which satisfy the other axioms but don't satisfy the parallel postulate. Any such a collection of things is called a non-Euclidean geometry.
There are many examples. Most concretely, if you do geometry on a curved surface instead of on a flat plane (where now "line" refers to the shortest path between two points, which obviously will not be straight if you are on a curved surface), you typically end up with a non-Euclidean geometry.
We already have on this web site a detailed description of one kind of non-Euclidean geometry called projective geometry. You can refer to that description for more details. It also includes a more precise description of Euclid's parallel postulate and the other axioms of geometry.
As a followup to my earlier question on non-Euclidean geometries, I would like to know how many types of non-Euclidean geometries there are. If possible, I'd also like to know a bit about each, and some source where I can find information about non-Euclidean geometries. Thanks a bundle!!!!!!!In one sense there are infinitely many types. If you take a surface, then any way in which you choose to bend it will give you a different geometry. You could take any surface, for example, a sphere, or an ellipsoid (football-shaped surface), or a hyperboloid (hourglass-shaped surface), or any variation of any of these by bending in any way you like, and you will get a different geometry (the "lines" will be the paths of shortest distance between two points, even though they're not straight lines in the usual sense of the word). Hyperbolic geometry is probably the most important of these.
You can also extend geometric concepts to any number of dimensions.
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