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Non-Euclidean Geometry

Asked by Brent Potteiger on April 5, 1997:
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.

Followup question by Brent Potteiger on April 10, 1997:
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.

Mathematically, any function which assigns a non-negative number to each pair of points determines a geometry: you just consider the distance between two points P and Q to be the whatever the function value f(P,Q) is. There are a few restrictions on the function, like the fact that the distance between a point and itself has to be zero, the distance between two distinct points has to be greater than zero, the distance from P to R has to be less than the sum of the distance from P to Q and the distance from Q to R, and the distance between two points must not change too much if you move one of the points slightly. However, there are still infinitely many functions that satisfy these properties, so there are infinitely many such geometries. Geometries defined in this way are called Riemannian Geometries.

I don't know offhand of any reference that would be suitable for your level and that you'd be able to find in your school library. If I find one I will let you know.

Followup question by Noah Potvin, student, East Meadow high School on November 26, 1997:
Hi, I'm back. I know you mentioned axioms on your website but what exactly are they? What is the fifth postulate and also please explain absolute geometry?I would also appreciate if you would explain hyperbolic geometry too. I know it's a bundle of questions but thanks for answering them.
An axiom is something which is held to be true as the basis or starting point for a logical argument, rather than something that is proven true by the argument. You can't prove things from nothing; you have to start with certain basic assumptions then derive the desired consequences from those assumptions.

Euclid's axioms were of two types: five he called "axioms", being basic principles of logic and reasoning (such as, if two things are each equal to a third thing then they are equal to each other), and five he called "postulates", being basic principles of geometry. His five postulates were (paraphrased in modern language):

  1. for every pair of points, it is possible to construct a line segment joining them;

  2. every line segment can be extended indefinitely in a straight line in either direction;

  3. for every pair of points, it is possible to construct a circle centred at one point and passing through the other;

  4. any two line segments emanating from the same point determine an angle. There's a definition of what it means for this angle to be a "right angle", and any two right angles are equal to each other.

  5. the fifth postulate is much less self-evident. For a description, see our web page

It is impossible to prove things about points and lines without first knowing what points and lines are--or at least, knowing something about the nature of points and lines, even if the ambiguous philosophical question of what a point or a line actually is remains unaswered. Therefore, certain basic characteristics need to be assumed without proof. These characteristics are given by Euclid's postulates. (Modern mathematicians would tend to call them "axioms" rather than "postulates", for as language has evolved over time the word "axiom" in mathematics has come to mean what "postulate" used to mean).

If it seems unsatisfying to think of having to assume certain things without proof before you can prove other things, you can think of it in the following alternative way: the postulates give a definition of what one means by the words "point" and "line". These words mean any things that behave in the manner described by the postulates. Theorems in geometry then take the form

if you have any collection of things called "points" and any other collection of things called "lines", and if they have the properties given in the postulates, then they will also have the property that . . .
(followed by whatever the theorem in question is). In other words, the theorems in geometry are all logical consequences of the postulates, and don't depend on anything else about the nature of what points and lines are.

I am unfamiliar with what you mean by "absolute geometry."

Hyperbolic geometry is geometry in which, instead of Euclid's fifth postulate (which, roughly speaking, says that two lines starting out in the same direction will remain parallel to each other, staying just as far away from each other as they were at the start), has the property that lines starting out in the same direction will get further and further away from each other. The opposite extreme, where lines starting out in the same direction get closer to each other and eventually meet, is called elliptic geometry.

Hyperbolic geometry is similar in spirit (though different in detail) to geometry on a hyperboloid, an hourglass-shaped surface given by an equation like  (IMAGE) (where a, b, c > 0). If you start with two side-by-side vertical lines on the neck of this hourglass, and continue them as straight as you can while staying on the surface, the lines will end up getting farther and farther apart, as indicated below. That's the same kind of thing that happens in hyperbolic geometry.


Followup question by Kyle Shau, student, Conard High School on January 6, 1998:
In Euclidean Geometry, isn't a line means a straight line and a plane means a flat plane? What are the definitions of "LINE," "PLANE," and "SPACE" in both Euclidean Geometry and Non-Euclidean Geometry?
The question is, what do you mean by "straight line"? To put it more fundamentally, what do you mean by "point"? The early Greeks realized that they didn't have any satisfying answer to that question; typical attempts went something like "a point is something that is infinitely small", but how do you know such things actually exist when you can't actually draw them?

That is why the axiomatic approach was adopted: rather than defining points and lines by some philosophical definition of what a "straight line" actually is, they are defined by what their properties are.

Under any axiomatic approach, be it Euclidean or non-Euclidean, a "geometry" is defined to be any set of things together with any collection of subsets of this set, that satisfies various properties. The "points" of the geometry are the elements of the set, and the "lines" of the geometry are the subsets.

Those are the definitions of "points" and "lines" in any form of axiomatic geometry.

Your intuitive notion of "straight line" and "flat plane" cannot be precisely defined (after all, if you draw a so-called "straight line" on a piece of paper, it really has some thickness and some ink smudges, not to mention the spaces between the atoms and molecules that make up the ink and the paper). So instead, one either opts for the axiomatic approach, or else moves to analytic geometry, whereby a point on the plane is defined to mean an ordered pair of numbers, and a line is defined to be a set of numbers (x,y) satisfying and equation of the form ax + by = c where a, b, and c are constants.

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