Mystery of non-Euclidean geometry

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Mystery of non-Euclidean geometry

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1richardbsmith
Jul 6, 2009, 2:18pm

It may be that I am misunderstanding a good bit about this.

Could much of the apparent mystery of hyperbolic and elliptical geometry be removed from the start if they were introduced initially as geometry of positive and negative curved shapes?

This is as opposed to the hard to get mystery, that failed attempts to prove the 5th postulate produced abstract and hard to explain mathematical concepts of geometries that are divorced from experience.

If we explain that elliptical geometry is the everyday geometry of a positive curve, and hyperbolic geometry is the everyday geometry of a negative curve, is it possible to start with both types of geometries being part of our normal experience?

Then we can understand that because calculating distance is easier in Euclidean geometry without curves, so we convert from Euclidean to non Euclidean distance by using logarithms.

This seems to me to take some of the mystery out, to place non Euclidean geometry within the realm of experience, and to make easier explanations about how attempted proofs of the 5th postulate led to establishing consistent non Euclidean geometrical systems.

Not being a mathematician, I am probably asking this more as a check to make sure my nascent understanding of these geometries is grounded correctly.

It may also be that I have worked into non Euclidean geometry in a backwards way, and that it is actually taught in schools in ways that remove the initial mystery.

2bertilak
Jul 6, 2009, 2:38pm

I think you are on the right track, but what age students are you thinking of?

I suggest that Non-Euclidean geometry could be introduced in a High School geometry course by talking about triangles on the Earth's surface: why they always have > 180 degrees total.

For beginning students, talking about positive and negative curvature first would be too abstract, IMHO.

3jimroberts
Edited: Jul 6, 2009, 2:53pm

Richard, what you seem to be trying to do is to bring non-E geometries into the realm of our experience by modeling them in Euclidean geometry. Certainly that can be done, that's how we know that they are at least as consistent (i.e. lacking in contradiction) as Euclidean geometry. The situation is however symmetrical. Euclidean geometry can be modeled in other geometries: thus it is no more consistent than they are.

Two dimensional elliptical geometry is not foreign to our experience. It fits our experience of the surface of the Earth better than does the Euclidean plane.

Personally, I prefer to approach them all from projective geometry over the complex field.

ETA: bertilak, I hadn't seen your post before I posted mine. Since "angles of a triangle sum to 180°" is equivalent to the parallel postulate, that's a good approach.

4jimroberts
Jul 6, 2009, 2:56pm

Is Euclid still taught? If so, given that his approach is to present axioms and see what we can deduce, it shouldn't be too much of a stretch to suggest that we could try different axioms and see where they lead.

5richardbsmith
Jul 6, 2009, 3:00pm

I suggest that Non-Euclidean geometry could be introduced in a High School geometry course by talking about triangles on the Earth's surface: why they always have > 180 degrees total.

The student is a 51 year old man (me). And I am thinking of spheres and potato chips as part of the everyday objects that convey these geometries.

My recent introduction did not start with these everyday objects. I had to work through conformal and projective representations and discussion about proving the 5th postulate, before I figured out we are talking about basketballs and potato chips.

It would seem to me helpful to begin this geometry early (high school or so) and not to make it so mysterious by using straight forward objects readily available, like the earth. :)

Really just wanted to make sure I was tracking correctly with this simpler understanding of what the geometry is describing.

Thanks.

6bertilak
Jul 6, 2009, 3:05pm

jimroberts: yes, Euclid is still taught in a watered-down fashion (it is too hard to start over from scratch). For a delightful history, see Euclid and His Modern Rivals by Lewis Carroll, which shows absurd blunders in geometry texts by authors who did not quite get it.

As for axioms, it might help to distinguish postulates which seem undeniable (if you deny a=b & b=c => a=c, you haven't understood what '=' means) from arbitrary postulates such as the parallel postulate.

Properly taught, the parallel postulate should seem pretty weird and dubious. After all, who has ever followed parallel lines 'to infinity and beyond' to check if they stay the same distance apart?

7bertilak
Jul 6, 2009, 3:10pm

richardbsmith: yes, the problem of reorganizing one's mind as an adult (I know about this because I am older than you are)!

Another example: properly taught, special relativity would be derived from Maxwell's equations and the student would think it weird that anybody ever thought that space and time were absolutes. I can't think how to do this at the High School level, though.

8jimroberts
Jul 6, 2009, 3:23pm

#6: bertilak "For a delightful history, see Euclid and His Modern Rivals by Lewis Carroll,"

I've heard of that but never seen it. Is there a digital version available that you know of? (I could just google and try the usual suspects.)

9richardbsmith
Jul 6, 2009, 3:27pm

>6 bertilak: As for axioms, it might help to distinguish postulates which seem undeniable (if you deny a=b & b=c => a=c, you haven't understood what '=' means) from arbitrary postulates such as the parallel postulate.

Shifting subjects, at least for me, I do not understand how the 5th postulate is arbitrary postulate - for plane geometry. I downloaded a paper that I hope has some more detailed discussion about the proof attempts, to help me understand why it cannot be proved, at least in the special case of plane geometry.

But as I said, I am not a mathematician. :)

10bertilak
Jul 6, 2009, 3:36pm

richardbsmith: I started from a pedagogical angle before knowing what category of learner you were asking about. I think it is useful for beginners not to assume that plane geometry is 'obvious'. I was categorizing the parallel postulate as 'weird' because it does not appeal to any human experience like the other Euclidean postulates do.

Given this pedagogical approach, it is then plane geometry and not Non-Euclidean geometry that requires extra assumptions.

The Carroll book is online at http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;s...