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March 31st, 2017, 12:51 AM   #1
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Spherical geometry axioms

Hi everyone,

I would be interested in finding different axiomatic systems for spherical geometry.

By spherical geometry, I mean a system that treats the sphere as a plane, NOT the Euclidean body. Although, of course, the axioms should be compatible with the Euclidean sphere.

The basic idea is that the lines correspond to the Euclidean great circles of the sphere.

I have found a good text by M. Bolin from 2003: Exploration of Spherical Geometry, but it doesn't list an axiomatic system.

The book by W. Meyer: Geometry and its Applications gives a system, but the axioms are written in such a way that the content is merely described, instead of given in a formal mathematical form. Also the book uses an axiomatic system for Euclidean geometry that was invented for easier studying in the 1960s, so I wonder whether the system for spherical geometry uses the same kind of approach. A Hilbert-kind of approach would be preferred.

There is an article "Spherical Geometry" (Wilson 1904) that gives a system, but there I wonder whether it is complete or just a beginning of something.

All in all, I find it difficult to find stuff about it, whereas it is easier to find stuff about hyperbolic and elliptical geometry, and also more of a Euclidean based approach (for example spherical trigonometry). Could it be that the geometry is not that relevant for applications as the Euclidean, hyperbolic and elliptical geometries are?

The best thing would be to find a system that resembles Hilbert's axioms for Euclidean geometry or something else that is very much following the tradition started in ancient Greece.

Also, if there is an article about the development of different axiomatic systems for spherical geometry, that would be awesome.

But any recommendations will be gladly recieved.

Thanks in advance!


Lasse Candé
Helsinki, Finland
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Last edited by skipjack; March 31st, 2017 at 04:15 PM.
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March 31st, 2017, 05:16 PM   #2
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let me google that for you

The Three Geometries - EscherMath
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April 13th, 2017, 08:05 AM   #3
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Thanks for your reply!

Unfortunately this task isn't this easy, at least if you don't get very lucky with your google-jutsu. This can be seen if you check the stuff that I referred to. (Meaning I already have two axiomatic systems and I'm wondering if there is something more similar to Hilbert's system.)

According to that website the axioms for Euclidean geometry are:
Quote:
1. Any two points can be joined by a straight line. (This line is unique given that the points are distinct)
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
If I remember correctly these are the ones in Euclid's Elements. Later it became obvious, these are not enough to construct the Euclidean geometry. You need more. Hilbert's system is said to be one that follows the Euclidean mindset nicely but is also complete and independent.

I'm searching for something like that for spherical geometry. For spherical geometry the webpage gives five very similar axioms:

Quote:
1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. There are NO parallel lines.
This is very lacking and also a strange way to axiomatize spherical geometry. First of all, it doesn't have anything to say about the way how the "straight lines" are "round", using a Euclidean metaphor. It also doesn't say anything about the possibility that two points could determine more than one line. (The poles.)

Also, if you take a line-segment, that has the length of the distance between the poles, you will not construct a circle, but the opposite pole instead, in a funny way. (At least if you don't define a circle in such a way that every point is also a circle.)

Hilbert needed 20 axioms for constructing Euclidean geometry. My guess is that you then need around 20 axioms to construct spherical geometry too, if the approach is similar.

This stuff is very difficult to google successfully, so I'm asking if someone who knows geometry more in depth happens to know something about this and refer to my original post.

Thank you once again!

Last edited by skipjack; August 7th, 2017 at 06:32 PM.
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August 7th, 2017, 04:23 PM   #4
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TL;DR The only source for a rigorous and CORRECT axiomatization I have found is the following:

Quote:
Borsuk, Karol, and Wanda Szmielew, Foundations of geometry: Euclidean and Bolyai-Lobachevskian geometry. Projective geometry, New York: Interscience Publishers, 1960.
Another keyword to search for might be "double elliptic geometry" or "elliptic geometry".

(To the best of my understanding, spherical geometry is a subset of elliptic geometry: https://en.wikipedia.org/wiki/Spherical_geometry
Quote:
Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry.
That being said, spherical geometry is certainly very different from the geometry of the projective plane, although to the best of my understanding the projective plane is also studied in elliptic geometry: https://en.wikipedia.org/wiki/Ellipt...iptic_geometry Anyway my point being I wouldn't recommend disregarding elliptic geometry entirely if you are looking for axiom systems -- they could be relevant to spherical geometry.)

Basically, spherical geometry isn't compatible with either "absolute geometry" or even the more basic/general "betweenness geometry", two axioms systems which actually have been fleshed out rigorously and correctly (unlike Euclid's, which are incorrect), so any rigorous account of spherical geometry has to be truly from "the ground up".

And mathematicians are lazy and don't want to bother to spend the time and effort necessary to correctly and thoroughly understand a basic mathematical object. Instead the usual approach is just to consider spheres embedded inside of Euclidean 3-space, and then use Riemannian geometry to study and make conclusions about the sphere.

So if you want to learn about spherical geometry rigorously and correctly, it seems like your only two options are: (1) spend a bunch of time learning about metamathematics and formal logic so you can understand the works of Szmielew and her collaborators; (2) spend less time learning Riemannian geometry, which will also allow you to understand a lot of other spaces besides spheres, but at the cost of also learning a bunch of random stuff (analytic geometry) which actually isn't necessary to understand the structure, but provides a convenient enough framework to think about the subject mindlessly and do more work than necessary.

Most people choose (2) (I am definitely leaning towards that direction as well, despite my intense dissatisfaction at the extremely analytic and un-synthetic nature of Riemannian geometry -- metric geometry is somewhat of a palliative, but metric geometers also seem to have the habit of getting distracted by and proving pointless results about pointless spaces and introducing unnecessary complications, and thinking too much in terms of constructions and implementations rather than properties, but I digress).

If you're really fascinated by spherical geometry though and want to do a PhD in mathematics about it or something, then combining (1) or (2) would be a worthwhile approach, although it would probably take decades for the approach to bear fruit, and people around you will be criticizing you for not having published papers yet, since they don't care about the quality of what you publish nor about the importance of the problem you tackle, just that you publish something about some subject and often.

-----------------

Euclid's axiomatization of geometry was neither correct nor rigorous. For one thing, SAS is a postulate, not a theorem. Second, Pasch's axiom was used implicitly by Euclid, but never stated.

https://en.wikipedia.org/wiki/Pasch%27s_axiom

Euclid did a great thing to birth the subject, but propagating the myth that he gave either a correct or complete account of the subject encourages people to think lazily and uncritically, which is dangerous and arguably morally wrong. Even today most mathematicians, being overly reliant on Cartesian (analytic) geometry, have little understanding of the synthetic geometry of the Euclidean plane, a problem which is not helped by those few who do bother to study it, since they don't make an effort to structure the theory in a manner which is well-organized or easily understood.

The website quoted in the previous reply, in perpetuating the myth that Euclid's axioms can be trusted, is doing the public a great disservice.

Repeating the inaccurate information presented on that website is also a great disservice. The OP's question is NOT a question answered by a simple Google search -- most of Google's results are entirely extraneous to the subject of a rigorous axiomatization of spherical geometry.

To be fair to the original answerer though, Burago Burago Ivanov (unlike Hartshorne if the book review is to be believed) also seem to make the mistake of thinking that Euclid's postulates are actually a valid axiomatization of Euclidean geometry (they aren't) and that one can just replace one of the axioms of Euclidean geometry to get spherical geometry (one can't). They also mention/agree with me, however, that there are hardly any books discussing the subject. (I believe the reference to the axiomatic system of spherical geometry in Chapter 5, maybe section 5.3 but I don't quite remember.)

Spherical geometry is NOT an absolute geometry. ONLY Euclidean and hyperpolic geometries are.
https://en.wikipedia.org/wiki/Absolute_geometry

I have been looking personally for a rigorous treatment of spherical geometry along the lines of Tarski's axioms or Hilbert's axioms.

https://ncatlab.org/nlab/show/Euclid...y#TarskiAxioms
https://en.wikipedia.org/wiki/Tarski%27s_axioms

The closest I have been able to find which treats the issue seriously (unlike the original reply to your post) is this website.

Geometry: Euclid and Beyond

I quote:

Quote:
There is an axiom system for spherical geometry that is in the spirit of Hilbert's axioms (see for example, [Bor]), in this context spherical geometry is usually called double-elliptic geometry. However, the axiomatization has seemed not to be useful. However, there are many popular accounts that attempt to distinguish between Euclidean and spherical geometries on the basis of Euclid's Fifth (or Parallel) Postulate, which states:
If a straight line intersecting two straight lines makes the interior angles on the same side less than two right angles, then the two lines (if extended indefinitely) will meet on that side on which the angles are less than two right angles.
It can be easily checked that this Fifth Postulate is provable on the sphere. This also shows that, contrary to many accounts, Euclid's Fifth Postulate is not equivalent to the Playfair (Parallel) Postulate that is familiar from high school geometry:
Given a line and a point not on the line there is one and only one line through the point which is parallel to the given line.
The book under review does not fall into this trap of trying to distinguish spherical geometry through parallel postulates. Since spherical geometry does not fit easily into the Euclidean (Hilbert) axiomatic structure of this book, the author defines spherical geometry as the geometry of a sphere in Euclidean 3-space. Spherical geometry is relegated to Exercises 34.13 and 45.3-8 where it is claimed (and the reader is asked to show) that most of the first 26 propositions of the Elements are valid in spherical geometry, if one restricts spherical triangles to be those contained in a (open) hemisphere. However, in Exercise 45.8, the reader will find that in proving Angle-Angle-Side congruence one needs to further restrict spherical triangles to have each side less than of a great circle.
The only book quoted as giving a rigorous account of the axioms of spherical geometry:

Quote:
Borsuk, Karol, and Wanda Szmielew, Foundations of geometry: Euclidean and Bolyai-Lobachevskian geometry. Projective geometry, New York: Interscience Publishers, 1960.
is very difficult to find. Moreover, from what I have read of the author's other works (in particular Szmielew's joint work with Tarski), they do not make much effort to make the subject understandable, or to organize the theory into any coherent shape. (In particular I mean Szmielew and Tarski's book, Metamathematischen Methoden in der Geometrie -- there is no reason why one needs to study formal logic or metamathematics just to have a correct understanding of the basics of plane geometry. The insistence on writing everything with first-order logic only is absurd, as well as always using symbols rather than words, and never bothering to explain the ideas behind the theorems or proofs, instead just letting them sit their without any motivation. If one changes the setting to a complete metric space, and uses the notion of betweenness in the theory of convex metric spaces, the 11 axioms from Tarski's treatment reduce to 6, and one gets only the Euclidean plane that one wants, i.e. the one which is a manifold, and not degenerate objects coordinatized by the real algebraic or computable numbers or some other crappy real closed field.)

Tarski's axioms are better than Hilbert's, because they are much more concise and use only one primitive notion, betweenness, which can be applied to other geometries as well -- there is a whole field of betweenness geometry, and another reason why spherical geometry is difficult to axiomatize apparently is that it doesn't even satisfy the axioms of betweenness geometry either. See here:

http://academic.brcc.edu/ryanl/modul...20Geometry.pdf

Quote:
For a geometry that is totally weird when you look at it from a formal axiomatic perspective
(betweenness goes totally off the rails, for one thing – don’t ask!), it’s extremely natural from a real
world perspective.
Pasch's work in the late 19th century, including the discovery of his eponymous axiom, was central to the beginning of the study of betweenness geometry, necessary to understand Euclidean or hyperbolic geometry accurately, although my understanding was that the first thorough account of the subject was given by Veblen in the 20th century. I remember having seen at least one or two papers by Tarski and a collaborator on the subject, although they also possess the incomprehensibility and unnecessary pedantic fixation on unimportant issues like first-order logic typical of Tarski.
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Last edited by skipjack; August 7th, 2017 at 06:53 PM.
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August 7th, 2017, 11:56 PM   #5
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Thank you gentlemen for an interesting and enlightening thread.
I see that I have much to look and mull over that is currently beyond my knowledge and experience.
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August 8th, 2017, 06:06 PM   #6
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Thanks a lot horatiodudacus!

That will probably help me. I will dig into all of that!
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August 9th, 2017, 07:43 PM   #7
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Quote:
Originally Posted by studiot View Post
Thank you gentlemen for an interesting and enlightening thread.
I see that I have much to look and mull over that is currently beyond my knowledge and experience.
haha I appreciate it, although this gentleman is actually a girl!
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August 10th, 2017, 12:28 AM   #8
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Quote:
Originally Posted by horatiodudacus View Post
haha I appreciate it, although this gentleman is actually a girl!
Even better.

To rephrase the words of a famous mathematician of the past, Heaviside.

"Gentlemen, shall I refuse to communicate because I can't tell the difference?"

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