March 31st, 2017, 01:51 AM  #1 
Newbie Joined: Mar 2017 From: Helsinki, Finland Posts: 3 Thanks: 2  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 Hilbertkind 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 Last edited by skipjack; March 31st, 2017 at 05:15 PM. 
March 31st, 2017, 06:16 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,763 Thanks: 905  
April 13th, 2017, 09:05 AM  #3  
Newbie Joined: Mar 2017 From: Helsinki, Finland Posts: 3 Thanks: 2 
Thanks for your reply! Unfortunately this task isn't this easy, at least if you don't get very lucky with your googlejutsu. 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:
I'm searching for something like that for spherical geometry. For spherical geometry the webpage gives five very similar axioms: Quote:
Also, if you take a linesegment, 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 07:32 PM.  
August 7th, 2017, 05:23 PM  #4  
Newbie Joined: Aug 2017 From: Everywhere, all at once Posts: 3 Thanks: 1  TL;DR The only source for a rigorous and CORRECT axiomatization I have found is the following: Quote:
(To the best of my understanding, spherical geometry is a subset of elliptic geometry: https://en.wikipedia.org/wiki/Spherical_geometry Quote:
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 3space, 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 unsynthetic 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 wellorganized 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:
Quote:
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:
Last edited by skipjack; August 7th, 2017 at 07:53 PM.  
August 8th, 2017, 12:56 AM  #5 
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208 
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. 
August 8th, 2017, 07:06 PM  #6 
Newbie Joined: Mar 2017 From: Helsinki, Finland Posts: 3 Thanks: 2 
Thanks a lot horatiodudacus! That will probably help me. I will dig into all of that! 
August 9th, 2017, 08:43 PM  #7 
Newbie Joined: Aug 2017 From: Everywhere, all at once Posts: 3 Thanks: 1  
August 10th, 2017, 01:28 AM  #8 
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208  

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