Timeline for Is there an absolute geometry that underlies spherical, Euclidean and hyperbolic geometry?
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| Jan 14 at 17:39 | comment | added | Robert Bryant | @MoziburUllah: It's called conformal geometry because the conformal structure is the thing that's preserved by the symmetry group (at least in the standard model that includes elliptic, Euclidean, and hyperbolic geometry). It's only in dimension $n=2$ that the appropriate undefined terms are 'point' and 'circle'. True, there is no notion of length, but there's also no notion of 'segment', 'ray', or 'angle'. 'Betweenness' axioms have to be replaced by 'separation' axioms concerning 4 distinct concircular points, etc. There is also a notion of congruence, but for two 4-tuples of points. | |
| Jan 14 at 17:16 | comment | added | Mozibur Ullah | BTW, shouldn't your example be called circle geometry as I'm not sure where conformality comes into it. Did you choose that name on the basis that there is no notion of length? But then again, there is no notion of length in a vector space but we do not generally think of it as conformal. | |
| Jan 14 at 17:12 | comment | added | Mozibur Ullah | ... has as a primitive, a linear ordering relation. With just this one can define line, ray, segment; angle, triangle and plane. I'm curious as to how far one can push this if we swap a linear order relation for a cyclic order relation. | |
| Jan 14 at 17:07 | comment | added | Mozibur Ullah | Yes, it's along those lines you explained. There is a theory called incidence geometry which takes as its only primitive relation as incidence. Affine & projective geometry fits into this. As well as other geometries such as the Benz plane of which the three main examples are the Minkowski, Laguerre and Mobius plane. I don't recall an outline of conformal geometry along the lines you suggest, but there is likely to be one. Incidence geometry doesn't usually include axioms of order as you do - as well as Hilbert. But there is also a notion of ordered geometry which only ... | |
| Jan 14 at 15:08 | comment | added | Robert Bryant | @MoziburUllah: Actually, the second axiom that I proposed above is redundant. I should have just written, "The first axiom might be "Any three distinct points are incident with a unique circle.", with pairs of circles classified as disjoint, tangent, or transverse depending on whether they meet in 0, 1, or 2 points. A second axiom might be, "Any circle is incident with at least 3 points." Another axiom would probably be "There exist 4 points that are not concircular." A small theorem then would be "There exist 4 circles that are pairwise transverse". | |
| Jan 13 at 20:38 | comment | added | Robert Bryant | @MoziburUllah: I see. So, when $n=2$, conformal geometry would take 'point' and 'circle' as primitive, with the simplest primitive relation being 'incidence'. The first axiom might be "Any three distinct points are incident with a unique circle" and another axiom could be "Two distinct circles have at most two incident points in common", with pairs of circles classified as disjoint, tangent, or transverse depending on whether they meet in 0, 1, or 2 points. There would be a separation axiom for four distinct concircular points and a notion of congruence for quadruples of distinct points, etc. | |
| Jan 13 at 18:59 | comment | added | Mozibur Ullah | Thanks for this. To my mind, Klien's notion of a geometry seems to be a generalisation of the construction of the 'analytic' geometries I mentioned in my question. What I mean by a 'synthetic' construction is a axiomatic system in the style of Euclid, so that one takes the notion of point and line as primitive and have primitive relations like incidence and axioms of congruence and the like. | |
| Jan 13 at 17:24 | history | answered | Robert Bryant | CC BY-SA 4.0 |