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This question is about "how badly can we 'break' the laws of geometry and still have something which is deserving of the name geometry?". It is named after something else I saw of the same name which also gives strange contortions of the laws of geometry.

But here goes. I am interested in two ideas from Euclidean geometry:

  1. all right angles are equal
  2. there is exactly one parallel to a line through a point not on that line

In particular, I'm curious to see how far one can push the failure of these principles. With regard to the first one, an example of a space in which it would fail would be a cone. In this case, the axiom fails when dealing with lines through the cone point. But is it possible for it to fail more dramatically? Is there anything that could deserve to be called a geometry in which no right angles are equal? If not that, a geometry in which in every neighborhood of the vertex of a right angle you could find another right angle which is not equal to it?

What about the second one? One geometry where it fails is hyperbolic geometry, in which there are infinitely many parallels through a point. Another is elliptic, and the related spherical, geometry, in which there are no parallels. In the elliptic and related geometries, other axioms of Euclidean geometry, besides the parallel postulate, have to give out in order for it to fail with no parallels through a point. This makes me wonder: could there be some kind of geometry in which there are exactly two parallels through a point? Exactly three? Countably many? What else must go to allow for this?

Could you have a "geometry" in which both of these properties fail? E.g. not all right angles are equal, and furthermore there are exactly two parallels through a point?

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    $\begingroup$ Do you insist on "each pair of lines has at most one intersection"? Because this doesn't apply on a cone. $\endgroup$
    – John Dvorak
    Jan 24, 2014 at 5:52
  • $\begingroup$ If you had two parallels, what would stop the line midway between the parallels from being a third parallel, or from existing altogether? $\endgroup$
    – Neil W
    Jan 24, 2014 at 5:54
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    $\begingroup$ @Neil discrete geometry could do. There's discrete projective geometry, but that has no parallels. $\endgroup$
    – John Dvorak
    Jan 24, 2014 at 5:56
  • $\begingroup$ Wrt the question of how badly we can violate classical axioms and still call the result “geometry”, I guess tropical geometry might be one of the more extreme yet still established examples. So far I haven't found a definition of right angles in tropical geometry, though. $\endgroup$
    – MvG
    Jan 24, 2014 at 8:18
  • $\begingroup$ some kind of variable curvature geometry ? also mention Florentin Smarandache " The most paradoxist mathematician of the world " in your question $\endgroup$
    – Willemien
    Jan 24, 2014 at 8:56

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