The way I view Euclid's postulates are as follows:

  1. A line segment can be made between any two points on surface A.

  2. A line segment can be continued in its direction infinitely on surface A.

  3. Any line segment can form the diameter of a circle on surface A.

  4. The result of an isometry upon a figure containing a right angle preserves the right angle as a right angle on surface A.

  5. If a straight line falling on two straight lines make the interior angles on the same side less than 180 degrees in total, the two straight lines will eventually intersect on the side where the sum of the angles is less than l80 degrees.

The way I view the five postulates is simple. Each postulate defines some quality a surface has.

For instance, 2 seems to define whether a surface is infinite/looped or finite/bounded, 3 seems to force a surface to be circular (or a union of circular subsets), and 5 I believe change the constant curvature of a surface (wether it is 0 or nonzero).

I want to determine what "quality" 1 and 4 define in the context of the surface itself. 1 seems to imply discontinuity vs continuity, and I think 4 would imply non-constant curvature. However, I am unsure. Ultimately I would like to assign each of these a quality of a surface that they define such that all surfaces can be "categorized" under some combination of postulates, but that is irrelevant.

I am merely asking:

What two surfaces individually violate the first postulate and violate the fourth postulate?


closed as primarily opinion-based by The Great Duck, астон вілла олоф мэллбэрг, Daniel W. Farlow, JonMark Perry, user91500 Jan 17 '17 at 6:10

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  • 1
    $\begingroup$ Violating 4 is quite easy, you just have to mess up the usual notion of the angle. Say, take the standard Euclidean plane and then declare that the lines $x=0$, $y=0$ not to be orthogonal; instead declare that $x=0$ is orthogonal to the line $x=y$. $\endgroup$ – Moishe Kohan Oct 20 '16 at 14:51
  • $\begingroup$ Note also that Axiom 5 is not Euclid's; maybe you did have in mind the original 5th postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." $\endgroup$ – Moishe Kohan Oct 20 '16 at 14:58
  • $\begingroup$ @MoisheCohen Axiom 5 is equivalent to Euclid's. You're right, it is a different wording but its an equivalent statement. Also, axiom 4 probably trickier than just that. From what I read regarding it, the 4th postulate is more about how isometries affect angles, and the 4th one declares that isometries preserve right angles. It's not what euclid originally wrote but from I understand it is equivalent to his meaning of "equality" of "right angles". Plus, what I'm looking for is more a surface, not just a redefinition. Sure, saying certain lines are orthogonal like that will break it. $\endgroup$ – The Great Duck Oct 20 '16 at 18:19
  • $\begingroup$ however, that ignores the tricky part: finding a surface that actually has that quality. After all, orthogonality is pretty well defined from what I understand, so then the issue is finding a surface messy enough to break that while not breaking the others. I admit, it might not exist, but it is still a worthy thought to pursue. $\endgroup$ – The Great Duck Oct 20 '16 at 18:21
  • $\begingroup$ As I said, the surface is $R^2$, but you change the notion of angles. As for Axiom 5, it is equivalent to Euclid's only if you keep other axioms and you asked about violating Axiom 4. If you use Euclid's 5th postulate instead of Playfair's Axiom then violating Axiom 4 becomes more difficult since Euclid's 5th postulate uses angles (unlike Playfair's Axiom). $\endgroup$ – Moishe Kohan Oct 21 '16 at 0:03

(1.) Two parallel planes in Euclidean space.

(4.) It's not clear to me what Euclid means by equality of angles. But it would seem to imply (in any interpretation) that the total angle around any point is equal (to 4 right angles), so it would eliminate the possibility of a cone point on the surface?

  • $\begingroup$ I actually thought 2 planes was a counter example, but that breaks postulate 5. You can draw more than one line through some point that does not intersect a line in the other plane (and thereby multiple parallels). Number 4 I always assumed meant that (as an example) given a right triangle, there is some other triangle with the same distance between the 3 points, but without a right angle. I assume you know what an isometry is. I got that interpretation from googling around and seeing that as what wolfram alpha had as it's version. Since I had no real viable interpretation, I went with it. $\endgroup$ – The Great Duck Oct 14 '16 at 4:47
  • $\begingroup$ Thank you for the answer though. Those are great thoughts! $\endgroup$ – The Great Duck Oct 14 '16 at 4:48
  • $\begingroup$ I just realized... your answer is more right than I thought! With the correct version of postulate 5 being used as Moishe pointed out, 2 planes doesn't break postulate 5, because you cannot have a straight line falling between two straight lines on two different planes. I guess in this context, we wouldn't have a clear or immediate notion of comparing two lines where one is in each plane. Good job sir! Very well done. $\endgroup$ – The Great Duck Oct 24 '16 at 3:04

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