How many of the 32 potential geometric systems are actually useful or practical to study/understand?

Question

I know there are five Euclidean axioms. Each of them can be taken as true or false. For instance a system accepting the negation of the fifth axiom to be true is hyperbolic geometry. There are 5 axioms total allowing for 2^5 = 32 different geometric systems.

Which of these systems are notably useful, and what are their names?

You do not have to go into extreme detail describing each geometric system; however, a brief description (around 5-6 sentence paragraph) would be appreciated. I leave this here to make it clear that this question is not too broad and can be answered in a brief manner.

(If you wish (for convenience) you can denote the truth or falsehood of each axiom by T and F. So standard Euclidean geometry would be TTTTT, whereas hyperbolic geometry would be TTTTF.)

Answer 1

First of all the Euclids postulates are not complete as many commenters allready said.

But then some ideas in an attempt to answer your question:

The first postulate: "To draw a straight line from any point to any point."

Negating this: There is at least one pair of points that are not on a common line.

Difficult to imagine, some kind of black hole between the two points, I am not sure what to make of it.

The second postulate: "To produce [extend] a finite straight line continuously in a straight line."

There is discussion what this means, does this means that lines have an infinite length or that they are just boundless (a circle for example, you can go round forever)

But then negating this, not sure what to make of it but it would nicely fit with a negation of the first postulate two points are just to far away of each other to be able to be connected)

The third postulate: "To describe a circle with any centre and distance [radius]."

This can be negated in two different ways:

• There are points where a certain circle can not be drawn around , seems hopeless.
• There is a maxmum size of circle (for example spherical geometry)

The fourth postulate: "That all right angles are equal to one another."

Negating this: Some right angles are not equal to each other? Not sure what to make of this. (some geometers would even claim this postulate was unneeded in the first place)

More general

It is note worthy that these first 4 postulates are much shorter than the fifth postulate and much more difficult to deny.(what for strange geometry have you left?) But on the other hand just try the (surface) geometry of a torus it breaks postulates 2 and 3 but then doing geometry on a torus is rather difficult.

In the Heath's translation of Euclid's elements (Dover paperback) there are long sections on each postulate. maybe best to study it all

Good luck