Straightedge and compass constructions and absolute geometry

Question

Straightedge and compass constructions only use Euclid's first three axioms (that is "a line connects two points," "a line can be extended forever," and "a circle is defined by its center and radius").

As the parallel postulate is not used here, does that tell us that anything we can construct holds in absolute geometry as well?

Some simple examples:

1. We can bisect any angle
2. We can construct equilateral triangles
3. We can construct a perpendicular bisector to a line segment

It makes sense that all three of these things would work in absolute geometry.

That said, I can also easily build a square (equal sides and right angles) using a straightedge and compass as well, but squares don't exist hyperbolic geometry (because I can't have a quadrilateral with four right angles). What am I missing here?

Answer 1

As long as you don't use those axioms, you are fine. But sometimes you use them in a more "hidden" way.

For example to construct the perpendicular bisector, you use that two lines are parallel if and only if they have a common perpendicular. In hyperbolic geometry, two lines are ultraparallel if and only if they have a common perpendicular. But there exist other parallel lines, that are not ultraparallel. In other words, for two lines to be parallel, they do not need to share a common perpendicular.

Additionaly note that for ultraparallel lines, this perpendicular is unique by the ultraparallel theorem, so your construction will not work.

Finally note that to speak of circles, you usually need a notion of distance. Distance in the hyperbolic plane works differently, so you will not be able to draw meaningful circles with a compass in hyperbolic geometry.