High School Axes on a hyperbolic plane

[BerryGo]

TL;DR

Question about how you would define axes when working with a hyperbolic plane.

Alright, I've been wondering this for a while now. Say you have an infinite grid of squares in hyperbolic geometry, such that the curvature makes it so each angle of each square is 72° (5 squares at each corner). At the very 'center' of the grid, or the origin, there would be 5 straight rays that go from that point out to infinity. Would you say those 5 rays are the axes? Can an axis even be a ray, and not a line? Would that be 5 axes, or 2.5? Can there be a fractional amount of axes? Or would you say that the 5 lines (the ones you would get from extending the rays to stretch out to infinity in both directions from the origin) are the axes? Or would you stay having 4 axes? And how would coordinates work with more than 2 axes on a 2D grid, anyway?

I honestly don't really know what to expect. I'm the kind of person that overcomplicates EVERYTHING, so whenever I decide on an answer, my brain finds some new technicality that makes me go back to being conflicted between both/all of the options. Some thoughts or help on this topic would be greatly appreciated.

-BerryGo

 

[jedishrfu]

If you flattened this hyperbolic plane in 3D then aren't you really describing something like polar coordinates on a plane.

 

[BerryGo]

If you flattened this hyperbolic plane in 3D then aren't you really describing something like polar coordinates on a plane.

(Sorry that it took so long to reply) Would the distance from the origin be in some sort of logarithmic scale?

 

[Hornbein]

TL;DR Summary: Question about how you would define axes when working with a hyperbolic plane.

Alright, I've been wondering this for a while now. Say you have an infinite grid of squares in hyperbolic geometry, such that the curvature makes it so each angle of each square is 72° (5 squares at each corner). At the very 'center' of the grid, or the origin, there would be 5 straight rays that go from that point out to infinity. Would you say those 5 rays are the axes? Can an axis even be a ray, and not a line? Would that be 5 axes, or 2.5? Can there be a fractional amount of axes? Or would you say that the 5 lines (the ones you would get from extending the rays to stretch out to infinity in both directions from the origin) are the axes? Or would you stay having 4 axes? And how would coordinates work with more than 2 axes on a 2D grid, anyway?

I honestly don't really know what to expect. I'm the kind of person that overcomplicates EVERYTHING, so whenever I decide on an answer, my brain finds some new technicality that makes me go back to being conflicted between both/all of the options. Some thoughts or help on this topic would be greatly appreciated.

-BerryGo

You may define as many axes as you like, at any angle you prefer.

 

[martinbn]

Look at a hexagonal tiling of the usual Eucleadian plane. It looks like a honey comb. From each vertex three rays come out. Would those be axes in the plane? Would that be 3 or 1.5 axes?