Axes on a hyperbolic plane

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Discussion Overview

The discussion revolves around the concept of defining axes in a hyperbolic plane, particularly in relation to an infinite grid of squares where each angle is 72°. Participants explore whether rays can be considered axes, the implications of having multiple rays, and how coordinates might function with more than two axes in a two-dimensional grid.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions if the 5 rays emanating from the origin in hyperbolic geometry can be considered axes and whether an axis can be a ray instead of a line.
  • There is uncertainty about whether to classify the 5 rays as 5 axes, 2.5 axes, or some other fractional amount.
  • Another participant suggests that flattening the hyperbolic plane into 3D might relate to polar coordinates, raising questions about the distance from the origin potentially being on a logarithmic scale.
  • A later reply proposes that one may define as many axes as desired at any angle, indicating flexibility in the definition of axes.
  • Another participant draws a parallel to a hexagonal tiling in Euclidean geometry, questioning if the three rays from each vertex could be considered axes and whether that would be classified as 3 or 1.5 axes.

Areas of Agreement / Disagreement

Participants express differing views on the definition and quantity of axes in hyperbolic geometry, with no consensus reached on whether rays can be classified as axes or how many axes should be recognized.

Contextual Notes

The discussion includes various assumptions about the nature of axes in different geometrical contexts and the implications of curvature on these definitions, which remain unresolved.

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

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