I'd like to say I need this community's help in clearing my mind of the clutter that leads me to this contradiction:
As an example, the Wikipedia article for the Infinite-order triangular tiling provides this image as a coloring of this tiling, but also provides this image, but with the comment "nonregular".
However still, the Wikipedia page for ideal triangles claims that
All ideal triangles are congruent to each other.
The interior angles of an ideal triangle are all zero.
As such, I see no reason for one tiling to claim to be regular, while the other one is supposed not to be regular.
It seems to me to answer this question would require a more sophisticated definition of "regular" than I hold. Additionally, I think I might also lack knowledge of how distinct ideal point are, if there is a notion of distance between them and how to represent them notationally.
Can someone help?
Best Answer
If it was just a matter of congruent ideal triangles, then there could be a lot of "regular" ideal triangle tesselations. The key idea, I think, is that the tiling should have symmetries such as hyperbolic reflections, rotations or translations which preserve distances and angles. The ideal triangle points all have angles of zero and distance between them is meaningless, but it is important for non-ideal points. Although the "alternated color tiling" and recursive non-regular tiling look somewhat similar, there is a key difference. Notice how the pattern along the boundary of the disk is different between them. In the non-regular tiling the pattern is very homogeneous, whereas in the other tiling there are different spacing of the ideal vertices along the boundary. The result of a hyperbolic reflection, rotation or translation of the non-regular tiling makes the boundary pattern non-homogeneous while the other tiling will just be similar to what it was before because it is symmetric under hyperbolic reflections, rotations and ranslations. That is what makes it "regular". The process that created the recursive non-regular tiling did not use hyperbolic reflections, rotations and translations while the other tiling did.