I was just wondering whether circles are considered "round" still in taxicab geometry. I know that "roundness" is probably not a well-defined term and I know what a circle /appears/ to look like in taxicab geometry, but the question is deeper than that. From within the taxicab geometry, would the notion of "roundness" as intuited in Euclidean geometry become meaningless or different? From within the perspective of the geometry, a circle is still a circle and it might therefore still be round.
In lieu of this question, I have a few others. Could we still work with curvature (differential geometrically)? If so, and if it has the same definition and we base roundness on curvedness, one might (I think) still be able to call circles "round" in taxicab. Also, in taxicab, can there be a shape that /appears/ to be like a circle (in Euclidean)? It would not be a circle, but if roundness is understood that way, then a taxicab circle would not be round.
Thank you!
One possible definition of “round” would be “convex and without corners”, where the definition of “corner” would be “discontinuity in the speed vector” for some parametrization. So what about this speed vector in a taxicab circle? Even there, you go one direction and the suddenly go a different direction. I can think of no definition which would still consider this continuous.
So I'd say no, a circle in taxicab geometry is not round.